On the practically interesting instances of MAXCUT
This work addresses the gap between theoretical worst-case complexity and practical relevance for MAXCUT, potentially making theory more applicable to real-world clustering problems, though it is incremental in improving specific stability results.
The paper tackles the problem of solving practically interesting instances of MAXCUT, a clustering problem, by showing how to solve various types of instances (distinguished, metric, expanding, dense) in polynomial time under mild stability assumptions, with specific improvements such as handling $(1+\\epsilon)$-stability for metric and dense cases and $\\Omega(\\sqrt{n})$-stable instances.
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, $(1+ε)$-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time $Ω(\sqrt{n})$-stable instances of MAXCUT, substantially improving the best previously known result.