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 and later by Goldberg and Karzanov. In this article, we introduce a separation problem, d -S kew -S ymmetric M ulticut , where we are given a skew-symmetric graph D , a family τ of d -size subsets of vertices, and an integer k . The objective is to decide whether there is a set X ⊑ A of k arcs such that every set J in the family has a vertex υ such that υ and σ(υ) are in different strongly connected components of D ′=( V ,A \ ( X ∪ σ( X )). In this work, we give an algorithm for d -S kew -S ymmetric M ulticut that runs in time O ((4 d ) k ( m + n +ℓ)), where m is the number of arcs in the graph, n is the number of vertices, and ℓ is the length of the family given in the input. This problem, apart from being independently interesting, also captures the main combinatorial difficulty of numerous classical problems. Our algorithm for d -S kew -S ymmetric M ulticut 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 A lmost 2-SAT is a special case of 1-S kew -S ymmetric M ulticut , resulting in an algorithm for A lmost 2-SAT that runs in time O (4 k k 4 ℓ), where k is the size of the solution and ℓ is the length of the input formula. Then, using linear-time parameter-preserving reductions to A lmost 2-SAT, we obtain algorithms for O dd C ycle T ransversal and E dge B ipartization that run in time O (4 k k 4 ( m + n )) and O (4 k k 5 ( m + n )), respectively, where k is the size of the solution, and m and n are the number of edges and vertices respectively. This resolves an open problem posed by Reed et al. and improves on the earlier almost-linear-time algorithm of Kawarabayashi and Reed. — We show that D eletion q-Horn B ackdoor S et D etection is a special case of 3-S kew -S ymmetric M ulticut , giving us an algorithm for D eletion q-Horn B ackdoor S et D etection that runs in time O (12 k k 5 ℓ), where k is the size of the solution and ℓ is the length of the input formula. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a work by Narayanaswamy et al. Using this result, we get an algorithm for S atisfiability that runs in time O (12 k k 5 ℓ), where k is the size of the smallest q-Horn deletion backdoor set, with ℓ being the length of the input formula.