Exact Matching and Top-k Perfect Matching Parameterized by Neighborhood Diversity or Bandwidth

Abstract: The Exact Matching (EM) problem asks whether there exists a perfect matching which uses a prescribed number of red edges in a red/blue edge-colored graph. While there exists a randomized polynomial-time algorithm for the problem, only some special cases admit a deterministic one so far, making it a natural candidate for testing the P=RP hypothesis. A polynomial-time equivalent problem, Top-k Perfect Matching (TkPM), asks for a perfect matching maximizing the weight of the $k$ heaviest edges. We study the above problems, mainly the latter, in the scenario where the input is a blown-up graph, meaning a graph which had its vertices replaced by cliques or independent sets. We describe an FPT algorithm for TkPM parameterized by $k$ and the neighborhood diversity of the input graph, which is essentially the size of the graph before the blow-up; this graph is also called the prototype. We extend this algorithm into an approximation scheme with a much softer dependency on the aforementioned parameters, time-complexity wise. Moreover, for prototypes with bounded bandwidth but unbounded size, we develop a recursive algorithm that runs in subexponential time. Utilizing another algorithm for EM on bounded neighborhood diversity graphs, we adapt this recursive subexponential algorithm to EM. Our approach is similar to the use of dynamic programming on e.g. bounded treewidth instances for various problems. The main point is that the existence of many disjoint separators is utilized to avoid including in the separator any of a set of ``bad'' vertices during the split phase.