On the approximability of clique and related maximization problems

http://www.elsevier.com/locate/jcss

Journal of Computer and System Sciences 67 (2003) 633–651

On the approximability of clique and relatedmaximization problems$

Aravind Srinivasan1

Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland,

A.V. Williams Building, College Park, MD 20742, USA

Received 26 March 2001; revised 17 October 2002

Abstract

We consider approximations of the form n1�oð1Þ for the Maximum Clique problem, where n is the numberof vertices in the input graph and where the ‘‘oð1Þ’’ term goes to zero as n increases. We show thatsufficiently strong negative results for such problems, which we call strong inapproximability results, haveinteresting consequences for exact computation. A simple sampling method underlies most of our results.r 2003 Published by Elsevier Inc.

Keywords: Inapproximability; Approximation algorithms; Clique; Independent set; Packing integer programs;

Random sampling

1. Introduction

By presenting a connection between interactive proof systems and approximation algorithms,the seminal work of [8] has led to several recent results on the hardness of approximating variousoptimization problems. These involve ingenious reductions from decision problems to suitableapproximation problems; see [2,3] and their several interesting follow-up results. Here, we take theapproach of reducing approximation to optimization/approximation: crucially, these reductionslead to instances that are much smaller than the original instance. In particular, this lets us show

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$A preliminary version of this work appears as ‘‘The value of strong inapproximability results for clique’’ in the

Proceedings of the ACM Symposium on Theory of Computing, 2000, pp. 144–152.

E-mail address: [email protected].

URL: http://www.cs.umd.edu/~srin.1Most of this work was done while at Bell Laboratories, Lucent Technologies; part of this work was supported by

NSF Award CCR-0208005.

0022-0000/03/$ - see front matter r 2003 Published by Elsevier Inc.

doi:10.1016/S0022-0000(03)00110-7

some interesting implications of sufficiently good hardness-of-approximation results for theMaximum Clique problem.The Maximum Clique problem is a classical problem in combinatorial optimization. Given an

undirected graph G; a clique in G is a subset of the vertices of G in which every vertex is adjacentto every other vertex. The Maximum Clique problem (henceforth referred to just as ‘‘clique’’) isthat of finding a clique of maximum cardinality in a given graph, and was one of the early onesshown to be NP-hard. Given a parameter rX1; recall that a r-approximation algorithm for amaximization problem is one that always returns a feasible solution that is at least 1=r timesoptimal. Such an algorithm is also said to approximate the problem to within r; and r is called theapproximation guarantee or approximation ratio of the algorithm. The current-best approximation

ratio for clique that is achievable in polynomial time is Oðn=log2 nÞ [5], where n denotes thenumber of vertices in the input graph. Recent years have seen much progress in understanding theinapproximability of the problem, culminating in the celebrated result of [12] that it cannot be

approximated in ZPP to within n1�e for any fixed e40; unless NPDZPP; see [16] for a simplifiedproof. Arguably, the Lovasz W-function may yield the ‘‘best’’ approximation guarantee for clique.It has been shown that the worst-case approximation guarantee of this function is at least as high

as n=2Oðffiffiffiffiffiffiffiffilog n

pÞ [7]. The reader is referred to [10] for a survey of the (in)approximability of clique.

It would be of much interest to understand the worst-case polynomial-time approximability ofclique. In particular, it is mentioned in [10] that ‘‘It is not as presumptuous now to conjecture thatthe ultimate ratio is n=polylogðnÞ as it was in 1991 y’’. It would be very interesting to resolve thisconjecture.Before proceeding to discuss our results, we now introduce some notation.

Notation. Zþ stands for the set of non-negative integers; log x denotes log2 x: Throughout, we letN denote the input size of an instance of an NP problem; for all of Section 1, n will denote thenumber of vertices in an input graph, and ‘‘oð1Þ’’ will denote some function of n that goes to zeroas n increases. DTIME; BPTIME and ZPTIME; respectively, denote deterministic time,bounded-error probabilistic time, and zero-error probabilistic time as usual. [Though these classesare defined for decision problems, we will also use them to describe (optimization) algorithms inthe obvious way.] For randomized algorithms, we will consider Las Vegas algorithms in somecontexts, and Monte Carlo algorithms in others. However, ‘‘randomized complexity of NP’’ willthroughout mean the worst-case expected running time of randomized algorithms that alwaysreturn the correct answer, for NP languages: i.e., Las Vegas algorithms. More specifically, in everyusage of the phrase ‘‘randomized complexity of NP’’, we will either say that the randomizedcomplexity of NP is OðT1ðNÞÞ; or that it is OðT2ðNÞÞ: The former would mean that every NPlanguage has a Las Vegas algorithm with worst-case expected running time being OðT1ðNÞÞ; thelatter means that there exists an NP language for which the worst-case expected running time ofevery Las Vegas algorithm is OðT2ðNÞÞ: Similar remarks hold when we speak of the deterministiccomplexity of NP:As seen in the above few paragraphs, the ‘‘best-possible’’ polynomial-time approximation ratio

for clique seems to be in the n1�oð1Þ range. One of the main themes of our work is showing why the

(in-)approximability of clique in the n1�oð1Þ range would be very useful to understand. Let us call aresult such as ‘‘Clique is hard to approximate to within n=f ðnÞ’’, where f ðnÞ is a function that

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grows as noð1Þ; a strong inapproximability result for clique. Strong inapproximability results forclique are already on their way; using different notation than in [6], one key result of [6] is:

Theorem 1.1 (Engebretsen and Holmerin [6]). Let EHA be the complexity-theoretic assumption

‘‘NPD/ ZPTIME½2Oðlog Nðlog log NÞ3=2Þ�’’; define EHða; xÞ ¼ xa=ffiffiffiffiffiffiffiffiffiffiffiffiffilog log x

p: Then, assuming EHA; there

is a constant g40 such that clique cannot be approximated to within n=EHðg; nÞ by any ZPPalgorithm.

Following this work, an even stronger inapproximability result for clique has been obtained[13]. The consequences of this result (and of any future strong inapproximability result) thatfollow from our work can be derived in a manner identical to how we derive some consequencesof Theorem 1.1; this issue is sketched in Section 6.Our approach reduces certain families of approximation problems to optimization or

approximation problems of much smaller size, using elementary (random) sampling. As a simplestarting point, suppose we want to approximate clique to within some factor r in an n-vertexgraph G: We can assume that the maximum clique size, say opt; in G is at least r (since otherwiseany single vertex will be a r-approximation). Fix a maximum clique C: Suppose, for someparameter 1ptpr; we choose a set S of nt=r vertices from G at random. Note that the expectednumber of vertices of C in S is opt t=r; if indeed S has this many vertices of C; then a t-approximation for the clique problem on the subgraph induced by S; will be a r-approximation ofclique in G; as desired. In other words, we have reduced r-approximation for a problem of size n

to t-approximation for a problem of size nt=r: Various ways of viewing and building on this idealead to the following applications.(a) A sufficient condition for a gap in the location of NP: A step toward further understanding the

complexity of NP would be to show the following, under no assumptions: for some explicitfunctions f1 and f2 such that limN-N f1ðNÞ=f2ðNÞ ¼ 0; NP’s (deterministic or randomized)complexity is either Oð f1ðNÞÞ or Oð f2ðNÞÞ: Our approach helps show that sufficiently good stronginapproximability results for clique lead to such results. Suppose a strong inapproximability resultis shown for clique, under some hardness assumption A: We show that any such result yieldsexplicit super-polynomial lower bounds on NP; under the assumption A: For instance, let f andF be any functions satisfying

limn-N

f ðnÞ ¼ N and f ðnÞ ¼ noð1Þ; ð1Þ

FðzÞ ¼ minfyAZþ : 3ð f ðyÞÞ2Xzg: ð2Þ

An example result of ours is as follows. Suppose clique cannot be approximated to within n=f ðnÞ in

