Greedy Localization and Color-Coding: Improved Matching and Packing Algorithms

Matching and packing problems have formed an important class of NP-hard problems. There have been a number of recently developed techniques for parameterized algorithms for these problems, including greedy localization, color-coding plus dynamic programming, and randomized divide-and-conquer. In this paper, we provide further theoretical study on the structures of these problems, and develop improved algorithmic methods that combine existing and new techniques to obtain improved algorithms for matching and packing problems. For the 3-SET PACKING problem, we present a deterministic algorithm of time O*(4.613 k ), which significantly improves the previous best deterministic algorithm of time O* (12.8 3k For the 3-D MATCHING problem, we develop a new randomized algorithm of running time O*(2.32 3k ) and a new deterministic algorithm of running time O* (2.77 3k ). Our randomized algorithm improves the previous best randomized algorithm of running time O*(2.52 3k ), and our deterministic algorithm significantly improves the previous best deterministic algorithm of running time O* (12.8 3k ). Our results also imply improved algorithms for various triangle packing problems in graphs.

[1]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[2]  Jeanette P. Schmidt,et al.  The Spatial Complexity of Oblivious k-Probe Hash Functions , 2018, SIAM J. Comput..

[3]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[4]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[5]  Noga Alon,et al.  Color-coding , 1995, JACM.

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

[7]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[8]  Weijia Jia,et al.  Using Nondeterminism to Design Efficient Deterministic Algorithms , 2004, Algorithmica.

[9]  Michael R. Fellows,et al.  Greedy Localization, Iterative Compression, Modeled Crown Reductions: New FPT Techniques, an Improved Algorithm for Set Splitting, and a Novel 2k Kernelization for Vertex Cover , 2004, IWPEC.

[10]  Peter Shaw,et al.  Packing Edge Disjoint Triangles: A Parameterized View , 2004, IWPEC.

[11]  Dimitrios M. Thilikos,et al.  Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems , 2004, ESA.

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

[13]  Michael R. Fellows,et al.  Finding k Disjoint Triangles in an Arbitrary Graph , 2004, WG.

[14]  Ioannis Koutis A faster parameterized algorithm for set packing , 2005, Inf. Process. Lett..

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

[16]  Joachim Kneis,et al.  Divide-and-Color , 2006, WG.

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