Approximation and Hardness Results for the Maximum Edges in Transitive Closure Problem

• Published in: Combinatorial Algorithms Vol. 8986; pp. 13 - 23
• Main Authors: Adamaszek, Anna, Blin, Guillaume, Popa, Alexandru
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2015; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Adamaszek; Algorithms & data structures; Discrete mathematics; Graph Motif Problem; Hardness Results; Mathematical theory of computation; Maximum Edge; Transitive Closure Problem
• ISBN: 3319193147; 9783319193144
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-19315-1_2

In this paper we study the following problem, named Maximum Edges in Transitive Closure, which has applications in computational biology. Given a simple, undirected graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V,E)$$\end{document} and a coloring of the vertices, remove a collection of edges from the graph such that each connected component is colorful (i.e., it does not contain two identically colored vertices) and the number of edges in the transitive closure of the graph is maximized. The problem is known to be APX-hard, and no approximation algorithms are known for it. We improve the hardness result by showing that the problem is NP-hard to approximate within a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|V|^{1/3 - \varepsilon }$$\end{document}, for any constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}. Additionally, we show that the problem is APX-hard already for the case when the number of vertex colors equals 3. We complement these results by showing the first approximation algorithm for the problem, with approximation factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{2 \cdot \text {OPT}}$$\end{document}.