Improved Exact Exponential Algorithms for Vertex Bipartization and Other Problems

We study efficient exact algorithms for several problems including Vertex Bipartization, Feedback Vertex Set, 3-Hitting Set, Max Cut in graphs with maximum degree at most 4. Our main results include: 1. an O(1.9526) 1 algorithm for Vertex Bipartization problem in undirected graphs; 2. an O(1.8384) algorithm for Vertex Bipartization problem in undirected graphs of maximum degree 3; 3. an O(1.945) algorithm for Feedback Vertex Set and Vertex Bipartization problem in undirected graphs of maximum degree 4; 4. an O(1.9799) algorithm for 4-Hitting Set problem; 5. an O(1.5541) algorithm for Feedback Arc Set in tournaments. To the best of our knowledge, these are the best known exact algorithms for these problems. In fact, these are the first known exact algorithms with the base of the exponent < 2. En route to these algorithms, we introduce two general techniques for obtaining exact algorithms. One is through parameterized complexity algorithms, and the other is a ‘colored’ branch-and-bound technique.