ZPP assuming A holds; then, assuming A; the randomized complexity of NP is OððFðNÞÞoð1ÞÞ;‘‘oð1Þ’’ refers to some function of N that goes to infinity as N increases. Similar analogs hold for

the deterministic complexity of NP: Note that F is super-polynomial since f ðnÞ ¼ noð1Þ: For

instance, if f ðnÞ ¼ 2Yððlog nÞaÞ for some constant 0oao1; then FðNÞ ¼ 2Yððlog NÞ1=aÞ; if f ðnÞ ¼ðlog nÞc; then FðNÞ ¼ 2YðN1=cÞ: An interesting case here is when A � ðNPD/ ZPPÞ: a proof (whichis likely to be very challenging to obtain) that clique cannot be approximated to within n=f ðnÞ in

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ZPP assuming NPD/ ZPP; will provably show that NP’s randomized complexity is either

polynomially bounded or is at least the explicit super-polynomial function ðFðNÞÞoð1Þ: Ouritalicized result above shows that any strong inapproximability result for clique has a partialconverse. For instance, our partial converse to Theorem 1.1 is: if clique cannot be approximated to

within n=EHðg; nÞ by any ZPP algorithm, then the randomized complexity of NP is at least

Noðffiffiffiffiffiffiffiffiffiffiffiffiffiffilog log N

pÞ:

(b) An approach to P vs. NP: Our results also show the following. Let the randomized unit-costRAM (random-access machine) be the standard unit-cost RAM with the additional power todraw any number of unbiased, independent random bits, each bit in unit time. We start with adefinition:

Definition 1.1. Let f be any function satisfying (1); let e be any positive constant. Consider theproblem of approximating clique to within n=f ðnÞ; using the notation ½X ;Y ;Z� for ‘‘lower boundof X for input graph represented in format Y ; in computational model Z’’ for this problem, wedefine four types of lower-bound assertions:

(LB1) denotes [n2þe; adjacency list, deterministic unit-cost RAM];

(LB2) denotes [n1þe; adjacency matrix, deterministic unit-cost RAM];

(LB3) denotes [n1þe; adjacency list, randomized unit-cost RAM]; and(LB4) denotes [ne; adjacency matrix, randomized unit-cost RAM].

The lower bounds (LB3) and (LB4) should be on the running time of any Monte Carlo algorithmthat produces an n=f ðnÞ-approximation with probability at least 1=2:

Fix any f and e that satisfy the hypotheses of Definition 1.1. We show that (LB1) impliesNPaP; (LB2) implies NPaP; (LB3) implies NPD/ BPP; and (LB4) implies NPD/ BPP: In fact,

given f ; let F be any function satisfying (2); we get super-polynomial lower bounds of ðFðNÞÞOð1Þon NP in all these cases. (Very small lower bounds such as in (LB4) may seem obvious: e.g., one

may think that it could take nOð1Þ time to even write out an n=f ðnÞ-approximate solution. This isnot the case. For instance, if the graph has a clique number of at least f ðnÞ; an n=f ðnÞ-approximation algorithm just needs to output some clique of size f ðnÞ ¼ noð1Þ; if the cliquenumber is less than f ðnÞ; the algorithm can just output a one-vertex ‘‘clique’’.)These seem potentially fruitful for two reasons. First, another result of [6] is that for a

certain constant l40; clique cannot be approximated to within n=EHðl; nÞ by anyMonte Carlo polynomial-time algorithm with probability at least 1=2; unless NPD

BPTIME½2Oðlog Nðlog log NÞ3=2Þ�: Thus, lower bounds such as ne and n2þe seem likely to hold forthis approximation problem. To us, the second interesting point is that these required lowerbounds are quite small; e.g., ne is sublinear. A simple argument in Section 4.2 shows that the

n=EHðl; nÞ-inapproximability result of [6] holds even when restricted to graph families with Yðn2Þedges; n1þe is also sublinear in these cases. These reasons suggest that this may be a worthwhilepath to explore for goals such as ‘‘NPD/ BPP’’ and ‘‘NPaP’’ in some restricted models ofcomputation.(c) Bootstrapping hardness results. Our approach leads to a bootstrapping of hardness-of-

approximation results. This yields higher lower bounds on the running time of clique

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approximation algorithms, by appropriately lessening the approximation ratio. Applying this toTheorem 1.1, we get, under the assumption EHA; that: (i) for any constants n4g and 0obo1=2;

clique cannot be approximated to within n1�n=ðlog log nÞb

in ZPTIME½nOððlog log nÞ1=2�bÞ�; (ii) for anyfixed e40; clique cannot be approximated to within n1�e in ZPTIME½nOð

ffiffiffiffiffiffiffiffiffiffiffiffiffilog log n

pÞ�: Thus, EHA

yields explicit super-polynomial lower bounds on the running time of even approximating clique

(say, to within n1�e). We remark that our bootstrapping only requires hardness results such asTheorem 1.1 as a black-box (i.e., we do not require the internal details of these proofs).(d) From hardness to almost-everywhere hardness. Our method (or, more accurately, reduction)

also leads to a way of strengthening hardness-of-approximation assumptions/results to certain‘‘almost-everywhere hardness’’ versions. Section 4.4 demonstrates an instance of this on then=EHðl; nÞ-hardness result of [6] defined in (b) above.(e) Time-approximation trade-offs. For an abstract family of maximization problems that

includes clique, hypergraph matching, etc., an earlier work of Halldorsson [11] presentsapproximation algorithms that achieve a certain trade-off between approximation ratio andrunning time. Our approach yields similar results. This trade-off is also shown to be essentiallyoptimal for general problems belonging to this family.After setting up some preliminary notions in Section 2, we present the basic approach in Section

3. Various ways of using and viewing this approach lead to the hardness results shown in Section4. Trade-offs between approximability and a measure of running time (the ‘‘query complexity’’)are studied in Section 5. Finally, concluding remarks are made in Section 6.

2. Preliminaries

Large-deviation bounds. We recall a large-deviation bound for a relative of the hypergeometricdistribution. Let ½s� denote the set f1; 2;y; sg: Suppose that we are given positive integers r; s withrps; and reals c1;c2;y;csA½0; 1�: Choose a random r-element subset T of ½s�; each r-elementsubset being equally likely. Consider any AD½s�; if X denotes

PiAðA-TÞ ci; then E½X � ¼

ðr=sÞ P

iAA ci: Tail bounds from, e.g., [17] show for any dA½0; 1� that

Pr½Xpð1� dÞE½X ��pe�E½X �d2=2; ð3Þ

here and from now on, e denotes the base of the natural logarithm.A class of optimization problems. In presenting our approximation algorithms, it will help to

consider the following generalization of the clique and maximum independent set (MIS) problemon graphs. This class also generalizes the weighted versions of these problems (e.g., given a non-negative weight for each vertex, weighted maximum clique asks for a clique of maximum total

weight). Given ~xx ¼ ðx1; x2;y; xnÞAf0; 1gn and ~yy ¼ ðy1; y2;y; ynÞAf0; 1gn; we will say that ~yy%~xxiff yipxi for each i: Call a Boolean function g : f0; 1gn-f0; 1g monotone decreasing iff: for every~xxAf0; 1gn such that gð~xxÞ ¼ 1; and for all ~yy%~xx; gð~yyÞ ¼ 1: (Such functions are sometimes alsocalled hereditary properties.) Let Rþ denote the set of non-negative reals. The abstract class ofmaximization problems we consider consists of maximization problems called ðn; g; ~wwÞ-OPT,

where g is a monotone decreasing Boolean function mapping f0; 1gn to f0; 1g; and ~wwARnþ: The

ðn; g; ~wwÞ-OPT problem is to find some ~xxAf0; 1gn that maximizes ~ww ~xx; subject to gð~xxÞ ¼ 1: We are

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also assured that gð0; 0;y; 0Þ ¼ 1; which is equivalent to saying that there exists some ~xx for

which gð~xxÞ ¼ 1: We assume oracle access to g: given any ~xxAf0; 1gn; an oracle for g gives us thevalue of gð~xxÞ:In the case of weighted clique, given a vector~vv that is the incidence vector of a subset S of the

vertices, we define gð~vvÞ ¼ 1 iff S is a clique. It is immediate that this g is polynomial-timecomputable, and that ðn; g; ~wwÞ-OPT here is the weighted Maximum Clique problem. Similarly, it iseasy to see how this framework captures the weighted MIS problem. Section 5 will also discussapplications to certain types of packing integer programs.

