Enhancing partition crossover with articulation points analysis

Abstract

Partition Crossover is a recombination operator for pseudo-Boolean optimization with the ability to explore an exponential number of solutions in linear or square time. It decomposes the objective function as a sum of subfunctions, each one depending on a different set of variables. The decomposition makes it possible to select the best parent for each subfunction independently, and the operator provides the best out of 2 q solutions, where q is the number of subfunctions in the decomposition. These subfunctions are defined over the connected components of the recombination graph: a subgraph of the objective function variable interaction graph containing only the differing variables in the two parents. In this paper, we advance further and propose a new way to increase the number of linearly independent subfunctions by analyzing the articulation points of the recombination graph. These points correspond to variables that, once flipped, increase the number of connected components. The presence of a connected component with an articulation point increases the number of explored solutions by a factor of, at least, 4. We evaluate the new operator using Iterated Local Search combined with Partition Crossover to solve NK Landscapes and MAX-SAT.

Quasi-Optimal Recombination Operator

“…In order to experimentally analyze the performance of DPX, we included it in the Deterministic Recombination and Iterated Local Search (DRILS) algorithm [7]. We think this allows us to explore the performance of the operator in a real scenario, rather than generating random solutions and providing them to the operator.…”

“…All of the variables in the same recombining component in the recombination graph must be inherited together from one of the two parents. Articulation Points Partition Crossover (APX) [7] goes further and finds the articulation points of the recombination graphs. They are variables whose removal increases the number of connected components.…”

“…Recently defined crossover operators similar to ours are Partition Crossover (PX) [6] and Articulation Points Partition Crossover (APX) [7]. Although they were proposed to work with pseudo-Boolean functions, they can also be applied to the more general representation of variables defined over a finite alphabet.…”

“…PX and APX also use the variable interaction graph of the objective function to efficiently compute a good offspring among a large number of them. PX and APX have O(n 2 + m) time complexity and both of them obtained excellent performance in different problems [8,7,9,10]. When combined with other gray box optimization operators, partition crossover was capable of optimizing instances with 1 million variables in seconds.…”

Inferring Future Landscapes: Sampling the Local Optima Level

“…The baseline and the proof-of-concepts were necessary to establish LONs as promising tools. They now have attracted attention Bożejko et al, 2018;Herrmann et al, 2018;Fieldsend, 2018;Chicano et al, 2018;Simoncini et al, 2018;Liu et al, 2019), and will likely be applied to increasing numbers of real (and much larger) problems. In anticipation of these requirements, refined LON sampling methods are needed.…”