Fine-Grained Algorithm Design for Matchin

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For m-edge and n-vertex graphs, it is well-known to be solvable in O(m √ n) time; however, for several applications this running time is still too slow. Improving this worst-case upper bound resisted decades of research. In this paper we mainly focus on parameterizations of the input with respect to several kinds of distance to triviality, i.e. how far is the input from some linear-time solvable cases. Our contribution is twofold. First we focus on linear-time fixed-parameter algorithms (with low polynomial parameter dependence). To this end we develop the first linear-time algorithm for maximum matching on cocomparability graphs; this algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS) and is of independent interest. Using this algorithm we derive an O(k(n + m))-time algorithm for general graphs, where k is the vertex deletion distance to cocomparability graphs. Second we focus on linear-time kernelization. We start a deeper and systematic study of various “distance to triviality”parameters for the maximum matching problem. We design linear (and almost linear) time computable kernels of size O(k), O(k), O(k), and 2, respectively, where k is the considered parameter in each case. Investigating linear-time kernelization of a polynomial-time solvable problem, such as maximum matching, leads to a rich number of new and combinatorially interesting challenges. Based on our results, we postulate that maximum matching has the clear potential to become the “drosophila” of “FPT in P studies” analogously to the path-breaking role vertex covering played for classical FPT studies for NP-hard problems.