A family of ‘‘noð1Þ’’ functions. Henceforth, a ‘‘oð1Þ’’ term will always be associated either with N

(typically, the size of an input instance for an NP language) or n (typically, associated with anðn; g; ~wwÞ-OPT problem). The oð1Þ term will be some value that goes to 0 as either N or n tends toinfinity; there will be no ambiguity here, as there will be no usage of ‘‘oð1Þ’’ in terms where both N

and n occur. We will define a family of functions, all of which are of the form ‘‘noð1Þ’’; we callthese desirable functions. The two required properties of a desirable function f ðnÞ are that

limn-N f ðnÞ ¼ N and that f ðnÞ ¼ noð1Þ; i.e., limn-N ðlog f ðnÞÞ=ðlog nÞ ¼ 0: (In particular, werequire that these two limits exist; however, these requirements can be weakened. Since thistechnicality is not important in our applications, we do not discuss how to weaken these

requirements.) Functions such as ðlog nÞc; 2Yððlog nÞaÞ or nYððlog log nÞ�cÞ; where c40 and aAð0; 1Þ areconstants, are desirable in this sense.

3. A simple sampling-based reduction

We now present an elementary sampling approach that motivates most of our results. Thediscussion in this section is defined by four parameters: a positive integer n; an arbitrary constantk40; and reals 1þ kprpn and 1ptpr=ð1þ kÞ: It may be helpful to keep the followinginterpretations of these parameters in mind. The parameter n will in general be a measure of theinput size of a given problem: e.g., we could be considering the maximum clique problem on n-vertex graphs. The parameter r will typically be our targeted approximation guarantee. Note thattpr; we will often be interested in the case where t5r: The goal of the discussion below is toreduce the given r-approximation problem on instances of size n to a t-approximation problem oninstances of size Yðnt=rÞ: In particular, note that if t5r; then we may have achieved a significantsize-reduction. Finally, k will typically be a constant in the range ð0; 1Þ:Suppose we wish to approximate a given instance of ðn; g; ~wwÞ-OPT to within r: We show a

simple way of reducing this to optimization/approximation on certain ‘‘much smaller’’ instancesof the problem. We first define what ‘‘much smaller’’ means here. Given SD½n� and ~vv ¼ðv1; v2;y; vnÞ; let~vvS be the vector ðv01; v02;y; v0nÞ; where v0i ¼ vi if iAS and vi ¼ 0 if ieS: Now, ifjSj is ‘‘small’’ compared to n; we informally consider ðn; g; ~wwSÞ-OPT to be a ‘‘much smaller’’problem than the original ðn; g; ~wwÞ-OPT. This is because the indices ieS have effectively beeneliminated from consideration: for any ~xx with gð~xxÞ ¼ 1; we have gð~xxSÞ ¼ 1 and ~wwS ~xx ¼ ~wwS ~xxS:So, we can restrict attention to those ðx1;x2;y;xnÞAf0; 1gn for which xi ¼ 0 for all ieS:When specialized to concrete problems such as maximum clique, we will get interestingconsequences.

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Suppose we are given an instance of ðn; g; ~wwÞ-OPT which we wish to approximate within r:Letting ~ww ¼ ðw1;w2;y;wnÞ; we will assume throughout by scaling that maxi wi ¼ 1: Given a bitvector~vv; its Hamming weight, denoted Hweight here, is the number of ‘‘1’’ bits in it. (Our usage ofthe term ‘‘Hweight’’ is to distinguish this usage of ‘‘weight’’ from usage such as ‘‘weightedclique’’.) Let ei be the n-bit vector of Hweight 1 with a ‘‘1’’ in precisely the ith coordinate. IfgðeiÞ ¼ 0; the ith coordinate is essentially irrelevant for the problem, since any ~xx for whichgð~xxÞ ¼ 1; will have a ‘‘0’’ in the ith coordinate (as g is monotone decreasing). So, letting I denotethe set of indices i for which gðeiÞ ¼ 1; our problem reduces to ðn; g; ~wwIÞ-OPT. To avoid extranotation, we will assume without loss of generality that I ¼ ½n�:Our reduction is given by

Theorem 3.1. Let k be some positive constant. Suppose we are given an instance of ðn; g; ~wwÞ-OPT, aswell as two reals r and t such that 1þ kprpn and 1ptpr=ð1þ kÞ: We assume that gðeiÞ ¼ 1 for

all i; and that ~ww ¼ ðw1;w2;y;wnÞ; where maxi wi ¼ 1; let i0 be an index such that wi0 ¼ 1: Let ~uu be

an optimal solution for the given ðn; g; ~wwÞ-OPT instance. Then:(i) maxfJntð1þ kÞ=rn;Jn=Ir=tmngpminf2ntð1þ kÞ=r; ng:(ii) Define C ¼ J2ð1þ kÞ lnð10Þ=ðtk2Þn: Independently choose C subsets S1;S2;y;SC of ½n�;

each from the uniform distribution over subsets of size Jntð1þ kÞ=rn: Then:

* If the optimal solution value for the given problem is at least r; then for each i;

Pr½~wwSi~uuoopt t=r�pe�tk2=ð2ð1þkÞÞ: ð4Þ

* Let xi be a t-approximation for ðn; g; ~wwSiÞ-OPT for iX1: Then, any vector among

fei0 ; x1;x2;y;xCg that maximizes ~ww xi; is a r-approximation for ðn; g; ~wwÞ-OPT, with

probability at least 0.9; this probability is only over the random choice of the Si:

(iii) Define c ¼ Ir=tm; and partition ½n� arbitrarily into c subsets A1;A2;y;Ac;each of cardinality at most Jn=cn: Let yi be a t-approximation for ðn; g; ~wwAi

Þ-OPT for iX1:Then, any vector among fei0 ; y1; y2;y; ycg that maximizes ~ww yi; is a r-approximation for ðn; g; ~wwÞ-OPT.

Proof. Part (i) is a simple consequence of the facts that 1þ kprpn and 1ptpr=ð1þ kÞ:Let opt denote the optimal objective function value for the given ðn; g; ~wwÞ-OPT instance. If

optpr; then it is easy to see that ei0 would provide the required r-approximation; this will thencomplete the proofs of parts (ii) and (iii). Thus, we are left with the (interesting) case whereopt4r: Recall that~uu is an optimal solution for the given ðn; g; ~wwÞ-OPT instance: so, gð~uuÞ ¼ 1 and~ww ~uu ¼ opt:(ii) For each i; we have gð~uuSi

Þ ¼ 1 with probability 1, since g is monotone decreasing. Simple useof (3) along with the fact that opt4r; shows (4). Thus, the probability that ‘‘~wwSi

~uuoopt t=r’’held for all i is at most

e�tCk2=ð2ð1þkÞÞpe�lnð10Þ ¼ 0:1:

Now, if ~wwSi~uuXopt t=r; then a t-approximation for ðn; g; ~wwSi

Þ-OPT is also a r-approximationfor the given ðn; g; ~wwÞ-OPT problem, completing the proof of part (ii).

