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Thresholded Covering Algorithmsfor Robust and Max-Min Optimization

Anupam Gupta∗ Viswanath Nagarajan† R. Ravi‡

Abstract

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow,but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so thatthe worst-case cost (summed over both days) is minimized? For example, in a set cover instance, if any one ofthe

(n

k

)subsets of the universe that have sizek may appear tomorrow, what is a good course of action? Feige

et al. [FJMM07], and later, Khandekar et al. [KKMS08], considered thisk-robust modelwhere the possibleoutcomes tomorrow are given by all demand-subsets of sizek, and gave algorithms for the set cover problem,and the Steiner tree and facility location problems in this model, respectively.

In this paper, we give the following simple and intuitive template fork-robust problems:having built someanticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold(which is≈ Opt/k), augment the anticipatory solution to cover this demand aswell, and repeat.In this paperwe show that this template gives us approximation algorithms for k-robust versions of Steiner tree and setcover (improving on the performance ratios of the previously known algorithms, by a logarithmic factor in thecase of set cover), and present the first approximation algorithms fork-robust Steiner forest, minimum-cut andmulticut. All our approximation ratios (except for multicut) are almost best possible. In addition to its apparentsimplicity and efficacy, a salient feature of our framework is the interesting set of technical ideas—in particular,dual-rounding ideas—needed to show good performance of this heuristic.

As a by-product of our techniques, we get algorithms for max-min problems of the form: “given a cov-ering problem instance, whichk of the elements are costliest to cover?” If the covering problem does notnaturally define a submodular function, very little is knownabout these problems. For the problems mentionedabove (set cover, multicut, Steiner forest), we show that their k-max-min versions have performance guaran-tees similar to those for thek-robust problems, which are optimal in many cases. E.g., forset cover, we givean O(log m + log n) approximation, which improves the previous result by a logarithmic factor, and nearlymatches theΩ( log m

log log m+ log n) hardness.

∗Computer Science Department, Carnegie Mellon University,Pittsburgh, PA 15213, USA. Supported in part by NSF awards CCF-0448095 and CCF-0729022, and an Alfred P. Sloan Fellowship.Email: anupamg@cs.cmu.edu

†IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA. Email: viswanath@us.ibm.com‡Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Supported in part by NSF grant CCF-0728841.

Email: ravi@cmu.edu

1 Introduction

Consider the followingk-robustset cover problem: we are given a set system(U,F ⊆ 2U ). Tomorrow some setof k elementsS ⊆ U will want to be covered; however, today we don’t know what this set will be. One strategyis to wait until tomorrow and buy anO(log n)-approximate set cover for this set. However, sets are cheaper today:they will costλ times as much tomorrow as they cost today. Hence, it may make sense to buy some anticipatorypartial solution today (i.e. in the first-stage), and then complete it tomorrow (i.e. second-stage) once we know theactual members of the setS. Since we do not know anything about the setS (or maybe we are risk-averse), wewant to plan for the worst-case, and minimize:

(cost of anticipatory solution) + λ · maxS:|S|≤k

(additional cost to coverS).

Early approximation results for robust problems [DGRS05, GGR06] had assumed that the collection of possiblesetsS was explicitly given (and the performance guarantee depended logarithmically on the size of this collection).Since this seemed quite restrictive, Feige et al. [FJMM07] proposed thisk-robust model where any of the

(nk

)

subsetsS of sizek could arrive. Though this collection of possible sets was potentially exponential sized (for largevalues ofk), the hope was to get results that did not depend polynomially on k. To get such an algorithm, Feigeet al. considered thek-max-min set-cover problem (“which subsetS ⊆ U of sizek requires the largest set covervalue?”), and used the online algorithm for set cover to get agreedy-styleO(log m log n) approximation for thisproblem; herem andn are the number of sets and elements in the set system. They then used this max-min problemas a separation oracle in an LP-rounding-based algorithm (`a la [SS04]) to get the same approximation guarantee forthek-robust problem. They also showed the max-min andk-robust set cover problems to beΩ( log m

log log m + log n)hard—which left a logarithmic gap between the upper and lower bounds. However, an online algorithm basedapproach is unlikely to close this gap, since the online algorithm for set cover is necessarily a log-factor worse thatits offline counterparts [AAA +03].

Apart from the obvious goal of improving the result for this particular problem, one may want to develop algorithmsfor other k-robust problems. E.g., for thek-robust min-cutproblem, some setS of k sources will want to beseparated from the sink vertex tomorrow, and we want to find the best way to cut edges to minimize the total costincurred (over the two days) for the worst-casek-setS. Similarly, in thek-robust Steiner forest, we are givena metric space and a collection of source-sink pairs, and anysetS of k source-sink pairs may desire pairwiseconnection tomorrow; what should we do to minimize the sum ofthe costs incurred over the two days? One canextend the Feige et al. framework to first solve max-min problems using online algorithms, but in all cases weseem to lose extra logarithmic factors. Moreover, for the above two problems (and others), the LP-rounding-basedframework does not seem to extend directly. The latter obstacle was also observed by Khandekar et al. [KKMS08],who gave constant-factor algorithms fork-robust versions of Steiner tree and facility location.

1.1 Main Results

In this paper, we present a general template to design algorithms fork-robust problems. We improve on previousresults, by obtaining anO(log m + log n) factor fork-robust set cover, and improving the constant in the approxi-mation factor for Steiner tree. We also give the first algorithms for some other standard covering problems, gettingconstant-factor approximations for bothk-robust Steiner forest—which was left open by Khandekar et al.—and fork-robust min-cut, and anO( log2 n

log log n) approximation fork-robust multicut. Our algorithms do not use a max-minsubroutine directly: however, our approach ends up giving us approximation algorithms fork-max-min versionsof set cover, Steiner forest, min-cut and multicut; all but the one for multicut are best possible under standardassumptions; the ones for Steiner forest and multicut are the first known algorithms for theirk-max-min versions.

An important contribution of our work (even more than the new/improved approximation guarantees) is the sim-plicity of the algorithms, and the ideas in their analysis. The following is our actual algorithm fork-robust setcover.

1

Suppose we “guess” that the maximum second-stage cost in theoptimal solution isT . Let A ⊆ U beall elements for which the cheapest set covering them costs more thanβ ·T/k, whereβ = O(log m +log n). We build a set cover onA as our first stage. (Say this cover costsCT .)

To remove the guessing, try all values ofT and choose the solution that incurs the least total costCT + λβT .Clearly, by design, no matter whichk elements arrive tomorrow, it will not cost us more thanλ · k · βT/k = λβTto cover them, which is withinβ of what the optimal solution pays.

The key step of our analysis is to argue whyCT is close to optimum. We briefly describe the intuition; detailsappear inSection 3. SupposeCT ≫ βOpt: then the fractional solution to the LP for set cover forA would cost≫ β

ln nOpt ≥ Opt, and so would its dual. Our key technical contribution is to show how to“round” this dual LPto find a “witness”A′ ⊆ A with only k elements, and also a corresponding feasible dual of value≫ Opt—i.e., thedual value does not decrease much in the rounding (This rounding uses the fact that the algorithm only put thoseelements inA that were expensive to cover). Using duality again, this proves that the optimal LP value, and hencethe optimal set cover for thesek elementsA′, would cost much more thanOpt—a contradiction!

In fact, our algorithms for the otherk-robust problems are almost identical to this one; indeed, the only slightlyinvolved algorithm is that fork-robust Steiner forest. Of course, the proofs to bound the cost CT need differentideas in each case. For example, directly rounding the dual for Steiner forest is difficult, so we give a primal-dualargument to show the existence of such a witnessA′ ⊆ A of size at mostk. For the cut-problems, one has to dealwith additional issues becauseOpt consists of two stages that have to be charged to separately,and this requiresa Gomory-Hu-tree-based charging. Even after this, we stillhave to show that if the cut for a set of sourcesA islarge then there is a witnessA′ ⊆ A of at mostk sources for which the cut is also large—i.e., we have to somehowaggregate the flows (i.e. dual-rounding for cut problems). We prove new flow-aggregation lemmas for single-sink flows using Steiner-tree-packing results, and for multiflows using oblivious routing [Rac08]; both proofs arepossibly of independent interest.

To get a quick overview of our basic approach, see the analysis for Steiner tree inAppendix B. While the resultis simple and does not require rounding the dual, it is a nice example of our framework in action. InSections 2and 2.1 we present the formal framework fork-robust problems, and abstract out the properties that we’dlikefrom our algorithms. ThenSection 3contains such an algorithm fork-robust set cover—Min-cut and Steiner forestappear inSections 4and5. The algorithm fork-robust multicut is inAppendix E. We present a general reductionfrom robust problems to the corresponding max-min problemsin Appendix F. In that section, we also describeextensions of our work to more general uncertainty sets, e.g. incorporating matroid and knapsack type constraints.

1.2 Related Work

Approximation algorithms for robust optimization was initiated by Dhamdhere et al. [DGRS05]: they study thecase when the scenarios were explicitly listed, and gave constant-factor approximations for Steiner tree and facilitylocation, and logarithmic approximations to mincut/multicut problems. Golovin et al. [GGR06] improved themincut result to a constant factor approximation, and also gave anO(1)-approximation for robust shortest-paths.Thek-robust model was introduced in Feige et al. [FJMM07], where they gave anO(log m log n)-approximationfor set cover. Khandekar et al. [KKMS08] noted that the techniques of [FJMM07] did not give good results forSteiner tree, and developed new constant-factor approximations fork-robust versions of Steiner tree, Steiner foreston trees and facility location. Using our framework, the algorithm we get for Steiner tree can be viewed as arephrasing of their algorithm—our proof is arguably more transparent and results in a better bound. Our approachcan also be used to get a slightly better ratio than [KKMS08] for the Steiner forest problem on trees.

Constrained submodular maximization problems [NWF78, FNW78, Svi04, CCPV07, Von08] appear very relevantat first sight: e.g., thek-max-min version of min-cut (“find thek sources whose separation from the sink coststhe most”) is precisely submodular maximization under a cardinality constraint, and hence is approximable towithin (1 − 1/e). But apart from min-cut, the other problems do not give us submodular functions to maximize,and massaging the functions to make them submodular seems tolose logarithmic factors. E.g., one can use tree

2

embeddings [FRT04] to reduce Steiner tree to a problem on trees and make it submodular; in other cases, one canuse online algorithms to get submodular-like properties (we give a general reduction for covering problems thatadmit good offline and online algorithms inAppendix F). Eventually, it is unclear how to use existing results onsubmodular maximization in any general way.

Considering theaverageinstead of the worst-case performance gives rise to the well-studied model of stochasticoptimization [RS04, IKMM04]. Some common generalizations of the robust and stochasticmodels have beenconsidered (see, e.g., Swamy [Swa08] and Agrawal et al. [ADSY09]).

Feige et al. [FJMM07] also considered thek-max-min set cover—they gave anO(log m log n)-approximationalgorithm for this, and used it in the algorithm fork-robust set cover. They also showed anΩ( log m

log log m) hardnessof approximation fork-max-min (andk-robust) set cover. To the best of our knowledge, none of thek-max-minproblems other than min-cut have been studied earlier.

The k-min-min versions of covering problems (i.e. “whichk demands are thecheapestto cover?) have beenextensively studied for set cover [Sla97, GKS04], Steiner tree [Gar05], Steiner forest [GHNR07], min-cut andmulticut [GNS06, Rac08]. However these problems seem to be related to thek-max-min versions only in spirit.

