Optimizing one million variable NK landscapes by hybridizing deterministic recombination and local search

Abstract

In gray-box optimization, the search algorithms have access to the variable interaction graph (VIG) of the optimization problem. For Mk Landscapes (and NK Landscapes) we can use the VIG to identify an improving solution in the Hamming neighborhood in constant time. In addition, using the VIG, deterministic Partition Crossover is able to explore an exponential number of solutions in a time that is linear in the size of the problem. Both methods have been used in isolation in previous search algorithms. We present two new gray-box algorithms that combine Partition Crossover with highly e cient local search. e best algorithms are able to locate the global optimum on Adjacent NK Landscape instances with one million variables. e algorithms are compared with a state-of-the-art algorithm for pseudo-Boolean optimization: GrayBox Parameterless Population Pyramid. e results show that the best algorithm is always one combining Partition Crossover and highly e cient local search. But the results also illustrate that the best optimizer di ers on Adjacent and Random NK Landscapes. CCS CONCEPTS• eory of computation →Random search heuristics;

Enhancing partition crossover with articulation points analysis

“…In order to experimentally analyze the performance of APX, we included it in the Deterministic Recombination and Iterated Local Search (DRILS) algorithm. DRILS [1] uses a first improving move hill climber to reach a local optimum. Then, it perturbs the solution by randomly flipping α N bits, where α is the so-called perturbation factor.…”

“…We will use F to denote the set of subfunctions f i whose sum is f . Given a set of variables C, we denote with F C the subset of subfunctions of F that depend on a variable in C. Let us call д C to the sum of subfunctions that depend on a variable 1…”

“…The perturbation factor (α) in DRILS was set to α = 0.05 in the case K = 2, 3 and α = 0.01 in the case K = 4, 5. These values were taken from the recommendations in [1]. Table 1 shows averages over all the recombinations appearing in all the runs for each combination of N and K. In the case of the number of explored solutions, we compute the binary logarithm and provide the average.…”

“…If q recombining components are found, Partition Crossover finds the best of 2 q offspring in linear time. Combinations of these two techniques have produced hybrid algorithms able to solve adjacent NK landscapes to optimality on instances with up to one million variables [1]. For random NK landscapes, optimality cannot be assessed, but the proposed hybrid algorithms outperform the previous state-of-the-art.…”

“…The presence of an articulation point in a connected component increases the number of explored solutions by a factor of, at least, 4, compared to the original Partition Crossover. The improvements of the proposed operator are evaluated within the recent state-of-the-art Gray-Box Iterated Local Search combined with Partition Crossover, DRILS algorithm [1], to solve NK landscapes of up to one million variables and a set of real-world MAX-SAT instances.…”