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(iii) It is immediate that there is some i for which ~wwAi~uuXopt=cXopt t=r: Hence, if we can

t-approximate all of the ðn; g; ~wwAiÞ-OPT, the best among the c solutions produced will be a

r-approximation for the given ðn; g; ~wwÞ-OPT instance. &

Thus, part (ii) is a simple randomized reduction which, with high probability, reduces theoriginal r-approximation problem to t-approximation problems on a constant number ofinstances ðn; g; ~wwS1

Þ-OPT, ðn; g; ~wwS2Þ-OPT, y; ðn; g; ~wwSC

Þ-OPT, where jS1j ¼ jS2j ¼ ? ¼ jSC j ¼Jntð1þ kÞ=rn: (Note that C is bounded by the constant J2ð1þ kÞ lnð10Þ=k2n:) In other words, ifwe can correctly t-approximate all of the constant number of randomly generated ðn; g; ~wwSi

Þ-OPT,we get a r-approximation for the original ðn; g; ~wwÞ-OPT instance with high probability. Part (iii) isa slightly slower deterministic version of this reduction. Also note from part (i) that the ‘‘sizes’’ ofthe size-reduced instances in our randomized as well as deterministic reductions, are at most2ntð1þ kÞ=r: Finally, if we plug t ¼ 1 into Theorem 3.1, we get a reduction from approximationto exact optimization.Specialized to the (unweighted) Maximum Clique problem, where we desire a r-approximation

for n-vertex graphs, we get the following. Part (ii) of Theorem 3.1 basically shows a randomizedreduction to a constant number of instances of t-approximation for graphs with at most 2ntð1þkÞ=r vertices; part (iii) presents a deterministic reduction to Ir=tm instances of t-approximationfor graphs with at most 2ntð1þ kÞ=r vertices. (Furthermore, the new graphs produced areinduced subgraphs of the given graph.)

4. Hardness results

We now move on to various negative results that follow from Theorem 3.1. Suppose f ðnÞ isan arbitrary desirable function; let A be some complexity-theoretic assumption (such as‘‘NPD/ ZPP’’). We will present consequences of results of the following type:(R1) ‘‘If A holds, then clique cannot be approximated to within n=f ðnÞ; in

DTIME½polyðnÞ�’’.(R2) ‘‘If A holds, then clique cannot be approximated to within n=f ðnÞ; in ZPTIME½polyðnÞ�’’.In both (R1) and (R2), n denotes the number of vertices in the input graph.Note that an unweighted clique problem on a graph with v vertices can be expressed with ‘‘input

size’’ at most v2=2: e.g., by writing down v and, for each of the v2

� �possible edges, a bit denoting

whether the edge is in the graph or not. For all of Section 4, any use of Theorem 3.1 will set

k ¼ffiffiffiffiffiffiffi1:5

p� 1: Thus, whenever we specialize this reduction to the unweighted clique problem on

an original graph that has n vertices, the size-reduced instances (which have at most 2ntð1þ kÞ=rvertices) can be expressed with input size at most ð2ntð1þ kÞ=rÞ2=2 ¼ 3ðnt=rÞ2:The results of this section will hold for any ðn; g; ~wwÞ-OPT problem, but we use the clique

problem for concreteness.

4.1. A sufficient condition for a gap in the location of NP

We now consider application (a) presented in Section 1.

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Theorem 4.1. Let f be an arbitrary desirable function; define hðzÞ6minfyAZþ : 3ð f ðyÞÞ2Xzg:Then:(i) h is super-polynomial; i.e., limz-N ðlog hðzÞÞ=log z ¼ N:(ii) Suppose (R1) is true for f and for some assumption A: Then under assumption A; the

deterministic complexity of NP is OððhðNÞÞsðNÞÞ; for some function sðNÞ such that

lim supN-NsðNÞ ¼ N:

(iii) Suppose (R2) is true for f and for some assumption A: Then, under assumption A;

the randomized complexity of NP is OððhðNÞÞsðNÞÞ; for some function sðNÞ such that

lim supN-NsðNÞ ¼ N:

Proof. (i) This easily follows from the fact that f ðnÞ ¼ noð1Þ:(ii) Suppose for a contradiction that A and (R1) are true, but that the deterministic complexity

of NP is at most ðhðNÞÞOð1Þ: Set r ¼ n=f ðnÞ; t ¼ 1; and k ¼ffiffiffiffiffiffiffi1:5

p� 1: Then, Theorem 3.1(iii)

shows that approximating clique (on an n-vertex graph) to within n=f ðnÞ reduces to Oðn=f ðnÞÞmany instances of clique, each instance having at most 2

ffiffiffiffiffiffiffi1:5

pf ðnÞ vertices; thus, each of these

instances can be expressed in polyðnÞ time with input size at most I ¼ 3ð f ðnÞÞ2: So, if the

deterministic complexity of NP is ðhðNÞÞOð1Þ; each of these ‘‘small’’ maximum clique instances can

be solved in time ðhðIÞÞOð1Þ ¼ nOð1Þ; this would show that we can approximate clique to withinn=f ðnÞ in DTIME½polyðnÞ�; contradicting (R1). This proves (ii); the proof of (iii) is essentiallyidentical. &

Some of the most interesting applications of Theorem 4.1 are when A is ‘‘NPD/ ZPP’’or ‘‘NPaP’’. For instance, suppose A is ‘‘NPD/ ZPP’’, and that result (R2) is shown.Then, as seen above, we will provably get a gap in the location of the randomized complexityof NP: either bounded by polyðNÞ; or being at least the explicit super-polynomial function

ðhðNÞÞoð1Þ:In addition to Theorem 1.1, there are known partial results toward goals such as (R1) and (R2).

For instance, Theorem 9.6 of [4] is as follows: ‘‘Suppose there is a (randomized) quasipolynomial-

time algorithm A for Independent Set with performance guarantee N=2ffiffiffiffiffiffiffiffiffiffiffic log N

pon N-vertex

graphs, then there is a randomized quasipolynomial-time algorithm B to color anyn-vertex k-colorable graph with OðneÞ colors, where e ¼ ð20 log kÞ=c’’. Thus, good lowerbounds for approximate graph coloring will lead to progress on showing results like (R1)and (R2).

4.2. An approach to NP vs. BPP and to NP vs. P

We now consider application (b) of Section 1.