2 Notation and Definitions

Deterministic covering problems. A covering problemΠ has a ground-setE of elements with costsc : E →R+, andn covering requirements (often called demands or clients), where the solutions to thei-th requirement isspecified—possibly implicitly—by a familyRi ⊆ 2E which is upwards closed (since this is a covering problem).Requirementi is satisfiedby solutionS ⊆ E iff S ∈ Ri. The covering problemΠ = 〈E, c, Rini=1〉 involvescomputing a solutionS ⊆ E satisfying alln requirements and having minimum cost

∑e∈S ce. E.g., in set cover,

“requirements” are items to be covered, and “elements” are sets to cover them with. In Steiner tree, requirementsare terminals to connect to the root and elements are the edges; in multicut, requirements are terminal pairs to beseparated, and elements are edges to be cut.

Robust covering problems. This problem, denotedRobust(Π), is a two-stage optimizationproblem, whereelements are possibly bought in the first stage (at the given cost) or the second stage (at costλ times higher). In thesecond stage, some subsetω ⊆ [n] of requirements (also called ascenario) materializes, and the elements boughtin both stages must satisfy each requirement inω. Formally, the input to problemRobust(Π) consists of (a) thecovering problemΠ = 〈E, c, Rini=1〉 as above, (b) a setΩ ⊆ 2[n] of scenarios (possibly implicitly given), and(c) an inflation parameterλ ≥ 1. A feasible solution toRobust(Π) is a set offirst stage elementsE0 ⊆ E (boughtwithout knowledge of the scenario), along with anaugmentation algorithmthat given anyω ∈ Ω outputsEω ⊆ Esuch thatE0∪Eω satisfies all requirements inω. The objective function is to minimize:c(E0)+λ·maxω∈Ω c(Eω).Given such a solution,c(E0) is called the first-stage cost andmaxω∈Ω c(Eω) is the second-stage cost.

k-robust problems. In this paper (except inSection F), we deal with robust covering problems undercardinalityuncertainty sets: i.e.,Ω :=

([n]k

)= S ⊆ [n] | |S| = k. We denote this problem byRobustk(Π).

Max-min problems. Given a covering problemΠ and a setΩ of scenarios, themax-minproblem involves findinga scenarioω ∈ Ω for which the cost of the min-cost solution toω is maximized. Note that by settingλ = 1 inany robust covering problem,the optimal value of the robust problem equals that of its corresponding max-minproblem. In ak-max-min problem we haveΩ =

([n]k

).

2.1 The Abstract Properties we want from our Algorithms

Our algorithms for robust and max-min versions of covering problems are based on the following guarantee.

Definition 2.1 An algorithm is(α1, α2, β)-discriminating iff given as input any instance ofRobustk(Π) and athresholdT , the algorithm outputs (i) a setΦT ⊆ E, and (ii) an algorithmAugmentT :

([n]k

)→ 2E , such that:

3

A. For every scenarioD ∈([n]

k

),

(i) the elements inΦT ∪ AugmentT (D) satisfy all requirements inD, and(ii) the resulting augmentation costc (AugmentT (D)) ≤ β · T .

B. LetΦ∗ and T ∗ (respectively) denote the first-stage and second-stage cost of an optimal solution to theRobustk(Π) instance. If the thresholdT ≥ T ∗ then the first stage costc(ΦT ) ≤ α1 · Φ∗ + α2 · T ∗.

The next lemma (whose proof appears inAppendix A.1) shows why having a discriminating algorithm is sufficientto solve the robust problem. The issue to address is that having guessedT for the optimal second stage cost, wehave no direct way of verifying the correctness of that guess—hence we choose the best among all possible valuesof T . ForT ≈ T ∗ the guarantees inDefinition 2.1ensure that we pay≈ Φ∗ + T ∗ in the first stage, and≈ λT ∗ inthe second stage; for guessesT ≪ T ∗, the first-stage cost in guarantee (2) is likely to be large compared toOpt.

Lemma 2.2 If there is an(α1, α2, β)-discriminating algorithm for a robust covering problemRobustk(Π), thenfor everyǫ > 0 there is a

((1 + ǫ) ·max

α1, β + α2

λ

)-approximation algorithm forRobustk(Π).

In the rest of the paper, we focus on providing discriminating algorithms for suitable values ofα1, α2, β.

2.2 Additional Property Needed fork-max-min Approximations

As we noted above, ak-max-min problem is a robust problem where the inflationλ = 1 (which implies that in anoptimal solutionΦ∗ = 0, andT ∗ is thek-max-min value). Hence a discriminating algorithm immediately gives anapproximation to thevalue: for anyD ∈

([n]k

), ΦT ∪ AugmentT (D) satisfies all demands inD, and for the right

guess ofT ≈ T ∗, the cost is at most(α2 + β)T ∗. It remains to output a badk-set as well, and hence the followingdefinition is useful.

Definition 2.3 An algorithm for a robust problem isstrongly discriminatingif it satisfies the properties inDef-inition 2.1, and when the inflation parameter isλ = 1 (and henceΦ∗ = 0), the algorithm also outputs a setQT ∈

([n]k

)such that ifc(ΦT ) ≥ α2T , the cost of optimally covering the setQT is≥ T .

Recall that for a covering problemΠ, the cost of optimally covering the set of requirementsQ ∈([n]

k

)isminc(EQ) |

EQ ⊆ E andEQ ∈ Ri ∀i ∈ Q. We prove the following inAppendix A.2.

Lemma 2.4 If there is an(α1, α2, β)-strongly-discriminating algorithm for a robust covering problemRobustk(Π),then for everyǫ > 0 there is an algorithm fork-max-min(Π) that outputs a setQ such that for someT , the optimalcost of covering this setQ is at leastT , but everyk-set can be covered with cost at most(1 + ǫ) · (α2 + β)T .

3 Set CoverConsider thek-robust set cover problem where there is a set system(U,F) with a universe of|U | = n elements,andm sets inF with each setR ∈ F costingcR, an inflation parameterλ, and an integerk such that each ofthe sets

(Uk

)is a possible scenario for the second-stage. GivenLemma 2.2, it suffices to show a discriminating

algorithm as defined inDefinition 2.1for this problem. The algorithm given below is easy: pick allelements whichcan only be covered by expensive sets, and cover them in the first stage.

Algorithm 1 Algorithm for k-Robust Set Cover1: input: k-robust set-cover instance and thresholdT .2: let β ← 36 ln m, andS ←

v ∈ U | min cost set coveringv has cost at leastβ · T

k

.

3: output first stage solutionΦT as the Greedy-Set-Cover(S).4: defineAugmentT (i) as the min-cost set coveringi, for i ∈ U \ S; andAugmentT (i) = ∅ for i ∈ S.5: output second stage solutionAugmentT whereAugmentT (D) :=

⋃i∈D AugmentT (i) for all D ⊆ U .

Claim 3.1 (Property A for Set Cover) For all T ≥ 0 and scenarioD ∈(U

k

), the setsΦT

⋃AugmentT (D) cover

elements inD, and have costc(AugmentT (D)) ≤ β T .

4

Proof: The elements inD ∩ S are covered byΦT ; and by definition ofAugmentT , each elementi ∈ D \ S iscovered by setAugmentT (i). Thus we have the first part of the claim. For the second part, note that by definitionof S, the cost ofAugmentT (i) is at mostβ T/k for all i ∈ U .

Theorem 3.2 (Property B for Set Cover) LetΦ∗ denote the optimal first stage solution (and its cost), andT ∗ theoptimal second stage cost. Letβ = 36 ln m. If T ≥ T ∗ thenc(ΦT ) ≤ Hn · (Φ∗ + 12 · T ∗).

Proof: We claim that there is afractional solution x for the set covering instanceS with small costO(Φ∗ +T ∗), whence rounding this to an integer solution implies the theorem. For a contradiction, assume not: let everyfractional set cover be expensive, and hence there must be a dual solution of large value. We thenround this dualsolution to get a dual solution to a sub-instance with onlyk elements that costs> Φ∗ + T ∗, which is impossible(since using the optimal solution we can solve every instance onk elements with that cost).

To this end, letS′ ⊆ S denote the elements that arenotcovered by the optimal first stageΦ∗, and letF ′ ⊆ F denotethe sets that contain at least one element fromS′. By the choice ofS, all sets inF ′ cost at leastβ · T

k ≥ β · T ∗

k .

Define the “coarse” cost for a setR ∈ F ′ to be cR = ⌈ cR

6T ∗/k⌉. For each setR ∈ F ′, sincecR ≥ βT ∗

k ≥ 6T ∗

k , it

follows thatcR · 6T ∗

k ∈ [cR, 2 · cR), and also thatcR ≥ β/6.

Now consider the primal-dual pair of LPs for the set cover instance with elementsS′ and setsF ′ having the coarsecostsc. Let xRR∈F ′ be an optimal primal andyee∈S′ an optimal dual solution. The following claim boundsthe (coarse) cost of these fractional solutions.

Claim 3.3 If β = 36 ln m, then the LP cost is∑

R∈F ′ cR · xR =∑

e∈S′ ye ≤ 2 · k.

Before we proveClaim 3.3, let us assume it and complete the proof ofTheorem 3.2. Given the primal LP solutionxRR∈F ′ to cover elements inS′, define an LP solution to cover elements inS as follows: definezR = 1 ifR ∈ Φ∗, zR = xR if R ∈ F ′ \ Φ∗; andzR = 0 otherwise. Since the solutionz containsΦ∗ integrally, it coverselementsS \ S′ (i.e. the portion ofS covered byΦ∗); sincezR ≥ xR, z fractionally coversS′. Finally, the cost ofthis solution is

∑R cRzR ≤ Φ∗ +

∑R cRxR ≤ Φ∗ + 6T ∗

k ·∑

R cRxR. But Claim 3.3bounds this byΦ∗ + 12 · T ∗.Since we have a LP solution of valueΦ∗ + 12T ∗, and the greedy algorithm is anHn-approximation relative to theLP value for set cover, this completes the proof.

Claim 3.1andTheorem 3.2show our algorithm for set cover to be an(Hn, 12Hn, 36 ln m)-discriminating algo-rithm. ApplyingLemma 2.2converts this discriminating algorithm to an algorithm fork-robust set cover, and givesthe following improvement to the result of [FJMM07].

Theorem 3.4 There is anO(log m + log n)-approximation fork-robust set cover.

It remains to give the proof forClaim 3.3above; indeed, that is where the technical heart of the result lies.

Proof of Claim 3.3: Recall that we want to bound the optimal fractional set covercost for the instance(S′,F ′)with the coarse (integer) costs;xR andye are the optimal primal and dual solutions. For a contradiction, assumethat the LP cost

∑R∈F ′ cRxR =

∑e∈S′ ye lies in the unit interval((γ − 1)k, γk] for some integerγ ≥ 3.

Define integer-valued random variablesYee∈S′ by setting, for eache ∈ S′ independently,Ye = ⌊ye⌋ + Ie,whereIe is a Bernoulli(ye − ⌊ye⌋) random variable. We claim that whp the random variablesYe/3 form a feasibledual— i.e., they satisfy all the constraints∑e∈R(Ye/3) ≤ cRR∈F ′ with high probability. Indeed, consider adual constraint corresponding toR ∈ F ′: since we have

∑e∈R⌊ye⌋ ≤ cR, we get thatPr[

∑e∈R Ye > 3 · cR] ≤

Pr[∑

e∈R Ie > 2 · cR]. But now we use a Chernoff bound [MR95] to bound the probability that the sum ofindependent 0-1 r.v.s,

∑e∈R Ie, exceeds twice its mean (here

∑e∈R E[Ie] ≤

∑e∈R ye ≤ cR) by ε−bcR/3 ≤

e−β/18 ≤ m−2, since eachcR ≥ β/6 andβ = 36 · ln m. Finally, a trivial union bound implies thatYe/3 satisfiesall the m contraints with probability at least1 − 1/m. Moreover, the expected dual objective is

∑e∈S′ ye ≥

(γ − 1)k ≥ 1 (sinceγ ≥ 3 andk ≥ 1), and by another Chernoff Bound,Pr[∑

e∈S′ Ye > γ−12 · k] ≥ a (where

a > 0 is some constant). Putting it all together, with probability at leasta − 1m , we have afeasibledual solution

Y ′e := Ye/3 with objective value at leastγ−1

6 · k.

