Simplicity and Complexity in Combinatorial Optimization

TL;DR

This paper explores how complexity and optimization are linked, suggesting simpler solutions are more likely to be optimal in certain cases.

Contribution

The paper introduces novel theoretical connections between Kolmogorov complexity and optimization, proposing a new optimization method based on algorithmic probability.

Findings

Optima are more likely to have low complexity under certain conditions.

Sampling candidate solutions via algorithmic probability may be an effective optimization strategy.

Extrema in optimization problems are more likely to coincide than expected by random chance.

Abstract

Many problems in physics and computer science can be framed in terms of combinatorial optimization. Due to this, it is interesting and important to study theoretical aspects of such optimization. Here, we study connections between Kolmogorov complexity, optima, and optimization. We argue that (1) optima and complexity are connected, with extrema being more likely to have low complexity (under certain circumstances); (2) optimization by sampling candidate solutions according to algorithmic probability may be an effective optimization method; and (3) coincidences in extrema to optimization problems are a priori more likely as compared to a purely random null model.

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