Theorem 4.2. Let f be an arbitrary desirable function, and e be any positive constant. Consider thelower-bound assertions (LB1), (LB2), (LB3) and (LB4) of Definition 1.1 for this pair ð f ; eÞ: We have

the following implications: (LB1) implies NPaP; (LB2) implies NPaP; (LB3) implies NPD/ BPP;and (LB4) implies NPD/ BPP:

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Proof. We start by showing that (LB3) implies NPD/ BPP: In fact, letting h be as in Theorem 4.1and letting d be any positive constant smaller than e; we show that (LB3) implies

NPD/ BPTIME½ðhðNÞÞ1þd�;

recall that h is a super-polynomial function. The main observation is that the reduction ofTheorem 3.1(ii) runs very fast. Suppose we specialize the optimization problem of Section 3 to theproblem of approximating clique in an n-vertex graph G to within n=f ðnÞ: In the notation

of Theorem 3.1, let r ¼ n=f ðnÞ; t ¼ 1; and k ¼ffiffiffiffiffiffiffi1:5

p� 1: We randomly choose a set S of

Jntð1þ kÞ=rn vertices of G without replacement, and construct the subgraph G½S� of G that isinduced by S: This subgraph construction takes at most

T1ðnÞ ¼ Oðn1þoð1Þf ðnÞÞ ¼ n1þoð1Þ

time, for the following reason. Note that jSj ¼ Yð f ðnÞÞ ¼ noð1Þ: Generating the set S takes noð1Þ

time. Next, for each of the Yð f ðnÞÞ vertices in S; we need at most n1þoð1Þ time to examine itsadjacency list, and to extract out its neighbors which lie in S:

Now suppose NPDBPTIME½ðhðNÞÞ1þd�; for some positive constant doe; define e0 ¼ ðdþ eÞ=2:Then, by the standard reduction from the optimization version of clique to the decision version,there is a Monte Carlo algorithm for the (optimization version of ) clique, which runs within time,

say, ðhðNÞÞ1þe0 and succeeds with probability at least 1=2: Thus, since G½S� has input size at most

I ¼ 3ð f ðnÞÞ2; there is a Monte Carlo algorithm to solve the Maximum Clique problem on G½S�;with running time at most T2ðnÞ ¼ ðhðIÞÞ1þe0 ¼ Oðn1þe0 Þ and success probability at least 1=2: So,(4) shows that there is an algorithm with running time at most T1ðnÞ þ T2ðnÞ ¼ OðT2ðnÞÞ; whichcan approximate clique on G to within n=f ðnÞ with probability at least ð1� e�k2=ð2ð1þkÞÞÞ � ð1=2Þ:This constant success probability can be boosted to 1=2 by repeating the algorithm a constantnumber of times. Thus, we will have a Monte Carlo n=f ðnÞ-approximation algorithm for clique

with running time Oðn1þe0 Þ; which would contradict (LB3).Next, here is a sketch of how negative result ‘‘(LB4)’’ of Section 1 will imply NPD/ BPP:We will

largely follow the proof that (LB3) implies NPD/ BPP: if the input graph is expressed in adjacencymatrix form, we can implement the algorithm in that proof even faster, as follows. Instead of

defining S to be a random set of Jntð1þ kÞ=rn ¼ Jffiffiffiffiffiffiffi1:5

pf ðnÞn vertices of G chosen without

replacement, we choose these vertices independently and uniformly at random (i.e., with

replacement). Since f ðnÞ ¼ oðffiffiffin

pÞ; the probability that all these vertices are distinct is

YJffiffiffiffiffi1:5

pf ðnÞn�1

i¼1

ð1� i=nÞ ¼YJffiffiffiffiffi1:5

pf ðnÞn�1

i¼1

e�Yði=nÞ ¼ e�Yðð f ðnÞÞ2=nÞ ¼ 1� oð1Þ;

so, the analysis behind our above proof that (LB3) implies NPD/ BPP; is essentially unchanged.Moreover, since the elements of S are chosen with replacement, this random choice of S can be

done in noð1Þ time since jSj ¼ noð1Þ: Also, since G is expressed in adjacency matrix format, we can

construct G½S� in Oðð f ðnÞÞ2Þ ¼ noð1Þ time. We now use the above proof that (LB3) impliesNPD/ BPP; to conclude that (LB4) implies NPD/ BPP: In contrapositive form, we get thatNPDBPP implies very fast clique approximations: e.g., for all d40 there exist e40 and n0 such

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that for all graphs with nXn0 vertices input in adjacency matrix form, clique can be approximated

to within n1�e in nd time, by a Monte Carlo algorithm with probability at least 1=2 on therandomized unit-cost RAM.Proving the claimed consequences of (LB1) and (LB2) is very similar, and we merely point out

how the above proofs need to be modified. We use part (iii) of Theorem 3.1, instead of part (ii).Also, in the notation of Theorem 3.1(iii), we define each subset Ai to have consecutive-numbered

elements of ½n�: If G is input in adjacency list format, all the G½Ai� can be constructed in n2þoð1Þ time;

if G is expressed in adjacency matrix format, all the G½Ai� can be constructed in n1þoð1Þ time. &

Thus, we see some complexity-theoretic consequences of certain (quite low) polynomial lowerbounds for clique approximation. As mentioned in Section 1, work of [6] shows that for a

certain constant l40; clique cannot be approximated to within n=EHðl; nÞ ¼ n1�l=ffiffiffiffiffiffiffiffiffiffiffiffiffilog log n

p

by any Monte Carlo polynomial-time algorithm with probability at least 1=2; unless

NPDBPTIME½2Oðlog Nðlog log NÞ3=2Þ�: Thus, a lower bound as small as n1þe seems likely to hold for

this problem. Also, this result of [6] holds even when restricted to graph families with, e.g., Yðn2Þedges. [This can be seen as follows. We may assume that the input n-vertex graph G is non-empty;so its clique number is at least 2. Construct a new graph G0 by adding an n=2þ n=2 complete

bipartite graph as a new connected component to G: G0 has 2n vertices and Yðn2Þ edges; the cliquenumbers of G and G0 are the same. Thus, any polynomial-time n=EHðl; nÞ-approximation of cliquefor G0; is also such an approximation for G:] This may be an approach worth exploring for the Pvs. NP and NP vs. BPP questions, in some restricted models of computation.Also, recall that the clique problem on a graph G is identical to the MIS problem on G’s

complement. We observe that Theorem 4.2 holds if we replace clique by MIS in the assertions

(LB1)–(LB4). This observation is not interesting for the case where we seek an n2þe lower bound

for an n=f ðnÞ-approximation, since we can always construct a graph’s complement in Oðn2Þ time.

However, this observation may be useful for the issue of proving an n1þe or ne lower bound.

4.3. Bootstrapping hardness results

Recall the sample hardness results (R1) and (R2) defined in the beginning of Section 4. We nowshow how results such as (R1) and (R2) can be ‘‘bootstrapped’’ to yield new hardness results thathave a smaller approximation guarantee and stronger lower bounds on their running times.

Theorem 4.1 basically followed by using the reduction of Section 3 with k ¼ffiffiffiffiffiffiffi1:5

p� 1 and

t ¼ 1: The proof approach of Theorem 4.1, using Theorem 3.1 with k ¼ffiffiffiffiffiffiffi1:5

p� 1; r ¼ n=f ðnÞ;

and an arbitrary t ¼ tðnÞ; leads to

Theorem 4.3. Let f ðÞ be an arbitrary desirable function. Suppose n0; tðÞ and FðÞ are

such that for all nXn0: (a) 1ptðnÞpn=ðffiffiffi6

pf ðnÞÞ; and (b) Fð

ffiffiffi6

ptðnÞf ðnÞÞptðnÞ: Define

DðzÞ6minfyAZþ :ffiffiffi6

ptðyÞf ðyÞXzg: Then:

(i) Suppose (R1) is true for f and for some assumption A: Then, under assumption A; clique

cannot be approximated in v-vertex graphs to within FðvÞ; in DTIME½ðDðvÞÞOð1Þ�:

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(ii) Suppose (R2) is true for f and for some assumption A: Then, under assumption A; clique

cannot be approximated in v-vertex graphs to within FðvÞ; in ZPTIME½ðDðvÞÞOð1Þ�:

Proof. (i) Suppose (R1) and A hold, and that clique can be approximated in v-vertex graphs to

within FðvÞ; in DTIME½ðDðvÞÞOð1Þ�: Now suppose we are given an n-vertex graph G for which wewant to approximate clique to within n=f ðnÞ: We use Theorem 3.1(iii) to construct

