Combinatorial optimization in system configuration design
Automation and Remote Control
Convex Recoloring Revisited: Complexity and Exact Algorithms
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
A Kernel for Convex Recoloring of Weighted Forests
SOFSEM '10 Proceedings of the 36th Conference on Current Trends in Theory and Practice of Computer Science
Speeding up dynamic programming for some NP-hard graph recoloring problems
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
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A pair (T,C) of a tree T and a coloring C is called a colored tree. Given a colored tree (T,C) any coloring C′ of T is called a recoloring of T. Given a weight function on the vertices of the tree the recoloring distance of a recoloring is the total weight of recolored vertices. A coloring of a tree is convex if for any two vertices u and v that are colored by the same color c, every vertex on the path from u to v is also colored by c. In the minimum convex recoloring problem we are given a colored tree and a weight function and our goal is to find a convex recoloring of minimum recoloring distance. The minimum convex recoloring problem naturally arises in the context of phylogenetic trees. Given a set of related species the goal of phylogenetic reconstruction is to construct a tree that would best describe the evolution of this set of species. In this context a convex coloring corresponds to perfect phylogeny. Since perfect phylogeny is not always possible the next best thing is to find a tree which is as close to convex as possible, or, in other words, a tree with minimum recoloring distance. We present a (2+ε)-approximation algorithm for the minimum convex recoloring problem, whose running time is O(n 2+n(1/ε)241/ε ). This result improves the previously known 3-approximation algorithm for this NP-hard problem. We also present an algorithm for computing an optimal convex recoloring whose running time is $O(n^{2}+n\cdot n^{*}\cdot \Delta^{n^{*}+1})$, where n * is the number of colors that violate convexity in the input tree, and Δ is the maximum degree of vertices in the tree. The parameterized complexity of this algorithm is O(n2+n⋅k⋅2k).