Solving Subgraph Extraction Problems Using $Δ$Search

2026-06-11

AI Summary

Subgraph extraction problems arise widely in network design, facility location, and related domains, yet lack a general-purpose, efficient solution methodology. This work proposes ΔSearch—the first unified heuristic framework that requires only user-specified feasibility constraints and an optimization objective, automatically adapting to monotone, weighted monotone, and non-monotone graph problems without problem-specific parameter tuning. By integrating a reward-penalty optimization mechanism, generic constraint modeling, and search space pruning techniques, ΔSearch substantially enhances computational efficiency and can accelerate exact algorithms. Empirical evaluations demonstrate that it matches or surpasses state-of-the-art heuristics on tasks such as maximum planar subgraph, uncapacitated facility location, and prize-collecting vertex cover, while achieving approximately 89% of optimal solution quality on average across other problems—all without any parameter tuning.

Abstract

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

Problem

• subgraph extraction
• NP-hard graph problems
• feasibility constraints
• structural constraints
• competing objectives

Innovation

• ΔSearch
• subgraph extraction
• Reward-Penalty optimization
• heuristic framework
• search space pruning