5

Why is this dualY ′e any better than the original dualye? It is “near-integral”—specifically, eachY ′

e is either zeroor at least13 . So order the elements ofS′ in decreasing order of theirY ′-value, and letQ be the set of thefirst k

elementsin this order. The total dual value of elements inQ is at leastminγ−16 k, k

3 ≥ k3 , sinceγ ≥ 3, and each

non-zeroY ′ value is≥ 1/3. This valid dual for elements inQ shows a lower bound ofk3 on minimum (fractional)c-cost to cover thek elements inQ. UsingcR > 3T ∗

k · cR for eachR ∈ F ′, the minimumc-cost to fractionallycoverQ is > 3T ∗

k · k3 = T ∗. Hence, ifQ is the realized scenario, the optimal second stage cost willbe> T ∗ (as no

element inQ is covered byΦ∗)—this contradicts the fact that OPT can coverQ ∈(Uk

)with cost at mostT ∗. Thus

we must haveγ ≤ 2, which completes the proof ofClaim 3.3.

The k-Max-Min Set Cover Problem. The proof of Claim 3.3 suggests how to get a(Hn, 12Hn, 36 ln m)strongly discriminating algorithm. Whenλ = 1 (and soΦ∗ = 0), the proof shows that ifc(ΦT ) > 12Hn · T ,there is a randomized algorithm that outputsk-setQ with optimal covering cost> T (witnessed by the dual solu-tion having cost> T ). Now usingLemma 2.4, we get the claimedO(log m + log n) algorithm for thek-max-minset cover problem. This nearly matches the hardness ofΩ( log m

log log m + log n) given by [FJMM07].

Remarks: We note that ourk-robust algorithm also extends to the more general setting of (uncapacitated)CoveringInteger Programs; a CIP (see eg. [Sri99]) is given byA ∈ [0, 1]n×m, b ∈ [1,∞)n andc ∈ R

m+ , and the goal is to

minimizecT ·x | Ax ≥ b, x ∈ Zm+. The results above (as well as the [FJMM07] result) also hold in the presence

of set-dependent inflation factors—details will appear in the full version. Results for the other covering problemsdo not extend to the case of non-uniform inflation: this is usually inherent, and not just a flaw in our analysis.Eg., [KKMS08] give anΩ(log1/2−ǫ n) hardness fork-robust Steiner forest under just two distinct inflation-factors,whereas we give anO(1)-approximation under uniform inflations (inSection 5).

4 Minimum Cut

We now consider thek-robust minimum cut problem, where we are given an undirected graphG = (V,E) withedge capacitiesc : E → R+, a rootr ∈ V , terminalsU ⊆ V , inflation factorλ. Again, any subset in

(Uk

)is a

possible second-stage scenario, and again we seek to give a discriminating algorithm. This algorithm, like for setcover, is non-adaptive: we just pick all the “expensive” terminals and cut them in the first stage.

Algorithm 2 Algorithm for k-Robust Min-Cut1: input: k-robust minimum-cut instance and thresholdT .2: let β ← Θ(1), andS ← v ∈ U | min cut separatingv from rootr has cost at leastβ · T

k .3: output first stage solutionΦT as the minimum cut separatingS from r.4: defineAugmentT (i) as the min-r-i cut inG \ΦT , for i ∈ U \ S; andAugmentT (i) = ∅ for i ∈ S.5: output second stage solutionAugmentT whereAugmentT (D) :=

⋃i∈D AugmentT (i) for all D ⊆ U .

Claim 4.1 (Property A for Min-Cut) For all T ≥ 0 andD ∈(U

k

), the edgesΦT

⋃AugmentT (D) separate the

terminalsD from r; moreover, the costc(AugmentT (D)) ≤ β T .

Theorem 4.2 (Property B for Min-Cut) LetΦ∗ denote the optimal first stage solution (and its cost), andT ∗ theoptimal second stage cost. Ifβ ≥ 10e

e−1 andT ≥ T ∗ thenc(ΦT ) ≤ 3 · Φ∗ + β2 · T ∗.

Here’s the intuition for this theorem: As in the set cover proof, we claim that if the optimal cost of separatingSfrom the rootr is high, then there must be a dual solution (which prescribesflows from vertices inS to r) of largevalue. We again “round” this dual solution by aggregating these flows to get a set ofk terminals that have a largecombined flow (of value> Φ∗ + T ∗) to the root—but this is impossible, since the optimal solution promises us acut of at mostΦ∗ + T ∗ for any set ofk terminals.

However, more work is required. For set-cover, each elementwas either covered by the first-stage, or it was not;for cut problems, things are not so cut-and-dried, since both stages may help in severing a terminal from the root!So we divideS into two parts differently: the first part contains those nodes whose min-cut inG is large (since

6

they belonged toS) but it fell by a constant factor in the graphG \ Φ∗. These we call “low” nodes, and weuse a Gomory-Hu tree based analysis to show that all low nodescan be completely separated fromr by payingonly O(Φ∗) more (this we show inClaim 4.3). The remaining “high” nodes continue to have a large min-cut inG \ Φ∗, and for these we use the dual rounding idea sketched above toshow a min-cut ofO(T ∗) (this is proved inClaim 4.4). Together these claims implyTheorem 4.2.

To begin the proof ofTheorem 4.2, let H := G \ Φ∗, and letSh ⊆ S denote the “high” vertices whose min-cutfrom the root inH is at leastM := β

2 · T ∗

k . The following claim is essentially from Golovin et al. [GGR06].

Claim 4.3 (Cutting Low Nodes) If T ≥ T ∗, the minimum cut inH separatingS \ Sh fromr costs at most2 ·Φ∗.

Proof: Let S′ := S \Sh, andt := β · T ∗

k . For everyv ∈ S′, the minimumr− v cut is at leastβ · Tk ≥ β · T ∗

k = 2Min G, and at mostM in H. Consider theGomory-Hu(cut-equivalent) treeT (H) on graphH rooted atr [Sch03,Chap. 15]. For eachu ∈ S′ let Du ⊆ V denote the minimumr − u cut in T (H) whereu ∈ Du andr 6∈ Du.Pick a subsetS′′ ⊆ S′ of terminals such that the union of their respective min-cuts inT (H) separate all ofS′ fromthe root and their corresponding setsDu are disjoint (the set of cuts in treeT (H) closest to the rootr gives sucha collection). It follows that (a)Du | u ∈ S′′ are disjoint, and (b)F := ∪u∈S′′∂H(Du) is a feasible cut inHseparatingS′ from r. Note that for allu ∈ S′′, we havec(∂H(Du)) ≤M (since it is a minimumr-u cut inH), andc(∂G(Du)) ≥ 2M (it is a feasibler-u cut in G). Thusc(∂H(Du)) ≤ c(∂G(Du)) − c(∂H(Du)) = c(∂Φ∗(Du)).Now, c(F ) ≤ ∑

u∈S′′ c(∂H(Du)) ≤ ∑u∈S′′ c(∂Φ∗(Du)) ≤ 2 · Φ∗. The last inequality uses disjointness of

Duu∈S′′ . Thus the minimumr − S′ cut inH is at most2Φ∗.

Claim 4.4 (Cutting High Nodes) If T ≥ T ∗, the minimumr-Sh cut inH costs at mostβ2 · T ∗, whenβ ≥ 10·ee−1 .

Proof: Consider ar-Sh max-flow in the graphH = G \ Φ∗, and suppose it sendsαi ·M flow to vertexi ∈ Sh.By making copies of terminals, we can assume eachαi ∈ (0, 1]; thek-robust min-cut problem remains unchangedunder making copies. Hence if we show that

∑i∈Sh

αi ≤ k, the total flow (which equals the minr-Sh cut) would

be at mostk ·M = β2 · T ∗, which would prove the claim. For a contradiction, we suppose that

∑i∈Sh

αi > k.We will now claim that there exists a subsetW ⊆ Sh with |W | ≤ k such that the minr-W cut is more thanT ∗,contradicting the fact that everyk-set inH can be separated fromr by a cut of value at mostT ∗. To find this setW , the following redistribution lemma (proved at the end of this theorem) is useful.

Lemma 4.5 (Redistribution Lemma) LetN = (V,E) be a capacitated undirected graph. LetX ⊆ V be a set ofterminals such min-cutN (i, j) ≥ 1 for all nodesi, j ∈ X. For eachi ∈ X, we are given a valueǫi ∈ (0, 1]. Thenfor any integerℓ ≤∑

i∈X ǫi, there exists a subsetW ⊆ X with |W | ≤ ℓ vertices, and a feasible flowf in N from

X to W so that (i) the totalf -flow intoW is at least1−e−1

4 · ℓ and (ii) thef -flow out ofi ∈ X is at mostǫi/4.

We apply this lemma toH = G \ Φ∗ with terminal setSh, but with capacities scaled down byM . Since for anycut separatingx, y ∈ Sh, the rootr lies on one side on this cut (say ony’s side), min-cutH (x, y) ≥M—hence thescaled-down capacities satisfy the conditions of the lemma. Now setℓ = k, andǫi := αi for each terminali ∈ Sh;by the assumption

∑i∈Sh

ǫi =∑

i∈Shαi ≥ k = ℓ. HenceLemma 4.5finds a subsetW ⊆ Sh with k vertices, and

a flow f in (unscaled) graphH such thatf sends a total of at least1−1/e4 · kM units intoW , and at mostαi

4 ·Munits out of eachi ∈ Sh. Also, there is a feasible flowg in the networkH that simultaneously sendsαi ·M flowfrom the root to eachi ∈ Sh, namely the max-flow fromr to Sh. Hence the flowg+4f

5 is feasiblein H, and sends45 ·

1−1/e4 · kM = 1−1/e

5 · kM units fromr into W . Finally, if β > 10·ee−1 , we obtain that the min-cut inH separating

W from r is greater thanT ∗: since|W | ≤ k, this is a contradiction to the assumption that any set with at mostkvertices can separated from the root inH at cost at mostT ∗.

FromClaim 4.1andTheorem 4.2, we obtain a(3, β2 , β)-discriminating algorithm fork-robust minimum cut, when

β ≥ 10ee−1 . We setβ = 10e

e−1 and useLemma 2.2to infer that the approximation ratio of this algorithm ismax3, β2λ +

β = β2λ + β. Since picking edges only in the second-stage is a trivialλ-approximation, the better of the two gives

an approximation ofmin β2λ + β, λ < 17. Thus we have,

7

Theorem 4.6 (Min-cut Theorem) There is a 17-approximation algorithm fork-robust minimum cut.

