Half-integrality, Lp-branching and Fpt Algorithms∗

A recent trend in parameterized algorithms is the application of polytope tools to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Although this approach has yielded significant speedups for a range of important problems, it requires the underlying polytope to have very restrictive properties, including half-integrality and Nemhauser-Trotter-style persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view half-integrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of half-integral polytopes with the required properties. Our results unify and significantly extend the previously known cases, and yield a range of new and improved FPT algorithms, including an O∗(|Σ|2k)-time algorithm for node-deletion Unique Label Cover and an O∗(4k)-time algorithm for Group Feedback Vertex Set where the group is given by oracle access. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network-flow-based approach to solve several cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.