Integer programming formulations for the k$k$‐in‐a‐tree problem in graphs

• Published in: International transactions in operational research Vol. 31; no. 5; pp. 3090 - 3107
• Main Authors: Ferreira, Lucas Saldanha, Santos, Vinicius Fernandes dos, Valle, Cristiano Arbex
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01-09-2024
• Subjects: Algorithms; Apexes; Computing time; Graph theory; Graphs; induced subgraphs; Integer programming; k$k$‐in‐a‐tree; Mixed integer; Polynomials; Trees (mathematics)
• ISSN: 0969-6016; 1475-3995
• DOI: 10.1111/itor.13297

Chudnovsky and Seymour proposed the Three‐in‐a‐tree algorithm which solves the following problem in polynomial time: given three fixed vertices in a simple finite graph, check whether an induced tree containing these vertices exists. In this paper, we deal with a generalization of this problem, referred to henceforth as k$k$‐in‐a‐tree. The k$k$‐in‐a‐tree checks whether a graph contains an induced tree spanning k$k$ given vertices. When k$k$ is part of the input, the problem is known to be NP‐complete. If k≥4$k \ge 4$ is a fixed given number, its complexity is an open question, although there are efficient algorithms for restricted cases such as claw‐free graphs, graphs with a girth of at least k$k$ and chordal graphs. We present mixed‐integer programming formulations for this problem, and we show that instances with up to 25,000 vertices can be solved in reasonable computational time.