Finding maximum matchings in RDV graphs efficiently

TL;DR

This work studies maximum matching on RDV graphs, which are vertex-intersection graphs of downward paths in a rooted tree. It reduces matching to dynamic geometric queries that are solved via orthogonal ray shooting, enabling a maximum matching in $O(|T|+n\log n)$ time from a given RDV representation, and $O(n\log n)$ when $|T|=O(n)$. It further extends to the case where subtrees have at most $\Delta$ leaves, achieving $O(|T|+\Delta n\log n)$ time, and discusses limitations and open questions for broader chordal classes. Overall, the paper provides near-linear-time matching for RDV graphs by combining the delayed-greedy approach with efficient geometric data structures, highlighting the utility of geometric representations in graph algorithms.

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.

Figures (4)

• Figure 1: An RDV graph together with one possible RDV representation (for illustrative purposes the clique-tree is much bigger than needed). Nodes are drawn at their coordinates, and vertices are enumerated in bottom-up order. The graph is neither a circular arc graph nor a permutation graph.
• Figure 2: Mapping the vertices of the example in Figure \ref{['fig:RDV_example']} to horizontal and vertical segments (not all are shown).
• Figure 3: A directed path graph that is not strongly chordal. With the depicted bottom-up enumeration, the greedy-algorithm would choose matching $(v_5,v_2)$, $(v_6,v_1)$, $(v_8,v_3)$ and leave $v_4,v_7$ unmatched even though the graph has a matching of size 4.
• Figure 4: The graph $G$ of Figure \ref{['fig:RDV_example']} and a circular arc representation of $G\setminus \{v_2\}$.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5