A Parameterized Perspective on Packing Paths of Length Two

We study (vertex-disjoint) packings of paths of length two (i.e., of P 2 's) in graphs under a parameterized perspective. Starting from a maximal P 2 -packing $\mathcal {P}$ of size jwe use extremal combinatorial arguments for determining how many vertices of $\mathcal {P}$ appear in some P 2 -packing of size (j+ 1) (if it exists). We prove that one can 'reuse' 2.5jvertices. Based on a WIN-WIN approach, we build an algorithm which decides if a P 2 -packing of size at least kexists in a given graph in time ${\mathcal{O}}^*(2.482^{3k})$.

[1]  Jianer Chen,et al.  Improved algorithms for path, matching, and packing problems , 2007, SODA '07.

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

[3]  Weijia Jia,et al.  An efficient parameterized algorithm for m-set packing , 2004, J. Algorithms.

[4]  Pawel Zylinski,et al.  Packing Three-Vertex Paths in a Subcubic Graph , 2005 .

[5]  R. Ravi,et al.  Approximation algorithms for the test cover problem , 2003, Math. Program..

[6]  Michael R. Fellows,et al.  FPT is P-Time Extremal Structure I , 2005, ACiD.

[7]  Henning Fernau,et al.  2 Contents , 1996 .

[8]  Hannes Moser,et al.  A Problem Kernelization for Graph Packing , 2009, SOFSEM.

[9]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[10]  Henning Fernau,et al.  NONBLOCKER: Parameterized Algorithmics for minimum dominating set , 2006, SOFSEM.

[11]  Pablo Moscato,et al.  The k-FEATURE SET problem is W[2]-complete , 2003, J. Comput. Syst. Sci..

[12]  Henning Fernau,et al.  A Parameterized Perspective on P 2 -Packings , 2008 .

[13]  Refael Hassin,et al.  Approximation algorithms for some vehicle routing problems , 2005, Discret. Appl. Math..

[14]  David G. Kirkpatrick,et al.  On the completeness of a generalized matching problem , 1978, STOC.

[15]  Ge Xia,et al.  Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size , 2005, SIAM J. Comput..

[16]  Jianer Chen,et al.  An Improved Parameterized Algorithm for a Generalized Matching Problem , 2008, TAMC.

[17]  Christian Sloper,et al.  Either/Or: Using Vertex Cover Structure in Designing FPT-Algorithms - The Case of k-Internal Spanning Tree , 2003, WADS.

[18]  Refael Hassin,et al.  An Approximation Algorithm for Maximum Packing of 3-Edge Paths , 1997, Inf. Process. Lett..

[19]  Ján Plesník,et al.  Equivalence between the minimum covering problem and the maximum matching problem , 1984, Discret. Math..

[20]  Refael Hassin,et al.  An approximation algorithm for maximum triangle packing , 2006, Discret. Appl. Math..

[21]  Ioannis Koutis,et al.  Faster Algebraic Algorithms for Path and Packing Problems , 2008, ICALP.

[22]  Venkatesh Raman,et al.  Parameterized complexity of finding subgraphs with hereditary properties , 2000, Theor. Comput. Sci..

[23]  Michael R. Fellows,et al.  Parameterized Complexity: The Main Ideas and Connections to Practical Computing , 2000, Experimental Algorithmics.

[24]  Christian Sloper,et al.  Looking at the stars , 2004, Theor. Comput. Sci..

[25]  Jianer Chen,et al.  Greedy Localization and Color-Coding: Improved Matching and Packing Algorithms , 2006, IWPEC.

[26]  Jianxin Wang,et al.  An O*(3.523k) Parameterized Algorithm for 3-Set Packing , 2008, TAMC.

[27]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[28]  David Manlove,et al.  Vertex and Edge Covers with Clustering Properties: Complexity and Algorithms , 2009, ACiD.