1.5-approximation algorithm for the 2-Convex Recoloring problem

• Published in: Discrete Applied Mathematics Vol. 246; pp. 2 - 11
• Main Authors: Bar-Yehuda, Reuven, Kutiel, Gilad, Rawitz, Dror
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 10.09.2018; Elsevier BV
• Subjects: Algorithms; Approximation; Approximation algorithms; Color; Coloring; Convex analysis; Convex recoloring; Greedy; Greedy algorithms; Mathematical analysis; Weight; Approximation algorithms; Greedy; Convex recoloring
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2017.01.008

Given a graph G=(V,E), a coloring function χ:V→C, assigning each vertex a color, is called convex if, for every color c∈C, the set of vertices with color c induces a connected subgraph of G. In the Convex Recoloring problem a colored graph Gχ is given, and the goal is to find a convex coloring χ′ of G that recolors a minimum number of vertices. In the weighted version each vertex has a weight, and the goal is to minimize the total weight of recolored vertices. The 2-Convex Recoloring problem (2-CR) is the special case, where the given coloring χ assigns the same color to at most two vertices. 2-CR is known to be NP-hard even if G is a path. We show that weighted 2-CR cannot be approximated within any ratio, unless P=NP. On the other hand, we provide an alternative definition of (unweighted) 2-CR in terms of maximum independent set of paths, which leads to a natural greedy algorithm. We prove that its approximation ratio is 32 and show that this analysis is tight. This is the first constant factor approximation algorithm for a variant of CR in general graphs. For the special case, where G is a path, the algorithm obtains a ratio of 54, an improvement over the previous best known approximation. We also consider the problem of determining whether a given graph has a convex recoloring of size k. We use the above mentioned characterization of 2-CR to show that a problem kernel of size 4k can be obtained in linear time and to design an O(|E|)+2O(klogk) time algorithm for parameterized 2-CR.