Maximum Satisfiability: Anatomy of the Fitness Landscape for a Hard Combinatorial Optimization Problem

Abstract

Abstract-The fitness landscape of MAX-3-SAT is investigated for random instances above the satisfiability phase transition. This study includes a scaling analysis of the time to reach a local optimum, the number of local optima, the expected probability of reaching a local optimum as a function of its fitness, the expected fitness found by local search and the best fitness, the probability of reaching a global optimum, the size and relative positions of the global optima, the mean distance between the local and global optima, the expected fitness as a function of the Hamming distance from an optimum and their basins of attraction. These analyses show why the problem becomes hard for local search algorithms as the system size increases. The paper also shows how a recently proposed algorithm can exploit long-range correlations in the fitness landscape to improve on the state-of-the-art heuristic algorithms.

Quadratic assignment problem: a landscape analysis

“…As the Hamming distance decreases, the average cost of the local optima from the global optimum increases, but for most of the benchmark problems and all the random problems, taking the last steps to Hamming distance of n, the cost slightly decreases. This is similar (and may be for the same reason) to the behaviour we saw in the cost of the configurations at a Hamming sphere around the local optima in MAX-SAT [41] and Graph-Colouring [43] landscapes. Considering Figs.…”

“…(Note that this bell shaped distribution not only is seen in quadratic assignment problem but also has been observed in other problems [41][42][43]. Thus it probably means that this is a common behaviour of a variety of optimisation problems.)…”

“…This paper is a follow up to a group of other research on the landscape of some combinatorial optimisation problems including the Maximum-Satisfiability [41], Graph-Colouring [43] and Travelling Salesman problems [44]. The aim of this programme of research is to study a wide range properties of the fitness landscape for different combinatorial optimisation problems.…”

“…The qualitative behaviour of the curve is quite similar for both the random and benchmark problems. These two similar curves talk about an important property which is seen in all the benchmark and random problems we have studied, not only for the quadratic assignment problem, but also for the MAX-SAT [41], Graph-Colouring [43], and Travelling Salesman problems [44]. The property is that on average, there is a strong correlation between the cost of a local optimum and the size of its basin of attraction.…”

“…This is different from when there exist plateaux. See for example [41,42], where there could be huge number of steps to local optima when the plateaux exist.…”