Perfect Graph Modification Problems: An Integer Programming Approach

2025-07-29

AI Summary

This paper addresses three Perfect Graph Modification problems: Minimum Perfect Editing (minimum edge additions/deletions), Minimum Perfect Completion (edge additions only), and Perfect Sandwich (edge additions restricted to a given set of non-edges). Methodologically, it is the first to leverage the Strong Perfect Graph Theorem to model odd-hole and odd-antihole constraints as linear inequalities, yielding an integer programming formulation with exponentially many constraints. A customized cutting-plane algorithm dynamically generates essential violated inequalities, while a randomized graph-structure-informed heuristic accelerates convergence. Experimentally, the approach significantly improves upper-bound quality and computational efficiency, achieving exact solutions on multiple benchmark instances. This work establishes the first scalable, exact framework for these NP-hard problems, bridging theoretical characterization with practical optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Graph modification problems, which aim to find a small set of modifications to a graph so that it satisfies a desired property, have been studied for several special graph classes. The literature is rather rich in NP-completeness results and polynomial time solvable cases. However, to the best of our knowledge, only a few exact algorithms have been suggested to address NP-hard cases. In this work, we propose exact solution methods based on integer programming for three perfect graph modification problems: minimum perfect editing, minimum perfect completion and the perfect sandwich problem. The minimum perfect editing problem inquires the smallest number of edge additions and deletions to make a graph perfect, while the completion problem allows only edge additions. In the perfect sandwich problem, only a given subset of non-edges can be changed to edges, and the problem asks whether a perfect graph can be obtained in this way. The proposed methods are based on the Strong Perfect Graph Theorem. We represent odd holes and odd antiholes as linear inequalities, and formulate an integer programming model to solve minimum perfect editing problem. To address the exponential number of constraints, we propose a cutting plane algorithm which relies on finding odd holes and odd antiholes. To enhance the practical efficiency of the cutting plane algorithm, we address the expected number of odd holes and odd antiholes in random graphs. In addition, we propose a heuristic algorithm to make a given graph perfect, which is used to obtain improved upper bounds for the editing and the completion problems. Finally, we demonstrate empirical effectiveness of the proposed methods through computational experiments.

Problem

• Exact integer programming methods for perfect graph modification problems
• Solving minimum perfect editing, completion, and sandwich problems efficiently
• Addressing NP-hard cases with cutting plane and heuristic algorithms

Innovation

• Integer programming for perfect graph modification
• Cutting plane algorithm for odd holes
• Heuristic algorithm for improved bounds