Deterministic branching algorithms for parameterized Co-Path/Cycle Packing and three variants

Abstract

The Co-Path/Cycle Packing problem that tries to find a set of vertices that, when removed, leaves a graph of maximum degree 2 is a prominent problem in the graph theory field. The related Vertex Cover problem, which finds a deletion set where the remaining graph has maximum degree 0, is one of the most famous graph theory problems. In this thesis we describe a deterministic parameterized algorithm for Co-Path/Cycle Packing which uses branch-and-bound techniques. This algorithm is shown to have a time complexity of O*(3.0607^k), which improves upon the previous best known deterministic bound. A new problem which looks for a deletion set such that the remaining graph is 2-regular is also discussed, and a branching algorithm with time-bound O*(3^k) is shown for it. Additionally, two variants of these two problems that add the requirement that the remaining graph is a single connected component are introduced and shown to both have an algorithm that runs in O(2^k * n^3) time. For the three new problems, NP-completeness is also proven.