Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG

Abstract

We give improved parameterized algorithms for two “edge” problems MAXCUT and MAXDAG, where the solution sought is a subset of edges. MAXCUT of a graph is a maximum set of edges forming a bipartite subgraph of the given graph. On the other hand, MAXDAG of a directed graph is a set of arcs of maximum size such that the graph induced on these arcs is acyclic. Our algorithms are obtained through new kernelization and efficient exact algorithms for the optimization versions of the problems. More precisely our results include: (i) a kernel with at most αk vertices and βk edges for MAXCUT. Here 0 < α ⩽ 1 and 1 < β ⩽ 2 . Values of α and β depends on the number of vertices and the edges in the graph; (ii) a kernel with at most 4 k / 3 vertices and 2 k edges for MAXDAG; (iii) an O ∗ ( 1.2418 k ) parameterized algorithm for MAXCUT in undirected graphs. This improves the O ∗ ( 1.4143 k ) 1 1 The O ∗ notation suppresses polynomial terms. Thus we write O ∗ ( T ( x ) ) for a time complexity of the form O ( T ( x ) ) ⋅ poly ( | x | ) where T ( x ) grows exponentially with | x | , the input size. See [G. Woeginger, Exact algorithms for NP-hard problems: A survey, in: Combinatorial Optimization—Eureka! You Shrink! in: Lecture Notes in Comput. Sci., vol. 2570, 2003, pp. 185–207] for a detailed discussion on this. algorithm presented in [E. Prieto, The method of extremal structure on the k-maximum cut problem, in: The Proceedings of Computing: The Australasian Theory Symposium (CATS), 2005, pp. 119–126]; (iv) an O ∗ ( 2 n ) algorithm for optimization version of MAXDAG in directed graphs. This is the first such algorithm to the best of our knowledge; (v) an O ∗ ( 2 k ) parameterized algorithm for MAXDAG in directed graphs. This improves the previous best of O ∗ ( 4 k ) presented in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458]; (vi) an O ∗ ( 16 k ) parameterized algorithm to determine whether an oriented graph having m arcs has an acyclic subgraph with at least m / 2 + k arcs. This improves the O ∗ ( 2 k ) algorithm given in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458]. In addition, we show that if a directed graph has minimum out degree at least f ( n ) (some function of n) then Directed Feedback Arc Set problem is fixed parameter tractable. The parameterized complexity of Directed Feedback Arc Set is a well-known open problem.