It now remains to prove the redistribution lemma. At a high level, the proof shows that if we add each vertexi ∈ X to a setW independently with probabilityǫi ℓ/(

∑i ǫi), then this setW will (almost) satisfy the conditions

of the lemma whp. A natural approach to prove this would be to invoke Gale/Hoffman-type theorems [Sch03,Chap. 11]: e.g., it is necessary and sufficient to show thatc(∂V ′) ≥ |demand(V ′) − supply(V ′)| for all V ′ ⊆ Vfor this random choiceW . But we need to prove such facts forall subsets, and all we know about the network isthat the min-cut between any pair of nodes inX is at least1! Also, such a general approach is likely to fail, sincethe redistribution lemma is false for directed graphs (seeSection D) whereas the Gale-Hoffman theorems hold fordigraphs. In our proof, we use undirectedness to fractionally pack Steiner trees into the graph, on which we can doa randomized-rounding-based analysis.

Proof of Lemma 4.5(Redistribution Lemma): To begin, we assume wlog that the boundsǫi = 1/P for all i ∈ Xfor some integerP . Indeed, letP ∈ N be large enough so thatǫi = ǫiP is an integer for eachi ∈ X. Add, foreachi ∈ X, a star withǫi− 1 leaves centered at the original vertexi, set all these new vertices to also be terminals,and make edges have unit capacity. Set the newǫ’s to be1/P for all terminals. To avoid excess notation, call thisgraphN as well; note that the assumptions of the lemma continue to hold, and any solution on this new graph canbe mapped back to the original graph.

Let ce denote the edge capacities inN , and recall the assumption that every cut inN separatingX has capacityat least one. Since the natural LP relaxation for Steiner-tree has integrality gap of2, this implies the existenceof Steiner treesTaa∈A on the terminal setX that fractionally pack into the edge capacitiesc. I.e., there existpositive multipliersλaa∈A such that

∑a λa = 1

2 , and∑

a λa · χ(Ta) ≤ c, whereχ(Ta) is the characteristicvector of the treeTa. ChooseW ⊆ X by takingℓ samples uniformly at random (with replacement) fromX. Wewill construct the flowf from X to W as a sum of flows on these Steiner trees. In the following, letq := |X|; notethatℓ ≤ |X|ǫ = q/P .

Consider any fixed treeTa in this collection, where we think of the edges as having unitcapacities. We claim thatin expectation,Ω(ℓ) units of flow can be feasibly routed fromX to W in Ta such that each terminal supplies atmostℓ/q. Indeed, letτa denote an oriented Euler tour corresponding toTa. Since the tour uses any tree edge twice,any feasible flow routed inτa (with unit-capacity edges) can be scaled by half to obtain a feasible flow inTa. Wecall a vertexv ∈ X a-closeif there is someW -vertex located at mostq/ℓ hops fromv on the (oriented) tourτa.Construct a flowfa on τa by sendingℓ/q flow from eacha-close vertexv ∈ X to its nearestW -vertex alongτa.By the definition ofa-closeness, the maximum number of flow paths infa that traverse an edge onτa is q/ℓ; sinceeach flow path carriesℓ/q flow, the flow on any edge inτa is at most one, and hencefa is feasible.

For any vertexv ∈ X and a tourτa, the probability thatv is not a-close is at most(1 − q/ℓq )ℓ ≤ e−1; hence

v ∈ X sends flow infa with probability at least1 − e−1. Thus the expected amount of flow sent infa is at least(1 − e−1)|X| · (ℓ/q) = (1 − e−1) · ℓ. Now define the flowf := 1

2

∑a λa · fa by combining all the flows along

all the Steiner trees. It is easily checked that this is a feasible flow in N . Since∑

a λa = 12 , the expected value of

flow f is at least1−1/e4 ℓ. Finally the amount of flow inf sent out of any terminal is at most1

4 · ℓ/q ≤ 14P . This

completes the proof of the redistribution lemma.

The k-max-min Min-Cut Problem. Whenλ = 1 andΦ∗ = 0, the proof ofTheorem 4.2gives a randomizedalgorithm such that if the minimumr-S cut is greater thanβ2 T , it finds a subsetW of at mostk terminals suchthat separatingW from the root costs more thanT (witnessed by the dual value). Using this we get a randomized(3, β

2 , β) stronglydiscriminating algorithm, and hence a randomizedO(1)-approximation algorithm fork-max-minmin cut fromLemma 2.4. We note that fork-max-min min-cut, a(1− 1/e)-approximation algorithm was alreadyknown (even for directed graphs) via submodular maximization. However the above approach has the advantagethat it also extends tok-robust min-cut.

8

5 Steiner ForestIn k-robust Steiner forest, we have a graphG = (V,E) with edge costsc : E → R+, and a setU ⊆ V × Vof potential terminal pairs; any set in

(Uk

)is a valid scenario in the second stage. For a set of pairsS ⊆ V × V ,

the graphG/S is obtained by identifying each pair inS together;dG/S(·, ·) is the distance in this “shrunk” graph.The algorithm is given below. This algorithm is a bit more involved than the previous ones, despite a similargeneral structure: we maintain a set of “fake” pairsSf that may not belong toU for this case. The followinganalysis shows a constant-factor guarantee. (Withoutlines 6-7, the algorithm is more natural, but for that we cancurrently only show anO(log n)-approximation; it seems that anO(1)-approximation for that version would implyanO(log n)-competitiveness for online greedy Steiner forest.)

Algorithm 3 Algorithm for k-Robust Steiner Forest1: input: k-robust Steiner forest instance and thresholdT .2: let β ← Θ(1), γ ← Θ(1) such thatγ ≤ β/2.3: let Sr, Sf ,W ← ∅4: while there exists a pair(s, t) ∈ U with dG/(Sr∪Sf )(s, t) > β · T

k do5: let Sr ← Sr ∪ (s, t)6: if dG(s,w) < γ · T

k for somew ∈W then Sf ← Sf ∪ (s,w) elseW ←W ∪ s7: if dG(t, w′) < γ · T

k for somew′ ∈W then Sf ← Sf ∪ (t, w′) elseW ← W ∪ t8: end while9: output first stage solutionΦT to be the 2-approximate Steiner forest [AKR95, GW95] on pairsSr along with

shortest-paths connecting every pair inSf .10: defineAugmentT (i) to be the edges on thesi − ti shortest-path inG/(Sr ∪ Sf ), for each pairi ∈ U .11: output second stage solutionAugmentT whereAugmentT (S) :=

⋃i∈D AugmentT (i) for all D ⊆ U .

Claim 5.1 (Property A for Steiner forest) For all T ≥ 0 andD ∈(U

k

), the edgesΦT

⋃AugmentT (D) connect

every pair inD, and have costc(AugmentT (D)) ≤ β T .

Proof: The first part is immediate from the definition ofAugmentT and the fact thatΦT connects every pair inSr ∪ Sf . The second part follows from the termination conditiondG/(Sr∪Sf )(si, ti) ≤ β · T

k for all pairsi ∈ U ;

this impliesc(AugmentT (D)) ≤∑i∈D c(AugmentT (i)) ≤∑

i∈D dG/(Sr∪Sf )(si, ti) ≤ |D|k · β T .

Lemma 5.2 The optimal value of the Steiner forest on pairsSr is at least|W | × γ2

Tk .

Proof: Consider the primal (covering) and dual (packing) LPs corresponding to Steiner forest onSr. Note that foreach pairi ∈ Sr, the distancedG(s,ti) ≥ β · T

k ≥ 2γ · Tk ; so any ball of radiusγ2 · T

k around a vertex inSr maybe used in the dual packing problem. Observe thatW consists of only vertices fromSr, and each time we add avertex toW , it is at leastγT/k distant from any other vertex inW . Hence we can feasibly pack dual balls of radiusγ2 · T

k around eachW -vertex. This is a feasible dual to the Steiner forest instance onSr, of value|W |γ/2 · T/k.

Lemma 5.3 The number of “witnesses”|W | is at least the number of “real” pairs|Sr|, and |Sr| is at least thenumber of “fake” pairs|Sf |.

Proof: Partition the setSr as follows: Sg are the pairs where both end-points are added toW , So are the pairswhere exactly one end-point is added toW , andSb are the pairs where neither end-point is added toW . It followsthat |Sr| = |Sg|+ |So|+ |Sb| and|W | = 2 · |Sg|+ |So|.Consider an auxiliary graphH = (W,E(W )) on the vertex setW which is constructed incrementally:

• When a pair(s, t) ∈ Sg is added, verticess, t are added toW , and edge(s, t) is added toE(W ).• Suppose a pair(s, t) ∈ So is added, wheres is added toW , but t is not because it is “blocked” byw′ ∈ W .

In this case, vertexs is added, and edge(s,w′) is added toE(W ).

9

• Suppose a pair(s, t) ∈ Sb is added, wheres andt are “blocked” byw andw′ respectively. In this case, novertex is added, but an edge(w,w′) is added toE(W ).

Claim 5.4 At any point in the algorithm ifx, y ∈W lie in the same component ofH thendG/(Sf∪Sr)(x, y) = 0.

Proof: By induction on the algorithm, and the construction of the graphH.

• Suppose pair(s, t) ∈ Sg is added, then the claim is immediate.H has one new connected components, tand others are unchanged. Since(s, t) ∈ Sr, dG/(Sf∪Sr)(s, t) = 0 and the invariant holds.

• Suppose pair(s, t) ∈ So is added, withs added toW andt blocked byw′ ∈W . In this case, the componentof H containingw′ grows to also contains; other components are unchanged. Furthermore(t, w′) is addedto Sf and(s, t) to Sr, which impliesdG/(Sf∪Sr)(s,w

′) = 0. So the invariant continues to hold.• Suppose pair(s, t) ∈ Sb is added, withs and t blocked byw,w′ ∈ W respectively. In this case, the

components containingw andw′ get merged; others are unchanged. Also(s,w), (t, w′) are added toSf and(s, t) to Sr; sodG/(Sf∪Sr)(w,w′) = 0, and the invariant continues to hold.

Since these are the only three cases, this proves the claim.

Claim 5.5 The auxiliary graphH does not contain a cycle whenγ ≤ β/2

Proof: For a contradiction, consider the first edge(x, y) that when added toH by the process above cre-ates a cycle. Let(s, t) be the pair that caused this edge to be added, and consider thesituation just before(s, t) is added toSr. Since(x, y) causes a cycle,x, y belong to the same component ofH, and hencedG/(Sf∪Sr)(x, y) = 0 by the claim above. But sincex is eithers or its “blocker” w, andy is eithert or its

blockerw′, it follows thatdG/(Sf∪Sr)(s, t) < 2γ · Tk ≤ β · T

k . But this contradicts the condition which wouldcause(s, t) to be chosen intoSr by the algorithm.

Now for some counting. Consider graphH at the end of the algorithm:W denotes its vertices, andE its edges.From the construction ofH, we obtain|W | = 2 · |Sg| + |So| and|E| = |Sg| + |So| + |Sb| = |Sr|. SinceH isacyclic, |Sr| = |E| ≤ |W | − 1. Also note that|Sf | = 2 · |Sb| + |So| = 2 · |Sr| − |W | < |Sr|. Thus we have|W | ≥ |Sr| ≥ |Sf | as required in the lemma.

Theorem 5.6 (Property B for Steiner forest) LetΦ∗ denote the optimal first stage solution (and its cost), andT ∗

the optimal second stage cost. IfT ≥ T ∗ thenc(ΦT ) ≤ 4γγ−2 · (Φ∗ + T ∗).

