Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

Abstract

We give substantially improved exact exponential-time algorithms for a number of NP-hard problems. These algorithms are obtained using a variety of techniques. These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique. These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented. We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time. The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems. The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results. For example, for the Minimum Maximal Matching we give an O*(1.4425ⁿ) algorithm improving the previous best result of O*(1.4422^m) [35]. For the Odd Cycle Transversal problem, we give an O*(1.62ⁿ) algorithm which improves the previous time bound of O*(1.7724ⁿ) [3].