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ETH Hardness for Densest-K-Subgraph with Perfect Completeness

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ETH Hardness for Densest-K-Subgraph with Perfect Completeness ETH Hardness for Densest-k-Subgraph with Perfect Completeness Mark Braverman ∗ Young Kun Ko y Aviad Rubinstein z Omri Weinstein x November 1, 2016 Abstract Problem 1. (Approximate Max Clique, Informal) We show that, assuming the (deterministic) Exponential Given an n-vertex graph G, decide whether G con- Time Hypothesis, distinguishing between a graph with tains a clique of size k, or all induced cliques of G an induced k-clique and a graph in which all k-subgraphs are of size at most δk for some 1 > δ(n) > 0. have density at most 1−", requires nΩ(log~ n) time. Our re- sult essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this The second natural relaxation is to relax the \Clique" problem, and is the first one to rule out an additive PTAS requirement, replacing it with the more modest goal for Densest k-Subgraph. We further strengthen this re- of finding a subgraph that is almost a clique: sult by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by ~ Problem 2. (Densest k-Subgraph with per- a near-polynomial factor (k0 = k · 2−Ω(log n)) are assumed to be at most (1 − ")-dense. fect completeness, Informal) Our reduction is inspired by recent applications of the Given an n-vertex graph G containing a clique of \birthday repetition" technique [AIM14, BKW15]. Our size k, find an induced subgraphs of G of size k with analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of two- (edge) density at least (1 − "), for some 1 > " > 0. prover games in which the provers may choose to answer (More modestly, given an n-vertex graph G, decide some challenges multiple times, while completely ignoring whether G contains a clique of size k, or all induced other challenges. k-subgraphs of G have density at most (1 − ")). 1 Introduction Today, after a long line of research [FGL+96, k-Clique is one of the most fundamental problems AS98, ALM+98, H˚as99,Kho01, Zuc07] we have a in computer science: given a graph, decide whether it solid understanding of the inapproximability of Prob- has a fully connected induced subgraph on k vertices. lem 1. In particular, we know that it is NP-hard to Since it was proven NP-complete by Karp [Kar72], distinguish between a graph that has a clique of size extensive research has investigated the complexity of k, and a graph whose largest induced clique is of size relaxed versions of this problem. at most k0 = δk for δ = 1=n1−" [Zuc07]. The compu- This work focuses on two natural relaxations of tational complexity of the second relaxation (Prob- k-Clique which have received significant attention lem 2) remained largely open. There are a couple from both algorithmic and complexity communities: of (very different) quasi-polynomial algorithms that The first one is to relax \k", i.e. looking for a smaller guarantee finding a (1 − ")-dense k subgraph in every subgraph: graph containing a k-clique [FS97, Bar15], suggest- ing that this problem is not NP-hard. Yet we know neither polynomial-time algorithms, nor general im- ∗Department of Computer Science, Princeton University, email: [email protected]. Research supported in possibility results for this problem. part by an NSF CAREER award (CCF-1149888), a Turing In this work we provide a strong evidence that the Centenary Fellowship, a Packard Fellowship in Science and aforementioned quasi-polynomial time algorithms for Engineering, and the Simons Collaboration on Algorithms and Problem 2 [FS97, Bar15] are essentially tight, assum- Geometry. ing the (deterministic) Exponential Time Hypothesis yDepartment of Computer Science, Princeton University, email: [email protected] (ETH), which postulates that any deterministic algo- Ω(n) zDepartment of Electrical Engineering and Computer rithm for 3SAT requires 2 time [IP01]. In fact, we Sciences, University of California at Berkeley, email: show that under ETH, both parameters of the above [email protected]. This work was supported in part by relaxations are simultaneously hard to approximate: NSF grant CCF1408635 and by Templeton Foundation grant 3966. This work was done in part at the Simons Institute for the Theory of Computing. Theorem 1.1. (Main Result) There exists a uni- xDepartment of Computer Science, Courant Institute versal constant " > 0 such that, assuming the (de- (NYU), email: [email protected] terministic) Exponential Time Hypothesis, distin- guishing between the following requires time nΩ(log~ n), even sparser. In contrast, our result has perfect com- where n is the number of vertices of G. pleteness and provides the first additive inapproxima- bility for Densest k-Subgraph | the best one can Completeness G has an induced k-clique; and