Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph

Abstract

The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n 1/4 ) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest ksubgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge.In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include:• A lower bound of Ω n 1/4 / log 3 n on the integrality gap for Ω(log n/ log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are in fact Erdös-Renyi random graphs.• For every > 0, a lower bound of n 2/53− on the integrality gap of n Ω( ) rounds of the Lasserre SDP relaxation for Densest k-subgraph, and an n Ω (1) gap for n 1− rounds. Our construction proceeds via a reduction from random instances of a certain Max-CSP over large domains.In the absence of inapproximability results for Densest k-subgraph, our results show that beating a factor of n Ω(1) is a barrier for even the most powerful SDPs, and in fact even beating the best known n 1/4 factor is a barrier for current techniques. Our results indicate that approximating Densest k-subgraph within a polynomial factor might be a harder problem than Unique Games or Small Set Expansion, since these problems were recently shown to be solvable using n Ω(1) rounds of the Lasserre hierarchy, where is the completeness parameter in Unique Games and Small Set Expansion.

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.…”

Rounding sum-of-squares relaxations

“…There are several strong lower bounds (also known as integrality gaps) for these hierarchies, in particular showing that ω(1) levels (and often even n Ω (1) or Ω(n) levels) of many such hierarchies can't improve by much on the known polynomial-time approximation guarantees for many NP hard problems, including SAT, Independent-Set, Max-Cut and more [28,27,5,21,47,52,19,14,15]. Unfortunately, there are many fewer positive results, and many of them only show that these hierarchies can match the performance of Proceedings of the 2014 ACM Symposium on Theory of Computing 31 Proceedings of the 2014 ACM Symposium on Theory of Computing previously known (and often more efficient) methods, or give algorithms that can be converted into something much more combinatorial, rather than using hierarchies to get genuinely new algorithmic results.…”

Partitioning a Graph into Small Pieces with Applications to Path Transversal

“…In particular, a polynomial time O(log k)-approximation algorithm for k-Vertex Separator will imply O(log 2 n)-approximation algorithm for Densest kSubgraph. Given that the best approximation algorithm achieves ≈ O(n 1/4 )-approximation [4] and n Ω(1) -rounds of the Sum-of-Squares hierarchy have a gap at least n Ω(1) [5], such a result seems unlikely or will be considered as a breakthrough. .…”

“…The current best approximation algorithm achieves ≈ O(n 1/4 )-approximation [4]. While only PTAS is ruled out assuming NP ⊆ ∩ >0 BPTIME(2 n ) [22], there are strong gap instances for Sum-of-Squares hierarchies of convex relaxations (n Ω(1) gap for n Ω(1) rounds) [5], so having a polylog(n)-approximation algorithm for Densest k-Subgraph seems unlikely or will lead to a breakthrough. Therefore, it may be the case that achieving O(log k)-approximation for k-Vertex Separator requires superpolynomial dependence on k in the running time.…”