On the practically interesting instances of MAXCUT

TL;DR

This paper explores practically relevant instances of MAXCUT, demonstrating polynomial-time solutions for certain stable, metric, expanding, and dense cases, thus bridging the gap between theoretical hardness and practical solvability.

Contribution

It introduces polynomial-time algorithms for stable instances of MAXCUT, including metric and dense cases, under mild stability assumptions, extending previous results.

Findings

Polynomial-time solutions for $(1+\,\epsilon)$-stable MAXCUT instances.

Improved polynomial-time solvability for $\,\Omega(\,\sqrt{n})$-stable instances.

Identification of classes of MAXCUT instances that are efficiently solvable in practice.

Abstract

The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to be desired. In application areas, practically interesting instances very often occupy just a tiny part of an algorithm's space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, $(1+\epsilon)$-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time $\Omega(\sqrt{n})$-stable instances of MAXCUT, substantially improving the best previously known result.