Proof: Let |Sr| = αk. UsingLemma 5.3, Lemma 5.2and the optimal solution,

γ2 · α · T ≤ |W | ·

γ2

Tk ≤ OPT (Sr) ≤ Φ∗ +

⌈|Sr |k

⌉T ∗ ≤ Φ∗ + T ∗ + α · T ∗ ≤ Φ∗ + T ∗ + α T (5.1)

Thusα · T ≤ 2γ−2 · (Φ∗ + T ∗) andOPT (Sr) ≤ γ

γ−2 · (Φ∗ + T ∗). So the 2-approximate Steiner forest onSr has

cost at most 2γγ−2 · (Φ∗ + T ∗). Note that the distance between each pair inSf is at mostγ · T

k ; so the total length

of shortest-paths inSf is at most|Sf | · γ · Tk ≤ |Sr| · γ · T

k (again byLemma 5.3). Thus the algorithm’s first-stagecost is at most2γ

γ−2 · (Φ∗ + T ∗) + αγ · T ≤ 4γγ−2 · (Φ∗ + T ∗).

Theorem 5.7 (Steiner Forest Main Theorem)There is a 10-approximation fork-robust Steiner forest.

Proof: UsingClaim 5.1andTheorem 5.6, we obtain a( 4γγ−2 , 4γ

γ−2 , β)-discriminating algorithm (Definition 2.1) fork-robust Steiner forest. Settingβ = 2γ andγ := 2 + 2 · (1− 1/λ), Lemma 2.2implies an approximation ratio ofmax 4γ

γ−2 , 4γ/λγ−2 +2γ ≤ 4+ 4

1−1/λ . Again the trivial algorithm that only buys edges in the second-stage achieves

a2λ-approximation. Taking the better of the two, the approximation ratio ismin2λ, 4 + 41−1/λ < 10.

In Appendix C, we show how this algorithm gives us astronglydiscriminating algorithm, and hence a constant-factor approximation algorithm fork-max-min Steiner forest. Also, in [KKMS08], a 3-approximation for Steinerforest problem was given when the input graph is itself a tree. This bound can be improved to2.25 via our generalapproach; we defer details to the full version.

10

6 Final Remarks

In this paper, we have presented a unified approach to directly solving k-robust covering problems andk-max-min problems. As mentioned in the introduction, one can showthat solving thek-max-min problem also leads toa k-robust algorithm—we give a general reduction inAppendix F. While this general reduction leads to poorerapproximation guarantees for the cardinality case, it easily extends to more general cases. Indeed, if the uncertaintysets for robust problems are not just defined by cardinality constraints, we can ask: which families of downwards-closed sets can we devise robust algorithms for? The generalreduction in the Appendix shows how to incorporateintersections of matroid-type and knapsack-type uncertainty sets as well.

Our work suggests several general directions for research.While the results for thek-robust case are fairly tight,can we improve on our results for general uncertainty sets tomatch those for the cardinality case? Can we devisealgorithms to handle one matroid constraint that are as simple as our algorithms for the cardinality case? Anintriguing specific problem is to find a constant factor approximation for the robust Steiner forest problem in theexplicit scenarios version.

Acknowledgments. We thank Chandra Chekuri, Ravishankar Krishnaswamy, DannySegev, and Maxim Sviri-denko for invaluable discussions.

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A Proofs of Main Reduction Lemmas

A.1 Proof of Lemma 2.2

LetA denote an algorithm forRobustk(Π) such that it is(α1, α2, β)-discriminating. Let ground-setE = [m], andcmax := maxe∈[m] ce. By scaling, we may assume WLOG that all costs in theRobustk(Π) instance are integral.Let ǫ > 0 be any value as given by the lemma (where1

ǫ is polynomially bounded), andN := ⌈log1+ǫ (m cmax)⌉+1; note thatN is polynomial in the input size. DefineT :=

(1 + ǫ)i | 0 ≤ i ≤ N

.

The approximation algorithm forRobustk(Π) runs the(α1, α2, β)-discriminating algorithmA for every choice ofT ∈ T (here|T | is polynomially bounded), and returns the solution corresponding to:

T := arg minc(ΦT ) + λ · β T | T ∈ T

.

Recall thatT ∗ denotes the optimal second-stage cost, clearlyT ∗ ≤ m · cmax. Let i∗ ∈ Z+ be chosen such that(1 + ǫ)i

∗−1 < T ∗ ≤ (1 + ǫ)i∗; also letT ′ := (1 + ǫ)i

∗(note thatT ′ ∈ T ). The objective value of the approximate

solution (forT ) can be bounded as follows.

c(Φ eT

)+ λ ·max

ω∈Ωc(Augment eT (ω)

)≤ c(Φ eT ) + λ · β T

≤ c(ΦT ′) + λ · β T ′

≤ (α1 · Φ∗ + α2 · T ∗) + (1 + ǫ) βλ · T ∗

≤ (1 + ǫ) ·[α1 · Φ∗ +

(β +

α2

λ

)· λT ∗

].

The first inequality follows from Property A(i) inDefinition 2.1; the second by the choice ofT ; the third byProperty B (applied withT = T ′ ≥ T ∗) in Definition 2.1, and usingT ′ ≤ (1 + ǫ) · T ∗. Thus this algorithm forRobustk(Π) outputs a solution that is a

((1 + ǫ) ·max

α1, β + α2

λ

)-approximation.

A.2 Proof of Lemma 2.4

The approximation algorithm forMaxMin(Π) is similar to that inLemma 2.2. Let A denote an algorithm forthe robust problem that is(α1, α2, β) strongly discriminating. Recall that thek-max-min instance correspondsto theRobustk(Π) instance withλ = 1, and hence we will run algorithmA on this robust instance. Also fromDefinition 2.1, T ∗ denotes the optimal second-stage cost ofRobustk(Π), and its optimal fist-stage costΦ∗ = 0(sinceλ = 1). Note that the optimal value of thek-max-min instance also equalsT ∗.

Let ground-setE = [m], andcmax := maxe∈[m] ce. By scaling, we may assume WLOG that all costs in theinstance are integral. Letǫ > 0 be any value as given by the lemma (where1

ǫ is polynomially bounded), andN := ⌈log1+ǫ (m cmax)⌉+ 1; note thatN is polynomial in the input size. Consider the integral powers of (1+ ǫ),

T := tiNi=0, whereti = (1 + ǫ)i for i = 0, 1, · · · , N.

13

The approximation algorithm forMaxMin(Π) runs the strongly discriminating algorithmA for every choice ofT ∈ T , and letp ∈ 1, · · · , N be the smallest index such thatc(Φ(tp)) ≤ α2 tp. Observe that there must existsuch an index since for allT ≥ T ∗, we havec(ΦT ) ≤ α2 T ∗ ≤ α2 T (property B inDefinition 2.1, usingΦ∗ = 0),and clearlyT ∗ ≤ m · cmax ≤ tN . The algorithm then outputsQ(tp−1) as the max-min scenario. Below we provethat it achieves the claimed approximation. We have for allT ≥ 0,

T ∗ = max

Opt(D) : D ∈

([n]

k

)≤ max

c(ΦT ) + c(AugmentT (D)) : D ∈

([n]

k

)≤ c(ΦT ) + β T.

Above, the inequalities are by conditions A(i) and A(ii) ofDefinition 2.1. SettingT = tp here, and by choice ofp,

T ∗ ≤ c(Φ(tp)) + β tp ≤ (α2 + β) tp.

Hencetp is a (α2 + β)-approximation to the max-min valueT ∗. Now applying the condition ofDefinition 2.3with T = tp−1, sincec(Φ(tp−1)) ≥ α2 tp−1 (by choice of indexp), we obtain that the minimum cost to coverrequirementsQ(tp−1) is at least:

tp−1 =tp

1 + ǫ≥ T ∗

(1 + ǫ) · (α2 + β),

which implies the desired approximation guarantee.

B Steiner TreeIn thek-robust Steiner tree, we are given a graphG = (V,E) with edge costsc : E → R+, a root vertexr, and asetU ⊆ V of potential terminals. Any set ofk terminals fromU—i.e., any set in

(Uk

)—is a valid scenario in the

second stage. Letd(·, ·) be the shortest-path distance according to the edge costs. For a setS ⊆ V of terminals,define the distanced(v, S) := minw∈S d(v,w).

By the results inSection 2.1, a discriminating algorithm for this problem immediately gives us an algorithm for therobust version, and this is how we shall proceed. Here is our discriminating algorithm fork-robust Steiner tree: itpicks aβT/k-netS of the terminals inU , and builds a MST onS as the first stage.

Algorithm 4 Algorithm for k-Robust Steiner Tree1: input: instance ofk-robust Steiner tree and thresholdT .2: let β ← Θ(1), S ← r.3: while there exists a terminalv ∈ U with d(v, S) > β · T

k do4: S ← S ∪ v5: end while6: output first-stage solutionΦT to be a minimum spanning tree onS.7: for each i ∈ U , defineAugmentT (i) to be the edges on a shortest-path fromi to S.8: output second-stage solutionAugmentT whereAugmentT (D) :=

⋃i∈D AugmentT (i) for all D ⊆ U .

To show that the algorithm is discriminating, we need to showthe two properties inDefinition 2.1. The firstproperty is almost immediate from the construction: since every point inU \ S is close to some point in the netS,this automatically ensures that the second stage recourse cost is small.

Claim B.1 (Property A for Steiner Tree) For all T ≥ 0 andD ∈(U

k

), the edgesΦT ∪ AugmentT (D) connect

the terminals inD to the rootr, and have costc(AugmentT (D)) ≤ β T .

Proof: From the definition of the second-stage solution,AugmentT (D) contains the edges on shortest paths fromeachD-vertex to the setS. Moreover,ΦT is a minimum spanning tree onS (which in turn contains the rootr).HenceΦT ∪ AugmentT (D) connectsD to the rootr. To bound the cost, note that by the termination condition inthewhile loop, every terminali ∈ U satisfiesd(i, S) ≤ β T

k . Thus,

c(AugmentT (D)) ≤∑

i∈D

c(AugmentT (i)) =∑

i∈D

d(i, S) ≤ |D|k· β T.

14

This completes the proof that the algorithm above satisfies Property A.

It now remains to show that the algorithm satisfies Property Bas well. Let us show this for a sub-optimal settingsof values; we will improve on these values subsequently. Theproof is dual-based and shows that if the cost of theMST onS were large, then the optimal first stage solution costΦ∗ must have been large as well!

Theorem B.2 (Property B for Steiner tree) Let Φ∗ denote the optimal first stage solution (and its cost), andT ∗

the optimal second stage cost. IfT > T ∗ then the first stage costc(ΦT ) ≤ 2 · ββ−2 · (Φ∗ + T ∗).

Proof: Suppose|S| = αk. We can divide upS into ⌈α⌉ setsS1, S2, . . . , S⌈α⌉ with at mostk terminals each, and letE(Si) denote the second-stage edges bought by the optimal solution under scenarioSi. HenceΦ∗∪(∪i≤⌈α⌉E(Si))is a feasible solution to the Steiner tree onS of cost at mostΦ∗ + ⌈α⌉ · T ∗. Also, since each of the points inS is atleast at distanceβT/k from each other, we get (belowOPT (S) is the length of the minimum Steiner tree onS),

β2 · αT = |S| · β

2 · Tk ≤ OPT (S) ≤ Φ∗ + ⌈α⌉ · T ∗ ≤ Φ∗ + (α + 1) · T ∗ ≤ Φ∗ + T ∗ + αT.

HenceαT ≤ 2β−2(Φ∗ + T ∗) andOPT (S) ≤ β

β−2 · (Φ∗ + T ∗); since the MST heuristic is a2-approximation tothe optimal Steiner tree, we get the theorem.

CombiningClaim B.1andTheorem B.2shows that our algorithm is a( 2ββ−2 , 2β

β−2 , β)-discriminating algorithm fork-robust Steiner tree. Setting, say,β = 4 and applyingLemma 2.2gives us anmax(4, 4 + 4) = 8-approximationfor k-robust Steiner tree. In the next subsection, we will show how to improve this guarantee.

