Approximation Algorithms for Prize-Collecting Forest Problems with Submodular Penalty Functions Chaitanya Swamy University of Waterloo Joint work with

Approximation Algorithms for Prize-Collecting Forest Problems with Submodular

Penalty FunctionsChaitanya Swamy

University of Waterloo

Joint work withYogeshwer Sharma David

WilliamsonCornell University

Prize-collecting Steiner tree (PCST)

Given: graph G=(V,E), edge costs ce ≥ 0, root rV,

penalties pv ≥ 0 on vertices

Goal: choose a set of edges F E so as to

minimize ∑eF ce + ∑v not connected to r pvcost of edges picked + penalty of nodes disconnected from r





minimize ∑eF ce + ∑v not connected to r pv

r

cost of edges picked + penalty of nodes disconnected from r





minimize ∑eF ce + ∑v not connected to r pv

Bienstock et al.: gave a 3-approx. LP-rounding algorithm

Goemans-Williamson (GW): gave a primal-dual 2-approx. algorithm

r

cost of edges picked + penalty of nodes disconnected from r

PCST with submodular penalty f’n.


penalty is given by a set-function p : 2V

≥ 0 p(A): penalty if set AV is disconnected

from rp is submodular: p(A)+p(B) ≥ p(A B)

+p(A B)e.g., p(A) = min(|A|, M)

Goal: choose a set of edges F E so as to minimize

∑eF ce + p({v not connected to r})

r

•Generalizes penalty function of PCST

•Introduced by Hayrapetyan-S-Tardos: gave a 2-approximation algorithm by extending GW primal-dual algorithm

Prize-collecting Steiner forest (PCSF)

Given: graph G=(V,E), edge costs ce ≥ 0,source-sink pairs si-ti penalties pi ≥ 0 on each si-ti pair


minimize ∑eF ce + ∑i: si not connected to ti in F pi

Prize-collecting Steiner forest (PCSF)

Given: graph G=(V,E), edge costs ce ≥ 0,source-sink pairs si-ti penalties pi ≥ 0 on each si-ti pair


minimize ∑eF ce + ∑i: si not connected to ti in F pi

•Generalizes connectivity function of PCST

•Introduced by Jain-Hajiaghayi: gave a 3-approx. primal-dual algorithm

General framework for Prize-Collecting Forest

Problems

PCST with submodular

penalty function

Prize-collecting

Steiner forest

Prize-Collecting Forest (PCF) – connectivity function: arbitrary 0-1

function– penalty function: submodular function

on collections of sets of vertices

Prize-collecting Steiner tree

Prize-Collecting Forest (PCF)Given: graph G=(V,E) (|V|=n), edge costs ce ≥

0,•connectivity function f: 2V {0,1}

f(S)=1 need an edge from border of S, (S) := {(u,v)E: exactly one of u, v is

in S}

•penalty function p: 22V ≥ 0 p(S): penalty if collection S of subsets is

violated


minimize ∑eF ce + p({SV: f(S)=1, F(S)=})

Example: Prize-collecting Steiner forestf(S) = 1 iff there exists some i s.t. exactly one of si, ti S

p(S) = ∑ i:SS that separates si-ti pi

violated subsets

PCF: properties of p(.)

• p()=0

• Monotonicity: if ST then p(S) ≤ p(T)

• Submodularity: p(S) + p(T) ≥ p(S T) + p(S T)

• Complement property: for AV, p({A, Ac}) = p({A})

• Union property: for A,B V, p({A, B, A B})=p({A,B})

• Inactivity property: if f(A)=0, then p({A})=0

For any 0-1 connectivity f’n f, can define penalty function,pf(S) = M (very large #) if SS with f(S)=1; and 0 o/w.

Solving PCF with (f, pf) solving network design problem with connectivity f’n. f need certain restrictions on p(.)

If f()=0, then f is 0-1 proper iff pf satisfies above properties.p(.) will be given as an oracle (ground set has 2|V| elements)

Our Results• Give a primal-dual 3-approximation algorithm

– Requires novel ideas in implementation and analysis, to overcome difficulties caused due to the exponential size of the ground set of p(.)

