Are stable instances easy?

TL;DR

This paper introduces the concept of stable instances in discrete optimization, demonstrating that sufficiently stable instances of the NP-hard Max-Cut problem can be solved efficiently, highlighting practical relevance.

Contribution

The paper defines stability for instances of optimization problems and proves that stable instances of Max-Cut are solvable in polynomial time.

Findings

Stable instances of Max-Cut are easier to solve.

Sufficient stability guarantees polynomial-time solvability.

The concept applies to practical problem instances.

Abstract

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polynomial time all sufficiently stable instances of some NP--hard problem. The paper focuses on the Max--Cut problem, for which we show that this is indeed the case.