A General Regularized Continuous Formulation for the Maximum Clique Problem

TL;DR

This paper introduces a flexible continuous optimization framework with regularization for solving the maximum clique problem, ensuring equivalence to the original problem and analyzing specific regularizers for computational efficiency.

Contribution

It proposes a general regularized formulation for the maximum clique problem and establishes conditions for its equivalence, along with analysis of particular regularizers.

Findings

Regularization-based continuous formulation is equivalent to the original maximum clique problem.

Conditions for global and local equivalence are established.

Two specific regularizers are analyzed for computational effectiveness.

Abstract

In this paper, we develop a general regularization-based continuous optimization framework for the maximum clique problem. In particular, we consider a broad class of regularization terms that can be included in the classic Motzkin-Strauss formulation and we develop conditions that guarantee the equivalence between the continuous regularized problem and the original one in both a global and a local sense. We further analyze, from a computational point of view, two different regularizers that satisfy the general conditions.