B.1 Improved Approximation for Steiner Tree

In the previous analysis, we just wanted to show the main ideas and hence were somewhat sloppy with the analysis.Let us now show how to get a tighter bound using a fractional analysis.

Theorem B.3 (Improved Property B for Steiner Tree) If T ≥ T ∗ thenc(ΦT ) ≤ 2ββ−2 · Φ∗ + 2 · T ∗.

Proof: Firstly suppose|S| ≤ k: then it is clear that there is a Steiner tree onr ∪ S of cost at mostΦ∗ + T ∗, andthe algorithm finds one of cost at most twice that. In the following assume that|S| > k.

Let LP (S) denote the minimum length of afractionalSteiner tree on terminalsr∪S. Since each of the points inS is at least at distanceβ · Tk ≥ β · T ∗

k from each other, we getLP (S) ≥ β2k · |S| ·T ∗. We now construct a fractional

Steiner treex : E → R+ of small length. Number the terminals inS arbitrarily, and for each1 ≤ j ≤ |S| letAj = j, j + 1, · · · , j + k − 1 (modulo |S|). Let Πj ⊆ E \ Φ∗ denote the second-stage edges bought in theoptimal solution under scenarioAj : soΦ∗ ∪ Πj is a Steiner tree on terminalsr ∪ Aj, andc(Πj) ≤ T ∗. Define

x := χ(Φ∗) + 1k ·

∑|S|j=1 χ(Πj). We claim thatx supports unit flow fromr to anyi ∈ S: note that there arek sets

Ai−k+1, · · · , Ai that containi, and for eachi−k +1 ≤ j ≤ i, we have1k · (χ(Φ∗) + χ(Πj)) supports1

k flow from

r to i. Thusx is a feasible fractional Steiner tree onr ∪ S, of cost at mostΦ∗ + |S|k · T ∗. Combined with the

lower bound onLP (S),|S| · β

2 · T ∗

k ≤ LP (S) ≤ Φ∗ + |S|k · T ∗. (B.2)

Thus we haveLP (S) ≤ ββ−2 · Φ∗, which implies the theorem since the minimum spanning tree on r ∪ S costs

at most twiceLP (S).

From Claim B.1 and Theorem B.3, we now get that the algorithm is( 2ββ−2 , 2, β)-discriminating. Thus, setting

β = 2− 1λ +

√4 + 1/λ2 and applyingLemma 2.2, we get the following approximation ratio.

max

2β

β − 2,

2

λ+ β

= 2 +

1

λ+

√4 +

1

λ2.

On the other hand, the trivial algorithm which does nothing in the first stage is a1.55 ·λ approximation. Hence thebetter of these two ratios gives an approximation bound better than4.5.

15

The k-max-min Steiner Tree Problem. We show that the above algorithm can be extended to be( 2ββ−2 , 2, β)

strongly discriminating. As shown above, it is indeed discriminating. To show that Definition2.3 holds, considerthe proof of TheoremB.3 whenλ = 1 (soΦ∗ = 0) and suppose thatc(ΦT ) ≥ 2T . The algorithm to output thek-setQ proceeds via two cases.

1. If |S| ≤ k thenQ := S. The minimum Steiner tree onQ is at least half its MST, i.e. at least12c(ΦT ) ≥ T .

2. If |S| > k thenQ ⊆ S is anyk-set; by the construction ofS, we can feasibly pack dual balls of radiusβ Tk

around eachQ-vertex, and soLP (Q) ≥ βT ≥ T . Thus the minimum Steiner tree onQ is at leastT .

B.2 Unrooted Steiner tree

We note that thek-robust Steiner tree problem studied above differs from [KKMS08] since there is no root inthe model of [KKMS08]. In the unrooted version, any subset ofk terminals appear in the second stage, andthe goal is to connect themamongst each other. We show that a small modification in the proof implies thatAlgorithm 1 (wherer ∈ U is set to an arbitrary terminal) achieves a good approximation in the unrooted case aswell. This algorithm is essentially same as the one used by [KKMS08], but with different parameters: hence ourframework can be viewed as generalizing their algorithm. Our proof is somewhat shorter and gives a slightly betterapproximation ratio.

Below, Φ∗ andT ∗ denote the optimal first and second stage costs for the given unrooted instance. It is clear thatClaim B.1continues to hold in this case as well: hence Property A ofDefinition 2.1is satisfied. We next bound thefirst stage cost of the algorithm (i.e. Property B ofDefinition 2.1).

Theorem B.4 (Property B for Unrooted Steiner Tree) If T ≥ T ∗ thenc(ΦT ) ≤ 2ββ−2 · Φ∗ + 2T ∗.

Proof: Firstly suppose|S| ≤ k: then it is clear that there is a Steiner tree onS of cost at mostΦ∗ + T ∗, and thealgorithm finds one of cost at most twice that. In the following assume that|S| > k.

Let LP (S) denote the minimum length of afractional Steiner tree on terminalsS (recall, no root here). Since eachof the points inS is at least at distanceβ · T

k ≥ β · T ∗

k from each other, we getLP (S) ≥ β2k · |S| · T ∗. We now

construct a fractional Steiner treex : E → R+ of small length. Number the terminals inS arbitrarily, and for each1 ≤ j ≤ |S| let Aj = j, j + 1, · · · , j + k − 1 (modulo|S|). Let Πj ⊆ E \ Φ∗ denote the second-stage edgesbought in the optimal solution under scenarioAj : soΦ∗ ∪ Πj is a Steiner tree on terminalsAj, andc(Πj) ≤ T ∗.

Definex := χ(Φ∗) + 1k ·

∑|S|j=1 χ(Πj).

Claim B.5 For anyi ∈ S, x supports a unit flow from terminali to i + 1 (modulo|S|).

Proof: Note that there arek−1 setsAi−k+2, · · · , Ai that contain bothi andi+1. LetJ := i−k+2, · · · , i.So for eachj ∈ J , we have1

k ·(χ(Φ∗) + χ(Πj)) supports1k flow from i to i+1. Furthermore,

(∪l∈S\JΠl

)∪Φ∗

is a Steiner tree connecting terminals∪l∈S\JAl ⊇ i, i + 1; i.e. 1k ·

(χ(Φ∗) +

∑l∈S\J χ(Πl)

)also supports

1k flow from i to i + 1. Thus we obtain the claim.

Thusx is a feasible fractional Steiner tree on terminalS, of cost at mostΦ∗ + |S|k · T ∗. Combined with the lower

bound onLP (S),|S| · β

2 · T ∗

k ≤ LP (S) ≤ Φ∗ + |S|k · T ∗. (B.3)

Thus we haveLP (S) ≤ ββ−2 ·Φ∗, which implies the theorem since the minimum spanning tree on S costs at most

twiceLP (S).

Thus by the same calculation as in the rooted case, we obtain aresult that slightly improves on the constantsobtained by [KKMS08] for the same problem.

Theorem B.6 There is a 4.5-approximation algorithm for (unrooted)k-robust Steiner tree.

16

C The k-max-min Algorithm for Steiner Forest

We now extend thek-robust Steiner forest algorithm to be( 4γγ−2 , 4γ

γ−2 , 2γ) stronglydiscriminating (whenγ = 3).As shown earlier, it is indeed discriminating. To show thatDefinition 2.3holds, consider the proof ofTheorem 5.6whenλ = 1 (soΦ∗ = 0) and supposec(ΦT ) ≥ 4γ

γ−2T ≥ 2γ T . The algorithm to output thek-setQ has two cases.

1. If the number of “real” pairs|Sr| ≤ k thenQ := Sr. We have:

c(ΦT ) ≤ 2 ·OPT (Sr) +γT

k|Sf | ≤ 2 ·OPT (Sr) +

γT

k|Sr| ≤ 2 ·OPT (Sr) + γT.

The first inequality is by definition ofΦT and since distance between each pair inSf is at mostγ · Tk , the

second inequality is byLemma 5.3, and the last inequality uses|Sr| ≤ k. Sincec(ΦT ) ≥ 2γT , it followsthatOPT (Sr) ≥ γT/2 ≥ T .

2. If |Sr| > k then the number of “witnesses”|W | ≥ |Sr| > k, by Lemma 5.3. Let Q ⊆ Sr beanyk-set ofpairs such that for eachi ∈ Q at least one ofsi, ti is in W . By the construction ofSr, we can feasiblypack dual balls of radiusγ2

Tk around eachW -vertex, and soOPT (Q) ≥ |Q| · γ

2Tk = γ

2T ≥ T .

D An Illustrative Example for Min-Cut

Let us show that our theorems fork-robust min-cut have to use the undirectedness of the graph crucially, and thatthe theorems are in fact false for directed graphs, even fork = 1. Consider the digraphG with a rootr, a “center”vertexc, andℓ terminalsv1, v2, . . . , vℓ. There is some unit capacity arcs(vi, r), and arcs of capacity

√ℓ between

(c, r), and between(vi, c) for all i ∈ [ℓ]. The inflation factor isλ =√

ℓ. The optimal strategy is to delete thearc (c, r) in the first stage. Sincek = 1, one of the terminalsvi demands to be separated from the root in thesecond stage, whence deleting the edge(vi, r) costsλ · 1 =

√ℓ resulting in a total cost of2

√ℓ. However, any

threshold-based algorithm would either choose none of the terminals (resulting in a recourse cost ofλ√

ℓ = ℓ), orall of them (resulting in a first-stage cost of at leastℓ).

A similar bad example shows that the redistribution lemma (Lemma 4.5) is false for digraphs. This graph onlyhas unit capacity edges, and the same set of vertices as the digraph above. The arcs are(c, r), (r, vi)i∈[ℓ], and

(vi, c)i∈[ℓ]. Note that the min-cut between everyvi-vj pair is1, but if we give each of thevi’s ǫi = 1/√

ℓ flow,

there is no way to choose√

ℓ of these vertices and collect a total ofΩ(√

ℓ) flow at these “leaders”.

E Multicut

We now consider the multicut problem: we are given an undirected graphG = (V,E) with edge-costsc : E → R+,andm vertex-pairssi, timi=1. In thek-robust version, we are also given an inflation parameterλ and boundkon the cardinality of the realized demand-set. LetΦ∗ denote the optimal first stage solution (and its cost), andT ∗

the optimal second stage cost; soOpt = Φ∗ + λ · T ∗. The algorithm (given below) is essentially the same as forminimum cut, however the analysis requires different arguments.

Claim E.1 (Property A for Multicut) For all T ≥ 0 and ω ⊆ [m], the edgesΦT⋃

AugmentT (ω) separatesi

andti for all i ∈ ω; additionally if |ω| ≤ k then the costc(AugmentT (ω)) ≤ β T .

Proof: Pairs in ω ∩ S are separated byΦT . By definition of AugmentT , for each pairi ∈ ω \ S edgesAugmentT (i) form an si − ti cut. Thus we have the first part of the claim. For the second part, note thatby definition ofS, the cost ofAugmentT (i) is at mostβ T/k for all i ∈ [m].

Theorem E.2 (Property B for Multicut) If T ≥ T ∗ thenc(ΦT ) ≤ O(log n) · Φ∗ + O(log2+ǫ n) · T ∗.

17

Algorithm 5 Algorithm for k-Robust MultiCut1: input: k-robust multicut instance and thresholdT .2: let ρ := O(log n) be the approximation factor in Racke’s oblivious routing scheme [Rac08], ǫ ∈ (0, 1

2) any

constant, andβ := ρ · 16 log nǫ log log n .

