A Compressive Sensing Inspired Monte-Carlo Method for Combinatorial Optimization

TL;DR

This paper introduces a novel Monte-Carlo compressive sensing-based algorithm for combinatorial optimization, leveraging random sampling and greedy methods to efficiently find global optima with theoretical guarantees.

Contribution

It proposes a new algorithm combining compressive sensing and Monte-Carlo methods for combinatorial optimization, with theoretical analysis and practical parameter tuning.

Findings

Outperforms state-of-the-art dual annealing in numerical tests

Provides theoretical justification for the algorithm's success

Offers a tunable approach adaptable to computational resources

Abstract

In this paper, we present the Monte-Carlo Compressive Optimization algorithm, a new method to solve a combinatorial optimization problem that is assumed compressible. The method relies on random queries to the objective function in order to estimate generalized moments. Next, a greedy algorithm from compressive sensing is repurposed to find the global optimum when not overfitting to samples. We provide numerical results giving evidences that our methods overcome state-of-the-art dual annealing. Moreover, we also give theoretical justification of the algorithm success and analyze its properties. The practicality of our algorithm is enhanced by the ability to tune heuristic parameters to available computational resources.