一种用于图\(k\)-割问题的并行进化算法框架
A Parallel Evolutionary Algorithm Framework for Graph $k$-CUT Problems
AI总结：
研究图\(k\)-割问题，提出并行进化算法框架PEAF，它结合多种机制。实验表明在多个\(k\)-割问题上优于Gurobi，能改进MaxGCP中部分问题的解，验证解的质量，还揭示MinGCP中不同问题划分平衡性及结构相似性，是有效求解器和揭示结构特性的工具。
AI中文摘要：
图\(k\)-割问题包含许多重要变体，其目标以不同方式组合割值、体积和基数项。现有算法多针对单个公式设计，限制了跨相关模型的可转移性。本文根据优化方向和平衡相关结构将图划分问题分为MaxGCP和MinGCP两类。基于此分类，提出统一的并行进化算法框架（PEAF）。该框架结合结构继承交叉算子、基于多重变异启发式（MMH）和辅助割变异启发式（ACMH）的分层变异机制以及保持多样性的选择策略。在\(k\in\{2,3,4,5\}\)的G集上的大量实验表明，PEAF - ACMH在九个代表性\(k\)-割问题上始终优于Gurobi。对于MaxGCP，PEAF - ACMH改进了\(k\geq3\)时Max - \(k\)-割的几个最佳已知解，并验证了Judicious - \(k\)-划分和AntiCheeger - \(k\)-割所得解的高质量。结果还表明Judicious - \(k\)-划分通常比AntiCheeger - \(k\)-割产生更平衡的划分。对于MinGCP理论和计算比较表明，Cheeger - \(k\)-割和Sparsest - \(k\)-割分别比Normalized - \(k\)-割和Ratio - \(k\)-割产生更平衡的划分。PEAF - ACMH还在短运行时间内为Min - \(k\)-割和MinMax - \(k\)-割获得高度相似的划分，为它们的结构相似性提供了数值证据。这些结果表明PEAF既是有效的统一求解器，也是揭示图\(k\)-割模型结构特性的有用工具。
英文摘要：
Graph k-CUT problems include many important variants whose objectives combine cut value, volume, and cardinality terms in different ways. Most existing algorithms are designed for individual formulations, which limits their transferability across related models. In this paper, we organize a broad family of graph partitioning problems into two classes, MaxGCP and MinGCP, according to their optimization orientation and balance-related structure. Based on this classification, we propose a unified Parallel Evolutionary Algorithm Framework (PEAF). This framework combines structure-inheriting crossover operators, a hierarchical mutation mechanism based on the Multiple Mutation Heuristic (MMH) and the Auxiliary Cut Mutation Heuristic (ACMH), and a diversity-preserving selection strategy. Extensive experiments on G-set with k \in\{2, 3, 4, 5\} show that PEAF-ACMH consistently outperforms Gurobi on nine representative k-CUT problems. For MaxGCP, PEAF-ACMH improves several best-known solutions for Max-k-Cut with k \geq 3, and through numerical bounds derived from its relation to Max-k-Cut, verifies the high quality of the obtained solutions for Judicious-k-Partition and AntiCheeger-k-Cut. The results further indicate that Judicious-k-Partition usually yields more balanced partitions than AntiCheeger-k-Cut. For MinGCP, theoretical and computational comparisons show that Cheeger-k-Cut and Sparsest-k-Cut produce more balanced partitions than Normalized-k-Cut and Ratio-k-Cut, respectively. PEAF-ACMH also obtains highly similar partitions for Min-k-Cut and MinMax-k-Cut within short running times, providing numerical evidence for their structural affinity. These results demonstrate that PEAF is both an effective unified solver and a useful tool for revealing structural properties of graph k-CUT models.
