### Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts [article]

M. S. Ramanujan, Saket Saurabh
2013 arXiv   pre-print
A skew-symmetric graph (D=(V,A),σ) is a directed graph D with an involution σ on the set of vertices and arcs. 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 connected components of D'=(V,A∖ (X∪σ(X)). In this
more » ... r, we give an algorithm for this problem which runs in time O((4d)^k(m+n+ℓ)), where m is the number of arcs in the graph, n the number of vertices and ℓ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time O(4^kk^4ℓ) and we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4^kk^4(m+n)) and O(4^kk^5(m+n)) 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 also 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(12^kk^5ℓ). 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(12^kk^5ℓ) where k is the size of the smallest q-Horn deletion backdoor set, with ℓ being the length of the input formula.