Two Generalizations of Proper Coloring: Hardness and Approximability

• Published in: Computing and Combinatorics Vol. 13595; pp. 82 - 93
• Main Authors: Bliznets, Ivan, Sagunov, Danil
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2023; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Approximation algorithms; Coloring; Exact algorithms; Graph algorithms; Parameterized complexity; Unit disk graphs
• ISBN: 3031221044; 9783031221040
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-031-22105-7_8

We study two natural generalizations of q-Coloring. These problems can be seen as optimization problems and are mostly applied to graphs that are not properly colorable with q colors. One of them is known as Maximumq-Colorable Induced Subgraph, and asks to find the largest set of vertices inducing a q-colorable graph. While very natural, this generalization has a downside of that it does not assign any color to vertices outside of the solution, which limits its application. To address this issue, we introduce another natural generalization of q-Coloring. The main concept of this new problem is properly colored vertex, which is a vertex that has no neighbour colored with the same color as itself. The Maximum Properlyq-Colored Vertices asks to find a q-coloring of the input graph that maximizes the number of such vertices. Our work focuses on similarities and differences between these two problems. The first part of our work is the computational hardness of Maximum Properlyq-Colored Vertices in comparsion to Maximumq-Colorable Induced Subgraph. Here we first show that Maximum Properlyq-Colored Vertices admits a 1.4391n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.4391^n$$\end{document} exact algorithm for q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2$$\end{document}, and is NP-complete in this case even on unit-disk graphs. Following the parameterized complexity study of Maximumq-Colorable Induced Subgraph by Misra et al. [WG ’13], we then show that Maximum Properlyq-Colored Vertices and Maximum(q+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q+1)$$\end{document}-Colorable Induced Subgraph are basically the same problem when restricted to split graphs. In contrast to this, we show that Maximum Properlyq-Colored Vertices and Maximumq-Colorable Induced Subgraph behave differently on perfect graphs, as Maximum Properlyq-Colored Vertices is W[2]-hard on this graph class, while Maximumq-Colorable Induced Subgraph was known to be FPT. The second part of our work is dedicated to efficient approximation of both problems on unit-disk graphs. Namely, we design several approximation algorithms for these problems restricted to this graph class. The first kind of algorithms is based on (now classical) shifting technique and achieves an (1-ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\epsilon )$$\end{document}-approximate solution in nO(qϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\mathcal {O}(\frac{q}{\epsilon })}$$\end{document} time. The second kind of algorithms that we obtain aims to get rid of the dependency in q in the exponent. These two algorithms run in nO(1ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\mathcal {O}(\frac{1}{\epsilon })}$$\end{document}, though the approximation ratio they achieve is only (1-1e-ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\frac{1}{e}-\epsilon )$$\end{document}. Here we use the greedy (1-1e)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-\frac{1}{e})$$\end{document}-approximation algorithm for the Maximum Coverage problem. These algorithms rely heavily on the given geometric representation of the graph, so we propose the third kind of algorithms that does not require the geometric representation. They allow to achieve the same approximation ratios under a tradeoff of additional 1ϵlog2qϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\epsilon }\log ^2{\frac{q}{\epsilon }}$$\end{document} multiplier in the exponent. While the three methods used to design these schemes are not novel, our research extends boundaries of their applicability while showing how they can be efficiently combined together to achieve new algorithms.