Dynastic Potential Crossover Operator

TL;DR

This work introduces Dynastic Potential Crossover (DPX), a gray-box recombination operator designed to realize high-quality offspring in low-epistasis problems by exhaustively (within a bounded limit) exploring the dynastic potential defined by two parents. DPX uses the variable interaction graph to decompose the recombination problem into a chordal-graph structure and applies dynamic programming on a clique tree to compute the best offspring within a hyperplane determined by shared variables, with a tunable bound $eta$ controlling complexity. Theoretical results show DPX is at least as good as PX and typically at least as good as APX, and it generalizes to finite alphabets beyond binary. Empirically, DPX achieves superior offspring quality in crossover-only tests and substantially improves performance when embedded in trajectory-based DRILS and, to a lesser extent, in EA on NKQ Landscapes and MAX-SAT, albeit with higher runtime and memory demands; these findings suggest using DPX in combination with PX in rugged problems for best results.

Abstract

An optimal recombination operator for two parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions. Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time. In this paper, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.

Figures (10)

• Figure 1: Sample variable interaction graph (VIG).
• Figure 2: Recombination graph for the solutions (parents) $\text{red} = 000000000000000000$ and $\text{blue} = 111101011101110110$.
• Figure 3: Variable interaction graph (left) and recombination graph (right), generated by ChenWhitley2017, for instance atco_enc3_opt1_13_48 (1067657 variables) from the SAT Competition 2014. The recombination graph contains 1087 connected components.
• Figure 4: Maximum cardinality search applied to the third connected component of Figure \ref{['fig:recom2']} (left) and clique tree with the sets $S_i$ and $R_i$ (right).
• Figure 5: Connected component in a recombination graph (left) and its clique tree (right). DPX with $\beta = 1$ explores the articulation points in a different way as APX.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Theorem 3
• proof