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

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 k-subgraph 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/log³ 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.