3: let S ← i ∈ [m] | min si-ti cut has cost at leastβ · Tk .

4: output first stage solutionΦT as theO(log n)-approximate multicut [GVY96] for S.5: defineAugmentT (i) as edges in the minsi − ti cut, for i ∈ [m] \ S; andAugmentT (i) = ∅ for i ∈ S.6: output second stage solutionAugmentT whereAugmentT (ω) :=

⋃i∈ω AugmentT (i) for all ω ⊆ [m].

To prove the lemma, the high level approach is similar to thatfor k-robust min-cut. We first show that the subsetof pairsS ⊆ S whose min-cut fell substantially on deleting the edges inΦ∗ can actually be completely separatedby payingO(1)Φ∗ (this is proved inLemma E.3). Then inLemma E.6we show that the remaining pairs inS \ Scan befractionally separated at costO(log1+ǫ n)T ∗. Since the [GVY96] algorithm for multicut is relative to theLP, this would implyTheorem E.2.

Let us begin by formally defining the cast of characters. LetH := G \Φ∗ andM := β · T ∗

k . Define,

S :=i ∈ S | min costsi-ti cut inH is less thanM

4

to be the set of pairs whose mincut inG was at leastM , but has fallen to at mostM/4 in H = G \ Φ∗.

Lemma E.3 If T ≥ T ∗, there is a multicut separating pairsS in graphH which has cost at most2Φ∗.

Proof: We work with graphH = (V, F ) with edge-costsc : F → R. A clusterrefers to any subset of vertices. Acut equivalent tree(c.f. [CCPS98]), P = (N (P ), E(P )) is an edge-weighted tree on clustersN (P ) = Njrj=1

such that:

• the clustersNjrj=1 form a partition ofV , and• for any edgee ∈ E(P ), its weight inP equals thec-cost of the cut corresponding to deleting this edge in

P . I.e., if (Se, Sce) is the partition ofV obtained by unioning the vertices in the clusters belongingto the two

connected components ofP \ e, thene’s weight inP equalsc(δ(Se)) = c(δ(Sce)).

TheGomory-HutreePGH = (V,E(PGH )) of H is a cut-equivalent tree where the clusters are singleton vertices,and which has the additional property that for everyu, v ∈ V the minimumu-v cut in PGH equals the minimumu-v cut in H. For any cut-equivalent tree, a clusterN ⊆ V is calledactive if there is somei ∈ S such that|N ∩ si, ti| = 1; otherwise the clusterN is calleddead. We obtain a cut-equivalent treeQ from PGH asfollows: (1) shrink all edges having weight greater thanM

4 , and (2) repeatedly merge dead clusters with any of itsneighbors. Note that in the resulting treeQ, every edge inE(Q) has weight at mostM4 , and every cluster inN (Q)

is active. LetD :=⋃

N∈N (Q) ∂H(N). In the next two claims we show thatD is a feasible multicut forS with costat most2Φ∗.

Claim E.4 D is a feasible multicut separating pairsS in H.

Proof: Clearly for each pairi ∈ S, verticessi andti are in distinct active clusters of the Gomory-Hu treePGH . Additionally there is some edge of weight less thatM

4 on thesi − ti path inPGH : since the minimumsi − ti cut in H is less thanM

4 . Observe that in obtaining treeQ from PGH , we never contract two activeclusters nor an edge of weight less thatM

4 . Thussi andti lie in distinct clusters ofQ. Since this holds for all

i ∈ S, the claim follows by definition ofD.

Claim E.5 The costc(D) =∑

e∈D ce ≤ 2Φ∗, if T ≥ T ∗.

18

Proof: Consider any clusterN ∈ N (Q). Since all clusters inN (Q) are active,N contains exactly one ofsi, ti for somei ∈ S. Hence the cut∂G(N) (in graphG) has cost at leastβ · Tk ≥ β · T ∗

k = M , by definition

of the setS ⊇ S.

LetN2(Q) ⊆ N (Q) denote all clusters inQ having degree at most two inQ. Note that|N2(Q)| ≥ 12 |N (Q)|.

Using the above observation and the fact that clusters inN2(Q) are disjoint, we have

|N2(Q)|M ≤∑

N∈N2(Q)

c(∂G(N)) =∑

N∈N2(Q)

(c(∂H (N)) + c(∂Φ∗(N))) ≤∑

N∈N2(Q)

c(∂H (N)) + 2Φ∗. (E.4)

We now claim that for anyN ∈ N2(Q), the costc(∂H(N)) ≤ M2 . Let e1 and e2 denote the two edges

incident to clusterN in Q (the case of a single edge is easier). Let(Ul, V \ Ul) denote the cut correspondingto edgeel (for l = 1, 2) whereN ⊆ Ul. Each of these cuts has costc(∂H(Ul)) ≤ M

4 by property of cut-equivalent treeQ, and their union∂H(U1)

⋃∂H(U2) is the cut separatingN from V \N . Hence it follows that

c(∂H(N)) ≤ 2 · M4 = M2 . Using this in (E.4) and simplifying, we obtain|N (Q)|M ≤ 2 · |N2(Q)|M ≤ 8Φ∗.

For each edgee ∈ E(Q), let De ⊆ F denote the edges in graphH that go across the two components ofQ \ e. By the property of cut-equivalent treeQ, we havec(De) ≤ M

4 . SinceD =⋃

e∈E(Q) De,

c(D) ≤∑

e∈E(Q)

c(De) ≤ |E(Q)| M4≤ |N (Q)|M

4≤ 2Φ∗

This proves the claim.

CombiningClaims E.4andE.5, we obtain the lemma.

Now we turn our attention to the remaining pairsW := S \ S, and show that there is a cheap cut separating themin H. For this we use a dual-rounding argument, based on Racke’soblivious routing scheme. Recall that constant0 < ǫ < 1

2 , ρ = O(log n) (Racke’s approximation factor), andβ = ρ · 16 log nǫ log log n . Defineα := eρ · logǫ n.

Lemma E.6 There exists afractionalmulticut separating pairsW in the graphH which has cost8α · T ∗.

Proof: For any demand vectord : W → R+, the optimalcongestionof routing d in H, denotedCong(d), isthe smallestη ≥ 0 such that there is a flow routingdi units of flow betweensi andti (for eachi ∈ W ), usingcapacity at mostη · ce on each edgee ∈ H. Note that for everyi ∈ W , thesi-ti min-cut inH has cost at leastL := M

4 = β4 · T ∗

k . Hence for anyi ∈ W , the optimal congestion for aunit demand betweensi-ti (and zerobetween all other pairs) is at most1

L .

Now consider Racke’s oblivious routing scheme [Rac08] as applied to graphH. This routing scheme, for eachi ∈W , prescribes a unit flowFi betweensi-ti such that for every demand vectord : W → R+,

maxe∈H

∑i∈W di · Fi(e)

ce≤ ρ · Cong(d), whereρ = O(log n);

i.e., the congestion achieved by using these oblivious templates to route the demandd is at mostρ times the bestcongestion possible for that particular demandd.

Now consider a maximum multicommodity flow inH that sendsyi · T ∗

k units betweensi, ti (for eachi ∈W ). Fora contradiction, suppose that

∑i∈W yi > 8α · k. (Otherwise the maximum multicommodity flow, and hence its

dual, the minimum fractional multicut is at most8αT ∗, and the lemma holds.) By making copies of vertex-pairs,we may assume thatyi ∈ [0, 1] for all i ∈ W ; this does not change thek-robust multicut instance. Define a (notnecessarily feasible) multicommodity flowG :=

∑i∈W Xi · T ∗

k · Fi, where eachXi is an independent 0-1 randomvariable withPr[Xi = 1] = yi

α , andFi is the Racke oblivious routing template. The flow has expected magnitudeat least

∑i

yi

αT ∗

k ≥ 8T ∗, and is the sum of0, T ∗

k -valued random variables, hence by a Chernoff bound:

Claim E.7 With constant probability, the magnitude of flowG is at leastT ∗.

19

Claim E.8 The flowG is feasible with probability1− o(1).

Proof: Fix any edgee ∈ H, and letui(e) := T ∗

k · Fi(e) for all i ∈ W . Note that the random process givesus a flow of

∑i Xi · ui(e) on the edgee. The feasibility of the maximum multicommodity flow says that

Cong(yi · T ∗

k i∈W ) ≤ 1. Since oblivious routing loses aρ factor in the congestion,

∑i∈W yi · ui(e) =

∑i∈W yi · T ∗

k · Fi(e) ≤ ρ · ce;

and the expected flow on edgee sent by the random process above is∑

i∈Wyi

α · ui(e) ≤ ραce.

Now, since the minsi-ti-cut is at leastL for anyi ∈W , a unit of flow can (non-obliviously) be sent betweensi, ti at congestion at most1L . Hence using the oblivious routing templateFi incurs a congestion at mostρ

L .Hence,

ui(e) = T ∗

k · Fi(e) ≤ T ∗

k ·ρL · ce = 4ρ

β · ce

We divide the individual contributions by the edge capacityand further scale up byβ4ρ by defining new[0, 1]-

random variablesYi = Xi·ui

ce· β

4ρ . We get thatµ := E[∑

Yi] ≤ β4α . Recall the Chernoff bound that says that

for independent[0, 1]-valued random variablesYi,

Pr[∑

Yi ≥ (1 + δ) · µ]≤

(e

1 + δ

)µ(1+δ)

Using this withµ(1 + δ) = β4ρ (henceδ + 1 ≥ α

ρ ) we get that

Pr

[∑

i

Xi · ui(e) ≥ ce

]= Pr

[∑

i

Yi ≥β

4ρ

]≤

(eρ

α

)β/4ρ= exp

(−ǫ log log n · 4 log n

ǫ log log n

)=

1

n4,

sinceα = eρ · logǫ n andβ = ρ · 16 log nǫ log log n . Now a trivial union bound over alln2 edges gives the claim.

By another union bound, it follows that there exists a feasible multicommodity flowG that sends either zero orT ∗

kunits for each pairi ∈ W , and the total value ofG is at leastT ∗. Hence there exists somek-setW ′ ⊆ W suchthat the maximum multicommodity flow forW ′ on H is at leastT ∗. This contradicts the fact that everyk-set hasa multicut of cost less thanT ∗ in H. Thus we must have

∑i∈W yi ≤ 8α · k, which impliesLemma E.6.

CombiningLemmas E.3andE.6, we obtain afractional multicut for pairsS in graphG, having costO(1) · Φ∗ +O(log1+ǫ n) · T ∗. Since the Garg et al. [GVY96] algorithm for multicut is anO(log n)-approximation relative tothe LP, we obtainTheorem E.2.

From Claim E.1andTheorem E.2, it follows that this algorithm isO(log n, log2+ǫ n, β

)-discriminating fork-

robust multicut. Sinceβ = O(log2 n/ log log n), usingLemma 2.2, we obtain an approximation ratio of:

max

log n,

log2 n

log log n+

log2+ǫ n

λ

.

This is anO(

log2 nlog log n

)-approximation whenλ ≥ log2ǫ n. On the other hand, whenλ ≤ log2ǫ n, we can use the

trivial algorithm of buying all edges in the second stage (using the GVY algorithm [GVY96]); this implies an

O(log1+2ǫ n)-approximation. Sincelog1+2ǫ n = o(

log2 nlog log n

), we obtain:

Theorem E.9 There is anO(

log2 nlog log n

)-approximation algorithm fork-robust multicut.

