Combining Two Worlds: Parameterised Approximation for Vertex Cover

We explore opportunities for parameterising constant factor approximation algorithms for vertex cover. We provide a simple algorithm that works on any approximation ratio of the form \(\frac {2l+1}{l+1}\) and has complexity that outperforms an algorithm by Bourgeois et al. derived from a sophisticated exact parameterised algorithm. In particular, for l = 1 (factor 1.5 approximation) our algorithm runs in time \(\mathcal{O}^{*}(1.09^{k})\). Additionally, we present an improved polynomial-time approximation algorithm for graphs of average degree four.

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

[2]  Niklaus Wirth,et al.  Algorithms and Data Structures , 1989, Lecture Notes in Computer Science.

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

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

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

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

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

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

[9]  Ge Xia,et al.  Improved Parameterized Upper Bounds for Vertex Cover , 2006, MFCS.

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

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

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

[13]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 1999, J. Algorithms.

[14]  Rastislav Královič,et al.  Mathematical Foundations of Computer Science 2006, 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Proceedings , 2006, MFCS.

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

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

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

[18]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

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

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

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

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

[23]  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..

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

[25]  Vangelis Th. Paschos,et al.  Efficient Approximation of Combinatorial Problems by Moderately Exponential Algorithms , 2009, WADS.

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

[27]  Michael A. Langston,et al.  Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings , 2006, IWPEC.