• Give an LP-rounding 2.54-approximation algorithm– solving the LP relaxation poses a significant challenge– LP has 2n constraints and 22n

variables: not clear if even a basic solution has a polynomial description

– Reformulate LP as a convex program, solve via ellipsoid method; evaluating objective f’n and computing a subgradient both require solving an LP of size 2n22n

– overcome difficulty by proving certain structural properties; also required for the rounding procedure

An Integer Program

xe : indicates if edge e is picked

zS : indicates if penalty is incurred for collection S 2V

Minimize ∑e cexe + ∑S p(S)zS

subject to ∑e(S) xe + ∑S:SS zS ≥ f(S) for each SV

xe, zS {0,1}for each e, S

A Linear Program

xe : indicates if edge e is picked

zS : indicates if penalty is incurred for collection S 2V

Minimize ∑e cexe + ∑S p(S)zS (PCF-LP)

subject to ∑e(S) xe + ∑S:SS zS ≥ f(S) for each SV

xe, zS {0,1}for each e, S

xe, zS ≥ 0 for each e, S

• LP has 22n variables and 2n constraints

•Not clear if even a basic solution has a polynomial-size description – what does “solving the LP” mean?

A Compact Formulationxe : indicates if edge e is picked

zS : indicates if penalty is incurred for collection S 2VMinimize h(x) := ∑e cexe + g(x) s.t. 0

≤ xe ≤ 1 for each e

(PCF-CP)

where, g(x):=min ∑S p(S)zS

(Pen-P)

s.t. ∑S:SS zS≥ f(S) – ∑e(S) xe for each SV

zS ≥ 0 for each e, S

g(x) is convex, so (PCF-CP) is a convex programEquivalent to earlier LP.

The Overall Strategy

1. Get an optimal (or (1+)-optimal solution) x to the convex program using the ellipsoid method.

2. Round fractional solution x to integer solution– need that f is 0-1 proper f’n, or is weakly-submodular– use 2-approx. algorithm for the network-design problem

without penalties (Goemans-Williamson or Jain).

Obtain a 2.54-approximation algorithm for the prize-collecting forest problem.

The Ellipsoid MethodStart with ball containing polytope P.yi = center of current ellipsoid.

Min h(x) subject to xP.

P

The Ellipsoid Method

P

New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.


If yi is infeasible, use violated inequality to chop off infeasible half-ellipsoid.

Start with ball containing polytope P.yi = center of current ellipsoid.


New ellipsoid = min. volume ellipsoid containing “unchopped” half-ellipsoid.

P


If yi is infeasible, use violated inequality to chop off infeasible half-ellipsoid.


If yi P – how to make progress?



P

Start with ball containing polytope P.yi = center of current ellipsoid.If yi is infeasible, use violated inequality.If yi P – how to make progress?

add inequality h(x) ≤ h(yi)? Separation becomes difficult.yi

h(x) ≤ h(yi)

Let d = subgradient at yi.

use subgradient cut d.(x–yi) ≤ 0.Generate new min. volume

ellipsoid.



P



d m is a subgradient of h(.) at u, if for every v, h(v)-

h(u) ≥ d.(v-u).

add inequality h(x) ≤ h(yi)? Separation becomes difficult.

If yi is infeasible, use violated inequality.

d

yi

h(x) ≤ h(yi)



P



d m is a subgradient of h(.) at u, if for every v, h(v)-

h(u) ≥ d.(v-u).

Let d = subgradient at yi.

use subgradient cut d.(x–yi) ≤ 0.Generate new min. volume

ellipsoid.

x1, x2, …, xk: points in P. Can show, mini=1…k h(xi) ≤ OPT+.

x*

x1

x2

add inequality h(x) ≤ h(yi)? Separation becomes difficult.

If yi is infeasible, use violated inequality.

