Finding maximum matchings in RDV graphs efficiently

TL;DR

This paper presents an efficient $O(n\log n)$ algorithm for finding maximum matchings in RDV graphs by reducing the problem to orthogonal ray shooting queries, avoiding explicit edge examination.

Contribution

It introduces a novel reduction of the maximum matching problem in RDV graphs to geometric intersection queries, enabling faster computation.

Findings

Maximum matching in RDV graphs can be solved in $O(n\log n)$ time.

The problem reduces to orthogonal ray shooting queries.

Efficient data structures enable this geometric approach.

Abstract

In this paper, we study the maximum matching problem in RDV graphs, i.e., graphs that are vertex-intersection graphs of downward paths in a rooted tree. We show that this problem can be reduced to a problem of testing (repeatedly) whether a vertical segment intersects one of a dynamically changing set of horizontal segments, which in turn reduces to an orthogonal ray shooting query. Using a suitable data structure, we can therefore find a maximum matching in $O(n\log n)$ time (presuming a linear-sized representation of the graph is given), i.e., without even looking at all edges.