Quadratic time algorithm for maximum induced matching problem in trapezoid graphs

Maximum matching problem is one of the most fundamental and applicable problems in graph theory. Various algorithms to solve it are introduced from the mid-20th century. Recently, maximum induced matching problem, which is finding a matching of maximum size where every two distinct matches are not connected by any edge, has drawn a big attention among researchers because of its importance in marriage problems, artificial intelligence and VLSI design. First proposed by Cameron in 1989, the problem is proved to be NP-hard in general graphs. Nevertheless, a maximum induced matching can be found in polynomial time for many special graph classes such as co-comparability graphs and chordal graphs. Trapezoid graphs, which are a subclass of co-comparability graphs, arise in many applications, especially VLSI circuit design. In this paper, we design an O(n(2)) solution for finding a maximum induced matching in trapezoid graphs which are a subclass of co-comparability graphs, based on a dynamic programming method on edges with the aid of the sweep line technique on the geometry representation of trapezoid graphs. Our approach is to construct the longest chain of ordered edges starting from an arbitrary edge such that these edges form an induced matching. A sweep line moving from right to left correctly determines the order of dynamic processes. Our result is far better than the best known O(m(2)) algorithm proposed by Golumbic et al in 2000.