Hard Optimization Problems have Soft Edges

Abstract

Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erdös-Rényi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of N, the graph size, and K, the clique size sought. It displays a complex phase boundary, a staircase of steps at each of which 2log2 N and Kmax, the maximum size of a clique that can be found, increases by 1. Each of its boundaries has a finite width, and these widths allow local algorithms to find cliques beyond the limits defined by the study of infinite systems. We explore the performance of a number of extensions of traditional fast local algorithms, and find that much of the "hard" space remains accessible at finite N. The "hidden clique" problem embeds a clique somewhat larger than those which occur naturally in a G(N, p) random graph. Since such a clique is unique, we find that local searches which stop early, once evidence for the hidden clique is found, may outperform the best message passing or spectral algorithms.

arXiv admin note: text overlap with arXiv:1807.09091

References in corpus (3)

• Gibbs States and the Set of Solutions of Random Constraint Satisfaction Problems
• The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures
• Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs

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