Computing a subgradienth(x) := ∑e cexe + g(x)

g(x):=min. ∑S p(S)zS

s.t. ∑S:SS zS ≥ f(S) – ∑e(S) xe

SV

zS ≥ 0 S

Computing a subgradienth(x) := ∑e cexe + g(x)

g(x):=min. ∑S p(S)zS = max. ∑S (f(S) – ∑e(S)

xe) yS

s.t. ∑S:SS zS ≥ f(S) – ∑e(S) xe s.t. ∑SS yS ≤ p(S) S

SV

zS ≥ 0 S yS ≥ 0 SConsider point um. Let y optimal dual solution to g(u).So h(u) = ∑e ceue + ∑S (f(S) – ∑e(S) ue) yS = ∑e deue + ∑S

f(S)yS

where de = ce – ∑S:e(S) yS.

At any point vm, y is a feasible solution to dual of g(v), so

h(v) ≥ ∑e ceve + ∑S (f(S) – ∑e(S) ve) yS = ∑e deve + ∑S

f(S)yS

Lemma: For any point vm, we have h(v) – h(u) ≥ d.

(v-u). d is a subgradient of h(.) at point u.

Solving the dual

g(x) =max ∑S [f(S) – x((S))]yS (Pen-D)

s.t. ∑SS yS ≤ p(S) for all S2V

yS ≥ 0 for all S

Bad : Dual has 2n variables and 22n constraints

Good : It is a polymatroid: p(.) is a monotone submodular f’n. Edmonds’ greedy algorithm yields optimal solution

– Sort the sets S in decreasing order of [f(S)-x((S))]

– For the i-th set Si, if [f(Si)-x((Si))] > 0, set ySi =

p{S1,…Si-1}(Si)

Bad : Reduces complexity to 2n, but still not polytimeGood : Show that optimal solution where the sets S with yS > 0 form a laminar family – key structural lemma

Notation: x((S))= ∑e(S) xe

pS(A) = p(S{A}) – p(S)

Useful properties of p(.)

• If A, BS, then pS(T) = pS(Tc) = 0 for all sets T in {AB, AB, A\B, B\A, Ac, Bc} – due to complementarity and union properties

• If p({A}) = 0, then for any BV, pS{A}({B}) = pS({B}) – due to submodularity ordering of sets A with f(A)=0 is irrelevant

• If pS{A}({B}) = pS{B}({A}) = 0, then for any set TV, pS{A}({T}) = pS{B}({T}) – by submodularity

Solving the dual (contd.)

• Initialize yS = 0 for all sets S, laminar family L .

• While set S that does not cross any set of L– find T = argmin {x((S)): S does not cross L}

– if x((T)) ≥ 1 return; else set yT = pL({T}), L L{T}

Theorem: y is an optimal solution to (Pen-D).

LetL' = {TL: yT>0} = {T1,…,Tk},

Ti = maximal superset of {T1,…,Ti} s.t. p(Ti) = p({T1,…,Ti})

Theorem: Setting zTi = x((Ti+1)) – x((Ti)) (x((Tk+1)) :=

1) for i=1,…,k, and zS = 0 for all other S, yields an optimal solution to (Pen-P).

Structural lemma yields following algorithm:

Rounding procedure

Given: fractional solution x, sets T1,…, Tk – gives succinct description of

collections T1,…,Tk, and hence optimal soln. z to (Pen-P)

Let [0,1] be a parameter. – Define 0-1 connectivity function(S) = 1 if f(S) = 1 and ∑S:SS zS < ; 0

otherwise.– Solve network design problem with connectivity

function .

If f is proper or weakly-supermodular, then so is , therefore

cost of edges picked is boundedPenalty is at most p({S V: ∑S:SS zS ≥ }) ≤ [∑S

p(S)zS]/

Open Questions

• Is there a compact description of the LP? Or a more efficient procedure to solve it?

•Obtaining a 2-approximation algorithm: iterative rounding may be the way to go

•Applications to 2-stage stochastic network design: can the second-stage cost be captured by a “nice” penalty function?

•Extensions to higher connectivity reqmts.

Thank You.

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Approximation Algorithms for Prize-Collecting Forest Problems with Submodular Penalty Functions Chaitanya Swamy University of Waterloo Joint work with