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. 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 problem 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 a (Formula presented.) time algorithm for parametrized 2-CR.
|Title of host publication||Combinatorial Algorithms - 26th International Workshop, IWOCA 2015, Revised Selected Papers|
|Editors||William F. Smyth, Zsuzsanna Liptak|
|Number of pages||13|
|State||Published - 2016|
|Event||26th International Workshop on Combinatorial Algorithms, IWOCA 2015 - Verona, Italy|
Duration: 5 Oct 2015 → 7 Oct 2015
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||26th International Workshop on Combinatorial Algorithms, IWOCA 2015|
|Period||5/10/15 → 7/10/15|
Bibliographical noteFunding Information:
D. Rawitz—Supported in part by the Israel Science Foundation (grant no. 497/14).
© Springer International Publishing Switzerland 2016.