Constructing Concrete Hard Instances of the Maximum Independent Set Problem

TL;DR

This paper introduces a deterministic method to construct infinite graph sequences that serve as hard instances for the maximum independent set problem, demonstrating limitations of cycle-chain refutation algorithms.

Contribution

It presents a novel deterministic construction of challenging instances for MIS and introduces cycle-chain refutation as a general analytical tool.

Findings

Hard instances resemble sparse random graphs

Cycle-chain refutation cannot tightly bound MIS size

Constructed graphs are computationally challenging

Abstract

We provide a deterministic construction of hard instances for the maximum independent set problem (MIS). The constructed hard instances form an infinite graph sequence with increasing size, which possesses similar characteristics to sparse random graphs and in which MIS cannot be solved efficiently. We analytically and numerically show that all algorithms employing cycle-chain refutation, which is a general refutation method we introduce for capturing the ability of many known algorithms, cannot upper bound the size of the maximum independent set tightly.