Linear time parameterized algorithms via skew-symmetric multicuts

Abstract

A skew-symmetric graph (D = (V, A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skew-symmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, d-Skew-Symmetric Multicut, where we are given a skew-symmetric graph D, a family of T of d-sized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different strongly connected components of D' = (V, A (X ∪ σ(X)). In this paper, we give an algorithm for d-Skew-Symmetric Multicut which runs in time O((4d)k(m+n+e)), where m is the number of arcs in the graph, n the number of vertices and e the length of the family given in the input.This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for d-Skew-Symmetric Multicut paves the way for the first linear time parameterized algorithms for several problems. We demonstrate its utility by obtaining the following linear time parameterized algorithms.• We show that Almost 2-SAT is a special case of 1-Skew-Symmetric Multicut, resulting in an algorithm for Almost 2-SAT which runs in time O(4kk4e) where k is the size of the solution and e is the length of the input formula. Then, using linear time parameter preserving reductions to Almost 2-SAT, we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4kk4(m+n)) and O(4kk5(m+n)) respectively where k is size of the solution, m and n are the number of edges and vertices respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].• We show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time O(12kk5e) where k is the size of the solution and e is the length of the input formula. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013].Using this result, we get an algorithm for Satisfiability which runs in time O(12kk5e) where k is the size of the smallest q-Horn deletion backdoor set, with e being the length of the input formula.