Power of human-algorithm collaboration in solving combinatorial optimization problems

TL;DR

This paper explores how human-algorithm collaboration, through expert queries, can theoretically enable efficient approximate solutions to traditionally intractable combinatorial optimization problems.

Contribution

It introduces a theoretical framework showing that polynomial-time algorithms querying Gaussian priors from experts can solve certain hard problems efficiently.

Findings

Polynomial queries to Gaussian priors enable efficient approximate solutions.

The approach offers new theoretical insights into solving intractable problems.

Results are primarily theoretical, highlighting potential for future practical methods.

Abstract

Many combinatorial optimization problems are often considered intractable to solve exactly or by approximation. An example of such problem is maximum clique which -- under standard assumptions in complexity theory -- cannot be solved in sub-exponential time or be approximated within polynomial factor efficiently. We show that if a polynomial time algorithm can query informative Gaussian priors from an expert $poly(n)$ times, then a class of combinatorial optimization problems can be solved efficiently in expectation up to a multiplicative factor $\epsilon$ where $\epsilon$ is arbitrary constant. While our proposed methods are merely theoretical, they cast new light on how to approach solving these problems that have been usually considered intractable.