Solution-space structure of (some) optimization problems

TL;DR

This paper investigates the solution space structure of two NP-hard problems, vertex cover and number partitioning, revealing differing clustering behaviors and their relation to phase diagrams through numerical analysis.

Contribution

It provides a comparative numerical analysis of the solution space structures of vertex cover and number partitioning problems, highlighting their differing clustering patterns.

Findings

Vertex cover shows a transition from large single clusters to nested clusters.

Number partitioning maintains a simple, random-like solution distribution across phases.

Solution structure does not directly correlate with the phase diagram complexity.

Abstract

We study numerically the cluster structure of random ensembles of two NP-hard optimization problems originating in computational complexity, the vertex-cover problem and the number partitioning problem. We use branch-and-bound type algorithms to obtain exact solutions of these problems for moderate system sizes. Using two methods, direct neighborhood-based clustering and hierarchical clustering, we investigate the structure of the solution space. The main result is that the correspondence between solution structure and the phase diagrams of the problems is not unique. Namely, for vertex cover we observe a drastic change of the solution space from large single clusters to multiple nested levels of clusters. In contrast, for the number-partitioning problem, the phase space looks always very simple, similar to a random distribution of the lowest-energy configurations. This holds in the ``easy''/solvable phase as well as in the ``hard''/unsolvable phase.