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

Abstract

In the Densest k-Subgraph (DkS) problem, given an undirected graph G and an integer k, the goal is to find a subgraph of G on k vertices that contains maximum number of edges. Even though Bhaskara et al.'s state-of-the-art algorithm for the problem achieves only O(n 1/4+ε ) approximation ratio, previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer) and 2 O(log 2/3 n) (Alon et al.) respectively.In this work, we show, assuming the exponential time hypothesis (ETH), that there is no polynomialtime algorithm that approximates DkS to within n

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].…”

Jointly low-rank and bisparse recovery: Questions and partial answers

“…We provide a few incomplete results related to this situation. This is NP-hard even with the restriction that M is an adjacency matrix of a graph because it then reduces to the densest k-subgraph problem, which is known to be NP-hard [21].…”

Approximation and Hardness of Shift-Bribery

“…Although hardness of approximation of DkS within up to polynomial factor is not known, inapproximability up to almost polynomial factor is known assuming the exponential time hypothesis (ETH) and its gap version (Gap-ETH). 5 Specifically, Manurangsi [Man17] has shown that under the ETH assumption (the Gap-ETH assumption, respectively), DENSEST-k-SUBGRAPH is hard to approximate to within a factor of n 1/poly(log log n) (n o (1) , respectively). Together with Theorem 14, this implies the following corollary.…”

“…We first show FPT inapproximability of Copeland α -SHIFT-BRIBERY with unit prices, parameterized by the number of unit shifts (Theorem 16). To do so, let us recall the following hardness result regarding distinguishing a graph with a k-clique and one in which every k-vertex subgraph is sparse, as proved by Chalermsook et al [CCK + 17] (which, in turn, relies heavily on the reduction and the main lemma of Manurangsi [Man17]).…”