Oðn=ð f ðnÞtðnÞÞÞ graphs, each with at mostffiffiffi6

ptðnÞf ðnÞ vertices; a tðnÞ-approximation—in

particular, a Fðffiffiffi6

ptðnÞf ðnÞÞ-approximation—of clique applied to each of these graphs, yields an

ðn=f ðnÞÞ-approximation for clique in G: Let v ¼ffiffiffi6

ptðnÞf ðnÞ; and note that DðvÞpn: Thus, if

clique can be approximated in v-vertex graphs to within FðvÞ in DTIME½ðDðvÞÞOð1Þ� (i.e.,in time polyðnÞ), then we can get an ðn=f ðnÞÞ-approximation for clique in G in polyðnÞ time,violating (R1).The proof of (ii) is identical, except that we now need to use Theorem 3.1(ii). &

Corollary 1. In the notation of Theorem 1.1, the assumption EHA implies that: (a) Clique cannot

be approximated to within n1�e in ZPTIME½nOðffiffiffiffiffiffiffiffiffiffiffiffiffilog log n

pÞ�; for any fixed e40; and (b) Let n4g and

0obo1=2 be any constants. Then, clique cannot be approximated to within n1�n=ðlog log nÞb

in

ZPTIME½nOððlog log nÞ1=2�bÞ�:

Proof. Recall Theorem 1.1; we just need to combine Theorem 4.3 (setting f ðnÞ ¼ EHðg; nÞ) andTheorem 1.1 appropriately. Part (a) follows by using tðnÞ ¼ ð

ffiffiffi6

pEHðg; nÞÞð1�eÞ=e; FðxÞ ¼ x1�e; and

DðzÞ of the form zYðffiffiffiffiffiffiffiffiffiffiffiffiffilog log z

pÞ: For part (b), we choose tðnÞ ¼ nc=ðlog log nÞ1=2�b

; FðxÞ ¼ x1�n=ðlog log xÞb

;

and DðzÞ of the form zYððlog log zÞ1=2�bÞ; where c4g=n is an arbitrary constant. Let uðnÞ ¼ffiffiffi6

pEHðg; nÞtðnÞ; we need to show that FðuðnÞÞptðnÞ; i.e., that logðFðuðnÞÞÞplogðtðnÞÞ: Since

log logðuðnÞÞplog log n and since

logðuðnÞÞ ¼ 1þ gþ oð1Þcðlog log nÞb

!logðtðnÞÞ;

it suffices to show that

1� n

ðlog log nÞb

!1þ gþ oð1Þ

cðlog log nÞb

!p1;

which holds for all large enough n since c4g=n: &

4.4. From hardness to almost-everywhere hardness

Our approach can also be used to show certain ‘‘hardness on average’’-type results. We remarkthat ‘‘hardness on average’’ here does not necessarily mean that most instances are hard, but thatthere exist instances on which most ‘‘sub-instances’’ are hard. Suppose f ðnÞ is an arbitrary

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desirable function, and that A is some complexity-theoretic assumption. We will construct‘‘almost-everywhere hardness’’ results from results of the form(R3) ‘‘If A holds, then clique cannot be approximated to within n=f ðnÞ by any Monte Carlo

polynomial-time algorithm with success probability at least 1=2’’.We prove the following theorem:

Theorem 4.4. Suppose that (R3) holds for some desirable function f and some complexity-theoreticassumption A: Let WðnÞ be any desirable integer-valued function such that WðnÞXaf ðnÞ ln n for

some fixed a40: Define YðzÞ6minfyAZþ : WðyÞXzg; Y is super-polynomial since W is desirable.Then, there is no Monte Carlo algorithm M for the clique problem such that:

(i) given any v-vertex graph H for arbitrary v; M runs in time ðYðvÞÞOð1Þ; and

(ii) for each n-vertex graph G for arbitrary n; and for at least an n�a=4 fraction of the induced

subgraphs H of G that have WðnÞ vertices, M; when given H as input, finds a maximum clique in Hwith probability at least 1=2:

Proof. We specialize Theorem 3.1(ii) to the unweighted Maximum Clique problem. Suppose (R3)holds, but that there is indeed an algorithm M satisfying items (i) and (ii) in the statement of thetheorem. Then, given any n-vertex graph G; we now present a Monte Carlo polyðnÞ-time thatcomputes an n=f ðnÞ-approximation to the maximum clique of G with probability at least 1=2;contradicting (R3). This will prove the theorem.Let r ¼ n=f ðnÞ; let opt be the clique number of G; and let T denote an arbitrary

maximum clique of G: As in our discussion in the proof of Theorem 3.1, we can assume thatopt4r without loss of generality. Choose a set Z of WðnÞ vertices of G at random. Let X be thenumber of chosen vertices of T ; since optXn=f ðnÞ and WðnÞXaf ðnÞ ln n; (3) shows for all largeenough n that

Pr½Xoopt=r�pn�a=3: ð5Þ

Consider the subgraph G½Z� of G induced by Z: By item (ii) in the statement of Theorem 4.4, there

is at least an n�a=4 chance that M works correctly on G½Z� with probability at least 1=2: Thus, by(5), if we run M on G½Z�; we will compute a clique in G of size at least opt=r with probability at

least ð1=2Þðn�a=4 � n�a=3ÞBn�a=4=2: Further, by item (i) in the statement of Theorem 4.4, this will

only take ðYðWðnÞÞÞOð1Þ ¼ nOð1Þ time. So, by repeating this basic algorithm a polynomial numberof times, we can find a clique in G of size opt=r in G with probability at least 1=2: &

Thus, a hypothesis such as (R3) shows the following. Suppose we desire an algorithm thatcorrectly finds the maximum clique on even a certain negligible fraction of some inducedsubgraphs, for every input graph. Then, any such algorithm needs time at least an explicitsuper-polynomial function of its input size. As mentioned in Section 1, (R3) is shown in [6]with f ðnÞ ¼ EHðl; nÞ for some positive constant l; with A being the assumption

‘‘NPD/ BPTIME½2Oðlog Nðlog log NÞ3=2Þ�’’. We can plug these choices into Theorem 4.4 to see suchhardness-almost-everywhere results (e.g., if WðnÞ ¼ polyð f ðnÞÞ; then the super-polynomial lower

bound on the running time here is ðYðnÞÞoð1Þ ¼ noðffiffiffiffiffiffiffiffiffiffiffiffiffilog log n

pÞ).

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Furthermore, in the notation of Theorem 3.1, we have implicitly taken the approximationfactor t to be 1 above. (That is, item (ii) in the statement of Theorem 4.4 asks for an exactcomputation—not an approximation—of a maximum clique in H:) By choosing suitable othervalues for t; we can see that hypotheses such as (R3) imply that even approximating cliqueappropriately is ‘‘hard almost everywhere’’. We get the following result:

Theorem 4.5. Suppose that (R3) holds for some desirable function f and some complexity-theoreticassumption A: Suppose a and n0 are positive constants, and tðÞ and W ðÞ are functions such that for

all nXn0:

* 1ptðnÞpn=ðaf ðnÞ ln nÞ; and* WðnÞ is integer-valued and lies in the range ½af ðnÞtðnÞ ln n; n�:Define YðzÞ6minfyAZþ : WðyÞXzg: Then, there is no Monte Carlo algorithm M for the cliqueproblem such that:

(i) given any v-vertex graph H for arbitrary v; M runs in time ðYðvÞÞOð1Þ; and

(ii) for each n-vertex graph G for arbitrary n; and for at least an n�a=4 fraction of the inducedsubgraphs H of G that have WðnÞ vertices, M; when given H as input, approximates clique in H towithin tðnÞ with probability at least 1=2:

Proof (Sketch). The proof is very close to the proofs of Theorems 4.4 and 4.3; we merely give asketch here. Suppose we desire to approximate clique in a given n-vertex graph G; to withinn=f ðnÞ: We can assume that opt; the maximum clique size in G; is at least n=f ðnÞ: Choose a set Z

of WðnÞ vertices of G at random; as in the proof of Theorem 4.4, we can show that the maximum

clique size in G½Z� is at least opt tðnÞf ðnÞ=n; with probability at least 1� n�a=3: If the clique size inG½Z� is indeed this large, then we will be done if we can approximate clique in G½Z� to within tðnÞ:The rest of the proof is as for Theorem 4.4. &

5. Time-approximation trade-offs

A concrete measure of running time when working with a general ðn; g; ~wwÞ-OPT problem is thequery complexity: the number of queries to the oracle for g that we need to make, for a givenproblem. An earlier work of Halldorsson [11] shows an approach that yields a certain querycomplexity for approximating ðn; g; ~wwÞ-OPT problems to within a given bound; Theorem 5.1yields the same result (and uses the same type of algorithm). For completeness, we give the shortproof of Theorem 5.1. This query complexity is then shown to be essentially best-possible, byTheorem 5.2.