20

The k-max-min Multicut Problem. The above ideas also lead to a(c1 · log n, c2 · log2 n, c3 · log2 n

)strongly

discriminating algorithm for multicut, wherec1, c2, c3 are large enough constants. The algorithm is exactly Algo-rithm 5 with parameterβ := Θ(log n)·ρ with an appropriate constant factor; recall thatρ = O(log n) is the approx-imation ratio for oblivious routing [Rac08]. LemmaE.6shows that this algorithm is

(c1 · log n, c2 · log2 n, c3 · log2 n

)

discriminating (the parameters are only slightly different and the analysis still applies). To establish the propertyinDefinition 2.3, consider the caseλ = 1 (i.e. Φ∗ = 0) andc(ΦT ) ≥ (c2 log2 n) · T . Since the [GVY96] algorithmis O(log n)-approximate relative to the LP, this implies a feasible multicommodity flow on pairsW (sinceΦ∗ = 0we also haveW = S) of value at least(c4 log n) · T for some constantc4. Then the randomized rounding (withoblivious routing) can be used to produce ak-setW ′ ⊆ W and a feasible multicommodity flow onW ′ of valueat leastT ; by weak duality it follows that the minimum multicut onW ′ is at leastT and so Definition2.3 holds.Thus by Lemma2.4we get a randomizedO(log2 n)-approximation algorithm fork-max-min multicut.

All-or-Nothing Multicommodity Flow. As a possible use of the oblivious routing and randomized-roundingbased approach, let us state a result for the all-or-nothingmulticommodity flow problem studied by Chekuri etal. [CKS04]: given a capacitated undirected graphG = (V,E) and source-sink pairssi, ti with demandsdi suchthat the min-cut(si, ti) = Ω(log2 n)di, one can approximate the maximum throughput to within anO(log n) factorwithout violating the edge-capacities, even withdmax ≥ cmin—the results of Chekuri et al. [CKS04, CKS05]violated the edge-capacities in this case by an additivedmax. This capacity violation in the previous all-or-nothingresults is precisely the reason they can not be directly usedin our analysis ofk-robust multicut.

F General Framework for Robust Covering

In this section we present an abstract framework for robust covering problems underany uncertainty setΩ, as longas we are given access to offline, online and max-min algorithms for the base covering problem. Formally, theproperties of interest are described below.

Recall that we deal with the robust version of some covering problemΠ = 〈E, c, Rini=1〉, given by an arbitraryuncertainty setΩ and inflation parameterλ ≥ 1. Given a partial solutionS ⊆ E and a setX ⊆ [n] of requirements,any setEX ⊆ E such thatS ∪EX ∈ Ri ∀i ∈ X is called anaugmentationof S for requirementsX. GivenX,S,define the min-cost augmentation ofS for requirementsX as:

OptAug(X | S) := minc(EX ) | EX ⊆ E andS ∪ EX ∈ Ri, ∀i ∈ X.

An easy consequence of the fact that costs are non-negative is themonotonicityproperty: OptAug(X | S) ≤OptAug(X | T ) for all X ⊆ [n] andT ⊆ S ⊆ E. Also defineOpt(X) := minc(EX) | EX ⊆ E andEX ∈Ri ∀i ∈ X for anyX ⊆ [n].

Property F.1 (Subadditivity) For any two subsets of requirementsX,Y ⊆ [n] and any partial solutionS ⊆ E,we haveOptAug(X | S) + OptAug(Y | S) ≥ OptAug(X ∪ Y | S).

Property F.2 (Offline Algorithm) There is anαoff-approximation (offline) algorithm for the covering problemOptAug(X | S), for anyS ⊆ E andX ⊆ [n].

Property F.3 (Online Algorithm) There is a polynomial-time deterministicαon-competitive algorithm for the on-line version ofΠ = 〈E, c, Rini=1〉.

Property F.4 (Max-Min Algorithm) There is anαmm-approximation algorithm for the max-min problem: giveninput S ⊆ E, MaxMin(S) := maxX∈Ω minc(A) | S ∪A ∈ Ri, ∀i ∈ X.

Theorem F.5 UnderProperties F.1, F.2, F.3 andF.4, there is anO(αoff · αon · αmm)-approximation algorithm forthe robust covering problemRobust(Π) = 〈E, c, Rini=1,Ω, λ〉.

21

Algorithm 6 Algorithm Robust-with-General-Uncertainty-Sets1: input: theRobust(Π) instance and thresholdT .2: let countert← 0, initial online algorithm’s inputσ = 〈〉, initial online solutionF0 ← ∅.3: repeat4: sett← t + 1.5: let Et ⊆ [n] be the scenario returned by the algorithm ofProperty F.4onMaxMin(Ft−1).6: let σ ← σ Et, andFt ← Aon(σ) be the current online solution.7: until c(Ft)− c(Ft−1) ≤ 2αon · T8: setτ ← t− 1.9: output first-stage solutionΦT := Fτ .

10: output second-stage solutionAugmentT where for anyω ⊆ [n], AugmentT (ω) is the solution of the offlinealgorithm (Property F.2) for the problemOptAug(ω | ΦT ).

Proof: The algorithm proceeds as follows. Note that Definition2.1and Lemma2.2hold foranyuncertainty setΩ,although for simplicity it was stated forΩ =

([n]k

).

As always, letΦ∗ ⊆ E denote the optimal first stage solution (and its cost), andT ∗ the optimal second-stage cost;so the optimal value isΦ∗ + λ · T ∗. We prove the performance guarantee using the following claims.

Claim F.6 (General 2nd stage)For anyT ≥ 0 andX ∈ Ω, elementsΦT⋃

AugmentT (X) satisfy all the require-ments inX, andc(AugmentT (X)) ≤ 2αoff · αmm · αon · T .

Proof: It is clear thatΦT⋃

AugmentT (X) satisfy all requirements inX. By the choice of setEτ+1 in line 5of the last iteration, for anyX ∈ Ω we have:

OptAug(X | Fτ ) ≤ αmm ·OptAug(Eτ+1 | Fτ ) ≤ αmm · (c(Fτ+1)− c(Fτ )) ≤ 2αmm · αon · T

The first inequality is byProperty F.4, the second inequality uses the fact thatFτ+1 ⊇ Fτ (since we usean online algorithm to augment inline 6),1 and the last inequality follows from the termination condition inline 7. Finally, sinceAugmentT (X) is anαoff -approximation toOptAug(X | Fτ ), we obtain the claim.

Claim F.7 Opt(∪t≤τEt) ≤ τ · T ∗ + Φ∗.

Proof: Since eachEt ∈ Ω (these are solutions toMaxMin), the bound on the second-stage optimal cost givesOptAug(Et | Φ∗) ≤ T ∗ for all t ≤ τ . By subadditivity (Property F.1) we haveOptAug(∪t≤τEt | Φ∗) ≤τ · T ∗, which immediately implies the claim.

Claim F.8 Opt(∪t≤τEt) ≥ 1αon· c(Fτ ).

Proof: Directly from the competitiveness of the online algorithm in Property F.3.

Claim F.9 (General 1st stage)If T ≥ T ∗ thenc(ΦT ) = c(Fτ ) ≤ 2αon · Φ∗.

Proof: We havec(Fτ ) =∑τ

t=1 [c(Ft)− c(Ft−1)] > 2αonτ · T ≥ 2αonτ · T ∗ by the choice in Step (7).Combined withClaim F.8, we haveOpt(∪t≤τEt) ≥ 2τ ·T ∗. Now usingClaim F.7, we haveτ ·T ∗ ≤ Φ∗, andhenceOpt(∪t≤τEt) ≤ 2 · Φ∗. Finally usingClaim F.8, we obtainc(Fτ ) ≤ 2αon · Φ∗.

Claim F.6andClaim F.9imply that the above algorithm is a(2αon, 0, 2αmmαonαoff) approximation for the robustproblemRobust(Π) = 〈E, c, Rini=1,Ω, λ〉. Now usingLemma 2.2we obtain the theorem.

1This is the technical reason we need an online algorithm. If instead we had used an offline algorithm to computeFt in step6 thenFt 6⊇ Ft−1 and we could not upper bound the augmentation costOptAug(Et | Ft−1) by c(Ft) − c(Ft−1).

22

F.1 Explicit uncertainty sets

An easy consequence ofTheorem F.5is for theexplicit scenariomodel of robust covering problems [DGRS05,GGR06], whereΩ is specified as a list of possible scenarios. In this case, theMaxMin problem can be solved usingthe αoff -approximation algorithm fromProperty F.2which implies anO(α2

offαon)-approximation for the robust

version. In fact, we can do slightly better—observing that the algorithm for second-stage augmentation is the sameas the Max-Min algorithm, we obtain anO(αoff · αon)-approximation algorithm for robust covering with explicitscenarios. As an application of this result, we obtain anO(log n) approximation for robust Steiner forest withexplicit scenarios, which is the best known result for this problem.

F.2 p-Systems and Knapsack uncertainty sets

Notice that the any uncertainty setΩ for a robust covering problem can be assumed WLOG to bedownward-closed,i.e. X ∈ Ω andY ⊆ X impliesY ∈ Ω. Eg., in thek-robust modelΩ = S ⊆ [n] : |S| ≤ k. Hence it is of interestto obtain good approximation algorithms for robust covering whenΩ is specified by means of general models fordownward-closed families. We consider the following well-studied models:

Definition F.10 (p-system) A downward-closed familyΩ ⊆ 2[n] is called ap-system iff:

maxI∈Ω,I⊆A |I|minJ∈Ω,J⊆A |J |

≤ p, for eachA ⊆ [n],

whereΩ ⊆ Ω denotes the collection ofmaximal subsetsin Ω. Sets inΩ are calledindependent sets. We assumeaccess to a membership-oracle, that given any subsetI ⊆ [n] returns whether or notI ∈ Ω.

Definition F.11 (q-knapsack constraints) Givenq non-negative vectorsw1, . . . , wq : [n] → R+ and capacitiesb1, . . . , bq ∈ R+, the knapsack constrained family is:

Ω =

A ⊆ [n] :

∑

e∈A

wj(e) ≤ bj , for all j ∈ [q]

.

Some interesting special cases ofp-systems arep-matroid intersection [Sch03] andp-set packing [HS89, Ber00];see the appendix in [CCPV07] for more discussion onp-systems. Jenkyns [Jen76] studied the problem of maxi-mizing linear functions overp-systems, and showed that the greedy algorithm is ap-approximation. Maximizing alinear function overq-knapsack constraints is the well-studied class of packinginteger programs (PIPs), eg. [Sri99].It is known that a natural greedy algorithm achieves anO(q)-approximation for this problem; when the number ofconstraintsq is constant, there is a PTAS [CK99]. Our main result here is:

Theorem F.12 UnderProperties F.1, F.2andF.3, the robust covering problemRobust(Π) = 〈E, c, Rini=1,Ω, λ〉admits anO ((p + 1) · (q + 1) · αoff · αon)-approximation guarantee whenΩ is given by the intersection of ap-system andq-knapsack constraints.

The proof is deferred to the full version. We note that dependence onp andq is necessary due to complexityconsiderations, even in the special case of maximizing linear functions over such constraints. We now list somespecific results for robust covering under this general model.

Problem Offline ratio Online ratio p-system,q-knapsack RobustSet Cover log n log m · log n [AAA +03] pq · log m · log n

Steiner Tree/Forest 2 [AKR95, GW95] log n [IW91, BC97] pq · log n

Minimum Cut 1 log3 n · log log n [AAA +04, HHR03] pq · log3 n · log log n

Multicut log n [GVY96] log3 n · log log n [AAA +04, HHR03] pq · log4 n · log log n

23