An O(n(log n)³) algorithm for maximum matching in trapezoid graphs

摘要

Trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. Many graph problems that are NP-hard in general case have polynomial time algorithms for trapezoid graphs. A matching in a graph is a set of pairwise non-adjacent edges, and a maximum matching is a matching whose cardinality is maximum. In this paper, we define a modified range tree data structure, called S-Range tree, which allows to report the maximum label of points in a rectangular region and update the label of a point efficiently. We use this data structure to construct an O(n(log n)³) algorithm for finding a maximum matching in trapezoid graphs based on their box representation. In addition, we generalize this algorithm for a larger graph class, k-trapezoid graph by using multidimensional range tree. To the best of our knowledge, this is the first efficient maximum matching algorithm for trapezoid graphs.