Let us call an ðn; g; ~wwÞ-OPT problem unweighted, if ~ww is the n-bit vector~11 of all ones. (To avoid

extra notation, we will always assume ~11 to be an n-bit vector.)We start by recalling a known fact:

Fact 2. Suppose 0papbpc are integers. Then: (a)Pa

i¼0bi

� �pðbe=aÞa: (b) c�a

b�a

� �= c

b

� �pðb=cÞa:

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Proof. (a)Pa

i¼0bi

� �pPa

i¼0 ðb=aÞaða=bÞi bi

� �pðb=aÞa

Pbi¼0

bi

� �ða=bÞi¼ðb=aÞa ð1þ a=bÞbpðbe=aÞa:

(b) c�ab�a

� �= c

b

� �¼ ðQa�1

i¼0 ðb � iÞ=ðc � iÞÞpðb=cÞa: &

Theorem 5.1. (i) General ðn; g; ~wwÞ-OPT problems can be r-approximated deterministically using

2Oðlog nþn=rÞ queries to the oracle for g; for any rA½1; n�; in 2Oðlog nþn=rÞ time.

(ii) Suppose we are given, for an arbitrary ðn; g;~11Þ-OPT problem, that the optimal solution value is

at most c: Then for any rA½1; n�; we can deterministically find a r-approximation making

2Oðlog nþJc=rnlogð2n=cÞÞ queries to the oracle for g; and in 2Oðlog nþJc=rnlogð2n=cÞÞ time.

Proof. We start with the reduction of Theorem 3.1(iii), for both parts (i) and (ii). For both parts,

we do this reduction with k ¼ffiffiffiffiffiffiffi1:5

p� 1; say, and t ¼ 1; so, we basically reduce the problem to

Irm smaller instances, each of size at most M6Jn=Irmn: We now describe how to proceedfurther for parts (i) and (ii).

An optimal solution to an ðn; g; ~wwAiÞ-OPT instance can trivially be found using 2jAij queries to

the oracle for g; in 2Oðlog nþjAijÞ time. The claim of part (i) follows from this. For part (ii), havingdone the reduction to problems of size M; we only need to find a solution of value at most Jc=rn;simple enumeration thus yields a total query complexity of at most

O rXJc=rni¼0

M

i

� !¼ 2Oðlog nþJc=rnlogð2n=cÞÞ:

The time complexity follows similarly. &

Remark. We use the notation ‘‘logð2n=cÞ’’ instead of ‘‘logðn=cÞ’’ in Theorem 5.1(ii), to handle thecase where c ¼ nð1� oð1ÞÞ: In this case, logðn=cÞ ¼ oð1Þ; while we really want a term that is Oð1Þ:If n=c ¼ 1þ Oð1Þ; then logð2n=cÞ and logðn=cÞ are within a multiplicative constant of each other.

Also, for Theorem 5.1(ii), an algorithm usingPJc=rn

i¼0ni

� �¼ 2Oðlogð2n=Jc=rnÞJc=rnÞ queries is

obvious: just enumerate all n-bit vectors of Hweight at most Jc=rn: Theorem 5.1(ii) shows thatwe can develop algorithms that are faster than this obvious one.

Also, given the recent interest in massive data sets and, e.g., clique problems in very largegraphs [1], we remark that our observations of Section 4.2 imply certain types of very fast clique-approximation results. For instance, those observations show that if a graph is expressed inadjacency matrix format, then we can probabilistically approximate clique to within n=ðe log nÞ innce time, where c is an absolute constant and e is any given positive constant.Another interesting family of problems captured by the ðn; g; ~wwÞ-OPT model are the packing

integer programs with all variables constrained to lie in f0; 1g: given a system of linear inequalities

A~xxp~bb where all entries of A and ~bb are non-negative, we wish to find some ~xxAf0; 1gn which

maximizes ~ww ~xx subject to A~xxp~bb: This generalizes, e.g., the weighted MIS, (multiple) knapsack,and hypergraph matching problems. Letting m denote the number of constraints in the system

‘‘A~xxp~bb’’, the worst-case approximation guarantee for arbitrary packing integer programs is

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about Oðffiffiffiffim

pÞ [15,18]. Note that this is weak if m is large: even if, say, mXOðn2Þ: Theorem 5.1(i)

shows that we start getting benefits if m is significantly larger than n: Consider families of packing

integer programs where, e.g., m ¼ 2Oðn3=4Þ: For such families, we get r ¼ Oðn1=4Þ-approximation

algorithms that run in time polynomial in the length of the input (which is Oðm þ nÞ).Next, Theorem 5.2 shows that the query complexities of both parts of Theorem 5.1 are optimal

up to the constant factor in the exponent, if g is an arbitrary monotone decreasing functionavailable through its oracle. (To see why Theorem 5.2 implies that part (i) of Theorem 5.1 isessentially optimal, we can substitute, e.g., c ¼ n=2 in Theorem 5.2.)

Theorem 5.2. Let n; c; rX1 be arbitrary; define c0 ¼ Iminfc; n=2gm: Consider the class of ðn; g;~11Þ-OPT problems where we are also given that the optimal solution value is at most c: Any randomized

algorithm that produces a r-approximation with at least a positive constant probability for this class

of problems, should make Oðn þ ðn=c0ÞJc0=rnÞ queries to g’s oracle.

Proof. Suppose the desired success probability of the randomized algorithm is some constant

a40: By Yao’s theorem [19], it will suffice to present a randomized construction of an ðn; g;~11Þ-OPT problem, for which any deterministic algorithm needs to make Oðn þ ðn=c0ÞJc

0=rnÞ queries tohave a success probability of at least a: (This probability is only w.r.t. the random choice of theproblem.)

