MIP formulations for induced graph optimization problems: a tutorial

• Published in: International transactions in operational research 2023-11, Vol.30 (6), p.3159-3200
• Main Authors: Melo, Rafael A., Ribeiro, Celso C.
• Format: Article
• Language: English
• Subjects: Apexes; Combinatorial analysis; combinatorial optimization; feedback vertex set; Graph theory; induced graphs; induced paths; Integer programming; Linear programming; Literature reviews; Mixed integer; networks; Operations research; Optimization; quasi‐cliques; Trees (mathematics); Vertex sets
• ISSN: 0969-6016; 1475-3995
• DOI: 10.1111/itor.13299

Given a graph G=(V,E)$G=(V,E)$ and a subset of its vertices V′⊆V$V^{\prime }\subseteq V$, the subgraph induced by V′$V^{\prime }$ in G is that with vertex set V′$V^{\prime }$ and edge set E′$E^{\prime }$ formed by all the edges in E linking two vertices in V′$V^{\prime }$. Mixed integer programming (MIP) approaches are among the most successful techniques for solving induced graph optimization problems, that is, those related to obtaining maximum or minimum (weighted or not) induced subgraphs with certain properties. In this tutorial, we provide a literature review of some of these problems. Furthermore, we illustrate the use of MIP formulations and techniques for solving combinatorial optimization problems involving induced graphs. We focus on compact formulations and those with an exponential number of constraints that can be effectively solved using branch‐and‐cut procedures. More specifically, we revisit applications of their use for problems of finding induced forests (which correspond to the complement of feedback vertex sets), trees, paths, as well as quasi‐clique partitionings.