Hyperplane Elimination for Quickly Enumerating Local Optima

Abstract

Examining the properties of local optima is a common method for understanding combinatorial-problem landscapes. Unfortunately, exhaustive algorithms for finding local optima are limited to very small problem sizes. We propose a method for exploiting problem structure to skip hyperplanes that cannot contain local optima, allowing runtime to scale with the number of local optima instead of with the landscape size. We prove optimality for linear functions and Concatenated Traps, and we provide empirical evidence of optimality on NKq Landscapes and Ising Spin Glasses. We further refine this method to find solutions that cannot be improved by flipping r or fewer bits, which counterintuitively can reduce total runtime. While previous methods were limited to landscapes with at most $2^{34}$ binary strings, hyperplane elimination can enumerate the same problems with $2^{77}$ binary strings, and find all 4-bit local optima of problems with $2^{200}$ binary strings.