hope for as per the upper bound of [Bar15]. Soundness Every induced subgraph of G size k0 = −Ω( log n ) Planted Clique The Planted Clique problem k · 2 log log n has density at most 1 − ", is a special case of our problem, where the inputs Our result has implications for two major open come from a specific distribution (G (n; p) versus problems whose computational complexity remained G (n; p) + \a planted clique of size k", where p is 1 elusive for more than two decades: The (general) some constant ). The Planted Clique Conjecture + Densest k-Subgraph problem, and the Planted ([AAK 07, AKS98, Jer92, Kuc95, FK00, DGGP10]) Clique problem. asserts that distinguishing betweenp the aforemen- tioned cases for p = 1=2; k = o( n) cannot be done Densest k-Subgraph The Densest k-Subgraph in polynomial time, and has served as the underly- problem, DkS (η; "), is the same as (the decision ver- ing hardness assumption in a variety of recent ap- sion of) Problem 2, except that in the \complete- plications including machine-learning and cryptog- + ness" case, G has a k-subgraph with density η, and raphy (e.g. [AAK 07, BR13]) that inherently use in the \soundness" case, every k-subgraph is of den- the average-case nature of the problem, as well as sity at most ", where η ". Since Problem 2 is in reductions to worst-case problems (e.g. [HK11, + + + a special case of this problem, our main theorem AAM 11, KZ11, BBB 13, CLLR15, BPR 16b]). can also be viewed as a new inapproximability re- The main drawback of average-case hardness sult for DkS (1; 1 − "). We remark that the aforemen- assumptions is that many average-case instances tioned quasi-polynomial algorithms for the \perfect (even those of worst-case-hard problems) are in fact completeness" regime completely break in the sparse tractable. While a significant line of research in re- regime, and indeed it is believed that DkS n−α; n−β cent years has focused on obtaining lower bounds in + (for k = n") in fact requires much more than quasi- restricted models of computation [FGR 13, MPW15, + + polynomial time [BCV+12]. The best to-date approx- DM15, HKP 16, BHK 16], a general lower bound for imation algorithm for Densest k-Subgraph due to the average-case planted clique problem appears out Bhaskara et. al, is guaranteed to find a k-subgraph of reach for existing techniques. Therefore, an impor- whose density is within an ∼ n1=4-multiplicative fac- tant potential application of our result is replacing tor of the densest subgraph of size k [BCC+10], and average-case assumptions such as the planted-clique thus DkS (η; ") can be solved efficiently whenever conjecture, in applications that do not inherently rely η n1=4 · " (this improved upon a previous n1=3−δ- on the distributional nature of the inputs (e.g., when approximation of Feige et. al [FKP01]). Making fur- the ultimate goal is to prove a worst-case hardness ther progress on either the lower or upper bound fron- result). In such applications, there is a good chance tier of the problem is a major open problem. that planted clique hardness assumptions can be re- Several inapproximability results for Densest k- placed with a more \conventional" hardness assump- Subgraph were known against specific classes of al- tion, such as the ETH, even when the problem has gorithms [BCV+12] or under incomparable assump- a quasi-polynomial algorithm. Recently, such a re- tions of Unique Games with expansion [RS10] and placement of the planted clique conjecture with ETH hardness of random k-CNF [Fei02, AAM+11]. The was obtained for the problem of finding an approx- most closely related result is by Khot [Kho06], who imate Nash equilibrium with approximately optimal shows that the Densest k-Subgraph problem has social welfare [BKW15]. " no PTAS unless SAT can be solved in time 2n , as We also remark that, while showing hardness 1=2+" opposed to 2n in our paper. While Khot's work for Planted Clique from worst-case assumptions uses a slightly weaker assumption, an important ad- seems beyond the reach of current techniques, our vantage of our work is simplicity: our construction is result can also be seen as circumstantial evidence very simple, especially in contrast to Khot's reduc- that this problem may indeed be hard. In particular, tion. any polynomial time algorithm (if exists) would have We stress that the result of [Kho06], as well as other aforementioned works, focus on the sub- 1Planted Clique typically refers to p = 1=2, while our constant density regime, i.e. they show hardness hardness result is analogous to p = 1 − δ, for a small constant for distinguishing between a graph where every k- δ > 0. Nevertheless, in almost all applications of Planted subgraph is sparse, and one where every k-subgraph is Clique, hardness for any constant p suffices.
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