Many NP-hard graph problems can be modeled as optimal subgraph extraction problems with feasibility constraints. From Network Design to Facility Location, from Robotics to Graph Drawing, the subgraph extraction pattern emerges across diverse domains. Despite this commonality, these problems are typically solved with domain-specific heuristics. Usually, these problems balance competing objectives such as maximizing coverage or minimizing cost while satisfying structural constraints such as connectivity, planarity and reachability. In this work, we introduce $Δ$Search, a general and fast heuristic framework that exploits the insight of Reward-Penalty optimization for solving a large class of subgraph extraction problems. The framework is easy to use as it only requires feasibility constraints and optimality criteria to be provided by the user to express the subgraph extraction problem. We also show how exact methods can be augmented with $Δ$Search to improve their performance by aggressive pruning of the search space. We evaluate our framework on monotone graph problems such as Maximum Planar Subgraph (MPS) and Minimum Connected Dominating Set, Weighted Monotone problems such as Maximum Weighted Independent Set and Minimum Weighted Steiner Tree, and non-monotone graph problems such as Prize Collecting Vertex Cover (PCVC) and Uncapacitated Facility Location Problem (UFLP). Our results show that $Δ$Search matches or surpasses state of the art heuristics for MPS, UFLP and PCVC problems with similar runtime. For the remaining problems, $Δ$Search achieves approximately 89% of the solution quality of the state-of-the-art algorithms without any problem-specific tuning

翻译：许多NP难图问题可以建模为具有可行性约束的最优子图提取问题。从网络设计到设施选址，从机器人学到图形绘制，子图提取模式出现在不同领域中。尽管存在这种共性，这些问题通常使用特定领域的启发式算法来解决。通常，这些问题需要在满足连通性、平面性和可达性等结构性约束的同时，平衡诸如最大化覆盖或最小化成本等相互竞争的目标。在这项工作中，我们提出了 $Δ$Search，这是一个通用且快速的启发式框架，它利用奖励-惩罚优化的洞察力来解决一大类子图提取问题。该框架易于使用，因为它只需要用户提供表达子图提取问题的可行性约束和最优性标准。我们还展示了如何通过 $Δ$Search 增强精确方法，通过积极剪枝搜索空间来提高其性能。我们在单调图问题（如最大平面子图（MPS）和最小连通支配集）、加权单调问题（如最大加权独立集和最小加权斯坦纳树）以及非单调图问题（如奖励收集顶点覆盖（PCVC）和无容量设施选址问题（UFLP））上评估我们的框架。我们的结果表明，$Δ$Search 在 MPS、UFLP 和 PCVC 问题上以相似的运行时间匹配或超越了最先进的启发式算法。对于其余问题，$Δ$Search 在没有任何问题特定调优的情况下，达到了最先进算法约89%的解质量。