Worst-Case to Expander-Case Reductions

TL;DR

This paper investigates whether worst-case graph problems are as easy on expanders by introducing worst-case to expander-case reductions, revealing that many problems are not simplified by expander techniques.

Contribution

The paper introduces worst-case to expander-case self-reductions for key graph problems, demonstrating that expander decompositions do not universally simplify these problems.

Findings

Verified belief that some problems are as easy on expanders as on worst-case graphs

Showed that expander decompositions are ineffective for certain problems like $k$-Clique and Vertex-Cover

Contradicted the idea that expander techniques can break all complexity barriers

Abstract

In recent years, the expander decomposition method was used to develop many graph algorithms, resulting in major improvements to longstanding complexity barriers. This powerful hammer has led the community to (1) believe that most problems are as easy on worst-case graphs as they are on expanders, and (2) suspect that expander decompositions are the key to breaking the remaining longstanding barriers in fine-grained complexity. We set out to investigate the extent to which these two things are true (and for which problems). Towards this end, we put forth the concept of worst-case to expander-case self-reductions. We design a collection of such reductions for fundamental graph problems, verifying belief (1) for them. The list includes $k$-Clique, $4$-Cycle, Maximum Cardinality Matching, Vertex-Cover, and Minimum Dominating Set. Interestingly, for most (but not all) of these problems the proof is via a simple gadget reduction, not via expander decompositions, showing that this hammer is effectively useless against the problem and contradicting (2).