Suppose nXðn=c0ÞJc0=rn: Choose a random vector ~vv of Hweight 1; and define gð~vvÞ ¼

gð0; 0;y; 0Þ ¼ 1; gð~wwÞ ¼ 0 for all other ~ww: In this case, it is immediate that for any rX1; anydeterministic algorithm needs at least OðnÞ oracle queries to achieve a r-approximation withprobability a:We move on to showing an Oððn=c0ÞJc

0=rnÞ lower bound on the query complexity (in

the case where noðn=c0ÞJc0=rn): the random construction of a Boolean function g that we

now use is as follows. Let a ¼ Jc0=rn: We choose an n-bit vector ~rr of Hweight c0 at random.All n-bit vectors~ss such that: (i)~ss%~rr; or (ii) the Hweight of~ss is at most a � 1; have gð~ssÞ ¼ 1; allother ~ss have gð~ssÞ ¼ 0: It is easy to check that this random g is monotone decreasing. Recalling

that we have an ðn; g;~11Þ-OPT problem, we see that the optimal solution value is c0pc; with ~rrbeing the unique optimal solution; a r-approximation must thus produce some ~ss%~rr whoseHweight is at least a:Fix any deterministic r-approximation algorithm DA in this setting; we start with some

observations about the queries made by DA: Clearly, DA need not query the value of gð~ssÞ forany~ss of Hweight at most a � 1; since gð~ssÞ is known to be 1 in this case. We also argue that if DAqueries the value of gð~ssÞ for some~ss of Hweight more than a at some point in its execution, it can

do no worse by choosing an arbitrary~tt%~ss of Hweight exactly a; and querying the value of gð~ttÞ inplace of gð~ssÞ: To see this, note that if gð~ssÞ ¼ 1; then querying gð~ssÞ or gð~ttÞ will make DA

successfully terminate after this query. On the other hand, if gð~ssÞ ¼ 0; then by querying gð~ttÞ; DA

can only get more information. (If gð~ttÞ ¼ 1; DA will halt successfully; if gð~ttÞ ¼ 0; DA can inferthat gð~ssÞ ¼ 0:) So we may assume without loss of generality that DA only queries the value ofgð~ssÞ for vectors ~ss of Hweight precisely a:

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For a vector~ss of Hweight a; define Sð~ssÞ to be the set of vectors~tt of Hweight c0 such that~ss%~tt:Also define U to be the set of all n-bit vectors of Hweight exactly c0: Let Qi denote the ith n-bitvector whose gðÞ value is queried by DA; and let Ei be the event that gðQiÞ ¼ 0: Qi is a randomvariable, since the choice of g is random. Fix any iX0; and any sequence of vectors~qq1;~qq2;y;~qqiþ1;each of Hweight a: We argue that

Pr ~rreSð~qqiþ1Þi

j¼1

ððQj ¼ ~qqjÞ4EjÞ

" #¼

jU �Siþ1

j¼1 Sð~qqjÞjjU �

Sij¼1 Sð~qqjÞj

:

This is because, conditional on ‘‘Vi

j¼1 ððQj ¼ ~qqjÞ4EjÞ’’, ~rr is uniformly distributed in the set

U �Si

j¼1 Sð~qqjÞ: We have jSð~ttÞj ¼ z16 n�ac0�a

� �for each vector ~tt of Hweight a; thus, defining

z26 nc0

� �; we get

Pr ~rreSð~qqiþ1Þi

j¼1

ððQj ¼ ~qqjÞ4EjÞ

" #X 1� z1

jU �Si

j¼1 Sð~qqjÞj

X 1� z1

z2 � iz1

¼ z2 � ði þ 1Þz1z2 � iz1

:

Since this is true for all choices of ~qq1;~qq2;y;~qqiþ1; we get

Pr Eiþ1

i

j¼1

Ej

" #

Xz2 � ði þ 1Þz1

z2 � iz1;

Bayes’ theorem and a telescoping product yield

Pr½E14E24?4Ei�X1� iz1=z2:

Now, z1=z2pðc0=nÞJc0=rnÞ by Fact 2(b). Hence, forDA to have at least a positive constant success

probability, the number of queries should be at least Oðz2=z1Þ ¼ Oððn=c0ÞJc0=rnÞ: &

6. Conclusions

We have shown new benefits of mapping out the terrain of ‘‘n1�oð1Þ approximability’’ for theMaximum Clique or related maximization problems. Our approach makes elementary use of(random) sampling, and basically exploits the property that suitably chosen samples will haveapproximately the expected number of elements from some fixed optimal solution. However,especially for instances that have a ‘‘large’’ number of (near-)optimal solutions, we may be able todo better: with some reasonable probability, our sample may get many more elements thanexpected from some optimal solution. Could this idea be useful in some settings? For instance,consider the random graph model Gðn; 1=2Þ; where we take n labeled vertices and put an edgebetween every pair of distinct vertices, independently with probability 1=2: It is well-known thatwith high probability, the maximum clique size here is ð2� oð1ÞÞ log n: Thus, a Monte Carlo

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2 log n-approximation for clique is trivial for this model: just output a one-vertex clique.However, what if we want a Las Vegas r-approximation, i.e., an algorithm that alwaysproduces a r-approximation, and has polynomial expected running time? (This expectation istaken over the random choice of the graph, as well as the internal coin flips of the

algorithm if it is randomized.) In this case, the current-best value of r is n1=2�oð1Þ [14].Note that in Gðn; 1=2Þ; we expect a large number of near-optimal solutions: for instance, the

expected number of ðlog nÞ-sized cliques is nYðlog nÞ: It would be interesting to see if theabove remark on the sample holding many more elements than expected, would be applicable insuch settings.Our results show that: (i) if clique is NP-hard (under polynomial-time reductions)

to approximate within n=polylogðnÞ; then either NPDBPP or NP requires almostexponential time; and (ii) if clique is unconditionally shown to be inapproximable towithin n=polylogðnÞ; then NP requires almost-exponential time. On the contrapositive,evidence of the hardness of proving such results on NP; will show the hardness ofproving sufficiently good strong inapproximability results for clique. We have also shown anapproach to issues such as NP vs. P: that certain very low-degree polynomial lower bounds forsome clique/MIS approximation problems (which seem likely to need super-polynomial time)suffice to show results such as NPD/ BPP and NPaP: Can the recent improvements/proofmethods in time–space trade-offs (see, e.g., [9]) be put to use in this context for some restrictedmodels of computation?An interesting direction would be to strengthen Theorem 1.1. Recall that our results show the

following: ‘‘If, for some constant c40; clique cannot be approximated to within n=EHðc; nÞ byany ZPP algorithm, then the randomized complexity of NP is Noð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffilog log N

pÞ’’. Thus, if the

hypothesis EHA of Theorem 1.1 can be weakened to

EHA0 � ðNPD/ ZPTIME½NOðffiffiffiffiffiffiffiffiffiffiffiffiffiffilog log N

pÞ�Þ;

then we will have that EHA0 is equivalent to the existence of a constant c40 such that cliquecannot be approximated to within n=EHðc; nÞ by any ZPP algorithm. Such equivalences, if trueand provable, would be of much interest. A similar point can be made for any stronginapproximability result for clique: it is easy to observe that we can treat any stronginapproximability result for clique in a manner similar to the way we treat/utilize Theorem 1.1in this work. For instance, it has recently been shown in [13] that for a certain constant g40;

clique cannot be approximated to within n=2ðlog nÞ1�gunless NPDZPTIME½2ðlog NÞOð1Þ �: The partial

converse for this result that follows from our work is: if clique cannot be approximated to within

n=2ðlog nÞ1�g; then the randomized complexity of NP is at least Nðlog NÞOð1Þ :

Finally, although we have presented a near-optimal trade-off between approximation ratio andnumber of queries for ðn; g; ~wwÞ-OPT problems in general, it will be interesting to develop bettertrade-offs for specific problems such as Maximum Clique. One possible approach in this context isto do a ‘‘size reduction’’ via sampling, and to then apply some problem-specific method on thesize-reduced instance. Another direction in this context is to study interesting subfamilies of themonotone decreasing functions, and to prove trade-offs between approximation ratio and querycomplexity for such families.

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Acknowledgments

I thank Lars Engebretsen and Jonas Holmerin for clarifying issues about their results. Mythanks also to Avrim Blum, Johan Hastad, Seffi Naor, D. Sivakumar, Francis Zane and DavidZuckerman for helpful discussions. Finally, I thank the three referees for their detailed and helpfulsuggestions.

References

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On the approximability of clique and related maximization problems