Sherali-Adams Integrality Gaps Matching the Log-Density Threshold

TL;DR

This paper demonstrates that Sherali-Adams relaxations exhibit strong integrality gaps matching the log-density threshold, supporting the conjecture that certain approximation problems are inherently hard to distinguish from random structures.

Contribution

The paper proves that Sherali-Adams relaxations cannot distinguish between random hypergraphs and those with planted substructures, establishing tight integrality gaps for multiple combinatorial problems.

Findings

Sherali-Adams relaxations have integrality gaps matching the log-density threshold.

These gaps apply to hypergraph problems like Densest k-Subgraph and Small Set Vertex Expansion.

Results support the conjecture of inherent hardness in distinguishing random from planted structures.

Abstract

The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-$k$-Subgraph and Bipartite Small Set Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density". We bolster this conjecture by showing that in a random hypergraph with edge probability $n^{-\alpha}$, $\tilde\Omega(\log n)$ rounds of Sherali-Adams with cannot rule out the existence of a $k$-subhypergraph with edge density $k^{-\alpha-o(1)}$, for any $k$ and $\alpha$. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest $k$-Subgraph, Smallest $p$-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum $p$-Union). Previously, such integrality gaps were known only for Densest $k$-Subgraph for one specific parameter setting.