Quadratic Kernelization for Convex Recoloring of Trees

The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called “perfect phylogeny”. For an input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a subtree. The problem was introduced by Moran and Snir (J. Comput. Syst. Sci. 73:1078–1089, 2007; J. Comput. Syst. Sci. 74:850–869, 2008) who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k)kn4). The Moran and Snir result did not provide any nontrivial kernelization. In this paper, we show that CR has a kernel of size O(k2).

[1]  Michael R. Fellows,et al.  Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments , 2004, ALENEX/ANALC.

[2]  Saket Saurabh,et al.  Incompressibility through Colors and IDs , 2009, ICALP.

[3]  Rolf Niedermeier,et al.  Invitation to data reduction and problem kernelization , 2007, SIGA.

[4]  Stéphan Thomassé.  A quadratic kernel for feedback vertex set , 2009, SODA.

[5]  Michael R. Fellows,et al.  The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel , 2006, IWPEC.

[6]  Ge Xia,et al.  Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size , 2005, STACS.

[7]  Rolf Niedermeier,et al.  Speeding up Dynamic Programming for Some NP-Hard Graph Recoloring Problems , 2008, TAMC.

[8]  Michael R. Fellows,et al.  Quadratic Kernelization for Convex Recoloring of Trees , 2007, Algorithmica.

[9]  Michael R. Fellows,et al.  On problems without polynomial kernels , 2009, J. Comput. Syst. Sci..

[10]  Dror Rawitz,et al.  Improved Approximation Algorithm for Convex Recoloring of Trees , 2005, WAOA.

[11]  Rolf Niedermeier,et al.  Polynomial-time data reduction for dominating set , 2002, JACM.

[12]  Sagi Snir,et al.  Convex recolorings of strings and trees: Definitions, hardness results and algorithms , 2008, J. Comput. Syst. Sci..

[13]  Xiao-Dong Hu,et al.  Inapproximability and approximability of maximal tree routing and coloring , 2006, J. Comb. Optim..

[14]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[15]  Lance Fortnow,et al.  Infeasibility of instance compression and succinct PCPs for NP , 2007, J. Comput. Syst. Sci..

[16]  Bruno Courcelle,et al.  Graph grammars, monadic second-order logic and the theory of graph minors , 1991, Graph Structure Theory.

[17]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[18]  Falk Hüffner,et al.  Parametrisierte Ansätze für schwere Graphprobleme: Algorithmen und ExperimenteAlgorithms and Experiments for Parameterized Approaches to Hard Graph Problems , 2009, it Inf. Technol..

[19]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[20]  Jens Gramm,et al.  Fixed-parameter Algorithms in Phylogenetics , 2022 .

[21]  Frank Kammer,et al.  The complexity of minimum convex coloring , 2008, Discret. Appl. Math..

[22]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[23]  Sagi Snir,et al.  Efficient Approximation of Convex Recolorings , 2005, APPROX-RANDOM.

[24]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[25]  Rolf Niedermeier,et al.  Experiments on data reduction for optimal domination in networks , 2006, Ann. Oper. Res..