Exact Algorithms for Annotated Edge Dominating Set in Graphs with Degree Bounded by 3

Given a graph G = (V, E) together with a nonnegative integer requirement on vertices r : V → Z+, the annotated edge dominating set problem is to find a minimum set M ⊆ E such that, each edge in E − M is adjacent to some edge in M, and M contains at least r(v) edges incident on each vertex v ∈ V . The annotated edge dominating set problem is a natural extension of the classical edge dominating set problem, in which the requirement on vertices is zero. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the annotated edge dominating set problem in graphs with maximum degree 3 can be solved in O∗(1.2721n) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O∗(2.2306k)-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an annotated edge dominating set of size k or not. key words: edge dominating sets, exact algorithms, cubic graphs

[1]  Mingyu Xiao,et al.  New Parameterized Algorithms for the Edge Dominating Set Problem , 2011, MFCS.

[2]  Hiroshi Nagamochi,et al.  Parameterized edge dominating set in graphs with degree bounded by 3 , 2013, Theor. Comput. Sci..

[3]  Mingyu Xiao.  Exact and Parameterized Algorithms for Edge Dominating Set in 3-Degree Graphs , 2010, COCOA.

[4]  Fabrizio Grandoni,et al.  Measure and Conquer: Domination - A Case Study , 2005, ICALP.

[5]  Mihalis Yannakakis,et al.  Edge Dominating Sets in Graphs , 1980 .

[6]  David Eppstein,et al.  3-Coloring in Time O(1.3289^n) , 2000, J. Algorithms.

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

[8]  Hans L. Bodlaender,et al.  Exact Algorithms for Edge Domination , 2008, IWPEC.

[9]  Henning Fernau,et al.  Enumerate and Measure: Improving Parameter Budget Management , 2010, IPEC.

[10]  Igor Razgon.  Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3 , 2009, J. Discrete Algorithms.

[11]  Andreas Björklund,et al.  Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings , 2007, Algorithmica.

[12]  Robert E. Tarjan,et al.  Finding a Maximum Independent Set , 1976, SIAM J. Comput..

[13]  Mingyu Xiao.  A Simple and Fast Algorithm for Maximum Independent Set in 3-Degree Graphs , 2010, WALCOM.

[14]  Saket Saurabh,et al.  Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques , 2007, Theory of Computing Systems.

[15]  Hiroshi Nagamochi,et al.  A Refined Exact Algorithm for Edge Dominating Set , 2012, TAMC.

[16]  Fedor V. Fomin,et al.  Reports in Informatics , 2005 .

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

[18]  Kazuo Iwama,et al.  An Improved Exact Algorithm for Cubic Graph TSP , 2007, COCOON.

[19]  Fedor V. Fomin,et al.  On Two Techniques of Combining Branching and Treewidth , 2009, Algorithmica.

[20]  Igor Razgon,et al.  A Faster Solving of the Maximum Independent Set Problem for Graphs with Maximal Degree 3 , 2006, ACiD.

[21]  Igor Razgon.  Exact Computation of Maximum Induced Forest , 2006, SWAT.

[22]  Vangelis Th. Paschos,et al.  Maximum Independent Set in Graphs of Average Degree at Most Three in O(1.08537n){\mathcal O}(1.08537^n) , 2010, TAMC.

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

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

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

[26]  Henning Fernau,et al.  edge dominating set: Efficient Enumeration-Based Exact Algorithms , 2006, IWPEC.

[27]  Hiroshi Nagamochi,et al.  Further Improvement on Maximum Independent Set in Degree-4 Graphs , 2011, COCOA.