A New Conjecture on Hardness of Low-Degree 2-CSP's with Implications to Hardness of Densest $k$-Subgraph and Other Problems

TL;DR

This paper introduces a new conjecture on the hardness of low-degree 2-CSPs and demonstrates its implications for establishing new hardness of approximation results for problems like Densest k-Subgraph and others, bridging existing conjectures and techniques.

Contribution

It proposes a novel conjecture on low-degree 2-CSP hardness and shows its implications for multiple problems, also establishing their approximate equivalence.

Findings

New hardness of approximation results for Densest k-Subgraph and related problems.

Formalization of the approximate equivalence among several problems.

The conjecture bridges existing hardness assumptions and standard techniques.

Abstract

We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the $d$-to-$1$ conjecture, and hardness results for $2$-CSP's that can be obtained via standard techniques, such as Parallel Repetition combined with standard $2$-prover protocols for the 3SAT problem. We hope that this work will motivate further exploration of hardness of $2$-CSP's in the regimes arising from the conjecture. We believe that a positive resolution of the conjecture will provide a good starting point for further hardness of approximation proofs. Another contribution of our work is proving that the problems that we consider are roughly equivalent from the approximation perspective. Some of these problems arose in previous work, from which it appeared that they may be related to each other. We formalize this relationship in this work.