1.5-approximation algorithm for the 2-Convex Recoloring problem

Given a graph G = (V, E), a coloring function chi : V -> C, assigning each vertex a color, is called convex if, for every color c is an element of C, the set of vertices with color c induces a connected subgraph of G. In the CONVEX RECOLORING problem a colored graph G(chi) is given, and the goal is to find a convex coloring chi ' 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 chi 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 3/2 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 5/4, 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(vertical bar E vertical bar) + 2(O(klogk)) time algorithm for parameterized 2-CR. (C) 2017 Elsevier B.V. All rights reserved.