A novel parameterised approximation algorithm for minimum vertex cover

Parameterised approximation is a relatively new but growing field of interest. It merges two ways of dealing with NP-hard optimisation problems, namely polynomial approximation and exact parameterised (exponential-time) algorithms. We exemplify this idea by designing and analysing parameterised approximation algorithms for minimum vertex cover. More specifically, we provide a simple algorithm that works on any approximation ratio of the form 2l+1l+1, l=1,2,..., and has complexity that outperforms previously published algorithms based on sophisticated exact parameterised algorithms. In particular, for l=1 (factor-1.5 approximation) our algorithm runs in time O^*(1.0883^k), where parameter k@?23@t, and @t is the size of a minimum vertex cover. Additionally, we present an improved polynomial-time approximation algorithm for graphs of average degree at most four and a limited number of vertices with degree less than two.

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

[2]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[3]  Joachim Kneis,et al.  A Fine-grained Analysis of a Simple Independent Set Algorithm , 2009, FSTTCS.

[4]  Eran Halperin,et al.  Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs , 2000, SODA '00.

[5]  Dimitrios M. Thilikos,et al.  Kernels for the Vertex Cover Problem on the Preferred Attachment Model , 2006, WEA.

[6]  T. Gallai,et al.  Maximum-Minimum Sätze über Graphen , 1958 .

[7]  Jaikumar Radhakrishnan,et al.  Greed is good: Approximating independent sets in sparse and bounded-degree graphs , 1997, Algorithmica.

[8]  John Michael Robson,et al.  Algorithms for Maximum Independent Sets , 1986, J. Algorithms.

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

[10]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[11]  Rolf Niedermeier,et al.  On efficient fixed-parameter algorithms for weighted vertex cover , 2003, J. Algorithms.

[12]  Reuven Bar-Yehuda,et al.  One for the Price of Two: a Unified Approach for Approximating Covering Problems , 1998, Algorithmica.

[13]  Piotr Berman,et al.  On the Approximation Properties of Independent Set Problem in Degree 3 Graphs , 1999, WADS.

[14]  Ryan Williams,et al.  Confronting hardness using a hybrid approach , 2006, SODA '06.

[15]  Henning Fernau,et al.  An Exact Exponential Time Algorithm for Power Dominating Set , 2012, Algorithmica.

[16]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs , 2004, SODA '04.

[17]  Clifford Stein,et al.  Vertex Cover Approximations on Random Graphs , 2007, WEA.

[18]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs , 2005, JACM.

[19]  Fedor V. Fomin,et al.  Exact exponential algorithms , 2010, CACM.

[20]  Henning Fernau,et al.  Combining Two Worlds: Parameterised Approximation for Vertex Cover , 2010, ISAAC.

[21]  George Karakostas,et al.  A better approximation ratio for the vertex cover problem , 2005, TALG.

[22]  Vangelis Th. Paschos,et al.  Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms , 2011, Discret. Appl. Math..

[23]  Uriel Feige,et al.  Hardness results for approximating the bandwidth , 2011, J. Comput. Syst. Sci..

[24]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 2001, J. Algorithms.

[25]  Iyad Kanj,et al.  Vertex cover: exact and approximation algorithms and applications , 2001 .

[26]  Mingyu Xiao.  A Note on Vertex Cover in Graphs with Maximum Degree 3 , 2010, COCOON.

[27]  Liming Cai,et al.  On the existence of subexponential parameterized algorithms , 2003, J. Comput. Syst. Sci..

[28]  Clifford Stein,et al.  Vertex Cover Approximations: Experiments and Observations , 2005, WEA.

[29]  Dorit S. Hochbaum,et al.  Efficient bounds for the stable set, vertex cover and set packing problems , 1983, Discret. Appl. Math..

[30]  Ge Xia,et al.  Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems , 2003, Algorithmica.

[31]  Fabrizio Grandoni,et al.  Measure and conquer: a simple O(20.288n) independent set algorithm , 2006, SODA '06.

[32]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[33]  Michael R. Fellows,et al.  Parameterized Approximation via Fidelity Preserving Transformations , 2012, ICALP.

[34]  Ulrike Stege,et al.  Resolving conflicts in problems from computational biology , 1999 .

[35]  Dániel Marx,et al.  Parameterized Complexity and Approximation Algorithms , 2008, Comput. J..

[36]  Ge Xia,et al.  Improved upper bounds for vertex cover , 2010, Theor. Comput. Sci..

[37]  Vangelis Th. Paschos,et al.  New Results on Polynomial Inapproximability and Fixed Parameter Approximability of edge dominating set , 2012, IPEC.

[38]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[39]  Kord Eickmeyer,et al.  Approximation of Natural W[P]-Complete Minimisation Problems Is Hard , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[40]  Kazuo Iwama,et al.  Approximating vertex cover on dense graphs , 2005, SODA '05.

[41]  Henning Fernau,et al.  Searching Trees: An Essay , 2009, TAMC.

[42]  Rolf Niedermeier,et al.  Upper Bounds for Vertex Cover Further Improved , 1999, STACS.

[43]  Boris Konev,et al.  MAX SAT approximation beyond the limits of polynomial-time approximation , 2001, Ann. Pure Appl. Log..

[44]  Viggo Kann,et al.  Some APX-completeness results for cubic graphs , 2000, Theor. Comput. Sci..

[45]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[46]  Rolf Niedermeier,et al.  Experimental evaluation of a tree decomposition-based algorithm for vertex cover on planar graphs , 2005, Discret. Appl. Math..

[47]  Yijia Chen,et al.  On Parameterized Approximability , 2006, IWPEC.

[48]  Liming Cai,et al.  Fixed-Parameter Approximation: Conceptual Framework and Approximability Results , 2010, Algorithmica.

[49]  Fabrizio Grandoni,et al.  Refined memorization for vertex cover , 2005, Inf. Process. Lett..