Using Combinatorics to Prune Search Trees: Independent and Dominating Set

We introduce a surprisingly simple technique to design and analyze algorithms based on search trees, that significantly improves many existing results in the area of exact algorithms. The technique is based on measuring the progress of Branch & Bound algorithms by making use of a combinatorial relation between the average and maximum dual degrees of a graph. By dual degree of a vertex, we mean the sum of the degrees of its neighbors and the maximum dual degree of a graph is the dual degree of a vertex that has maximum dual degree. The technique is general enough to be applied for several problems, and the algorithms based on this technique are simple and do not require extensive case analyses. Using this technique, we give the fastest known polynomial space algorithms for Maximum Independent Set, #2SAT and Minimum Dominating Set. In fact our O(1.1795n) time algorithm for Maximum Independent Set is faster than the existing exponential space algorithms. Immediate consequences of our results include improved polynomial space algorithms for Chromatic Number, (d, 2)-CSP, #1-IN-k-SAT, #Exact Cover, #Exact Hitting Set, #Weighted Set Packing and parameterized Weighted Vertex Cover among others. Department of Informatics, University of Bergen, N-5020 Bergen, Norway. [email protected], [email protected], [email protected] The Institute of Mathematical Sciences, Chennai 600 113, India. [email protected]