ETH Hardness for Densest-k-Subgraph with Perfect Completeness

Abstract

We show that, assuming the (deterministic) Exponential Time Hypothesis, distinguishing between a graph with an induced k-clique and a graph in which all k-subgraphs have density at most 1−ε, requires nΩ (log n) time. Our result essentially matches the quasi-polynomial algorithms of Feige and Seltser [FS97] and Barman [Bar15] for this problem, and is the first one to rule out an additive PTAS for Densest k-Subgraph. We further strengthen this result by showing that our lower bound continues to hold when, in the soundness case, even subgraphs smaller by a near-polynomial factor (k = k · 2 −Ω(log n) ) are assumed to be at most (1 − ε)-dense.Our reduction is inspired by recent applications of the "birthday repetition" technique [AIM14, BKW15]. Our analysis relies on information theoretical machinery and is similar in spirit to analyzing a parallel repetition of twoprover games in which the provers may choose to answer some challenges multiple times, while completely ignoring other challenges.

Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph

“…Prior to our result, the best known ratio ruled out under any worst case assumption is only any constant [RS10] and the best ratio ruled out under any average case assumption is only 2 O(log 2/3 n) [AAM + 11]. In addition, our results also have perfect completeness, which was only achieved in [BKRW17] under ETH and in [AAM + 11] under the Planted Clique Hypothesis but not in [Kho06,Fei02,RS10].…”

“…Recently, Braverman et al [BKRW17], showed, under the exponential time hypothesis (ETH), which will be stated shortly, that, for some constant ε > 0, no nÕ (log n) -time algorithm can approximate Densest k-Subgraph with perfect completeness to within (1 + ε) factor. It is worth noting here that their result matches almost exactly with the previously mentioned Feige-Seltser algorithm [FS97].…”

Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis

“…To the best of our knowledge, this is the only known hardness of approximation result for DALkS. We remark that DALkS is a variant of the Densest k-Subgraph (DkS) problem, which is the same as DALkS except that the desired set S must have size exactly k. DkS has been extensively studied dating back to the early 90s [10,18,20,23,[42][43][44][45][46][47][48][49][50][51][52]. Despite these considerable efforts, its approximability is still wide open.…”

“…Despite these considerable efforts, its approximability is still wide open. In particular, even though lower bounds have been shown under stronger complexity assumptions [10,18,20,23,50,52] and for LP/SDP hierarchies [49,53,54], not even constant factor NP-hardness of approximation for DkS is known. On the other hand, the best polynomial time algorithm for DkS achieves only O(n 1/4+ε )-approximation [49].…”

Settling the complexity of computing approximate two-player Nash equilibria

“…Braverman et al [25] recently showed that the hardness Planted-Clique can be replaced by the Exponential Time Hypothesis, the NP-analog of the ETH for PPAD we use here. The work of Braverman et al, together with an earlier paper by Aaronson et al [1] inspired a line of works on quasi-polynomial hardness results via the technique of "birthday repetition" [15,24,61,22]. In particular [15] investigated whether birthday repetition can give quasi-polynomial hardness for finding any -Approximate Nash Equilibrium (our main theorem).…”