On the Exact Matching Problem in Dense Graphs

TL;DR

This work tackles the Exact Matching problem ($Em$), seeking deterministic polynomial-time algorithms for dense graphs where randomized methods are known to exist. It introduces two complementary frameworks: a simple local-search method that is guaranteed to succeed under the path-shortening property $ extsc{Pshort}(t)$, and a Karzanov-inspired characterization extended to chain graphs, unit interval graphs, and complete $r$-partite graphs via a chord property that reduces Em to Bcpm. The paper proves $Em$ is solvable in deterministic polynomial time for several dense graph classes (complete $r$-partite, unit interval, chain graphs, graphs with bounded neighborhood diversity, and those without a complete bipartite $t$-hole), and achieves quasi-polynomial time for $G(n,p)$ with $p$ constant (notably $p=1/2$). It also identifies important limitations, such as the failure of Local Search on chain graphs and the non-satisfaction of Karzanov's property by interval graphs, and discusses weak variants that may guide future derandomization. Overall, the results advance deterministic approaches for $Em$ in dense regimes and map out clear lines for future extension and open questions, including derandomization for $G(n,1/2)$ and broader classes with the weak Karzanov property.

Abstract

In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching Problem can be solved in randomized polynomial time. Despite numerous efforts, it is still not known today whether a deterministic polynomial-time algorithm exists as well. In this paper, we make substantial progress by solving the problem for a multitude of different classes of dense graphs. We solve the Exact Matching problem in deterministic polynomial time for complete r-partite graphs, for unit interval graphs, for bipartite unit interval graphs, for graphs of bounded neighborhood diversity, for chain graphs, and for graphs without a complete bipartite t-hole. We solve the problem in quasi-polynomial time for Erdős-Rényi random graphs G(n, 1/2). We also reprove an earlier result for bounded independence number/bipartite independence number. We use two main tools to obtain these results: A local search algorithm as well as a generalization of an earlier result by Karzanov.

Figures (13)

• Figure 1: An example of the property $\textsc{Pshort}(4)$ on a path of length 11. Matching edges are bold. Both possibilities of path shortening are highlighted.
• Figure 2: An example of a chain graph $G$ and a perfect matching $M$ in $G$ (left figure), where there exists an $M$-alternating path with $n$ edges from $M$ (right figure). Edges in $M$ are bold. The property $\textsc{Pshort}(n)$ is violated on this path.
• Figure 3: An example of a $10$-cycle with three chords. The chords $\{x, y\}$ and $\{u, v\}$ are adjacent and they are both odd chords. Conversely, $\{a, b\}$ is an even chord with split $2$. The split of $\{x, y\}$ is $3$ and the split of $\{u, v\}$ is $5$.
• Figure 4: On the left we have a colored interval graph on eight vertices which does not satisfy Karzanov's property. In particular, there are PMs with $0$, $1$, $3$, and $4$ red edges but there is no PM with exactly $2$ red edges. The interval representation of the graph is given on the right. Interval $I(v)$ corresponds to vertex $v$ from the left. Note that the vertical position of the intervals is irrelevant, only the relative horizontal position of the intervals matters.
• Figure 5: By deleting the even chords from the graph in Figure \ref{['fig:counterexample_interval']}, we obtain a bipartite interval graph. It admits the same PMs as the interval graph in Figure \ref{['fig:counterexample_interval']}. Hence, this colored graph also violates Karzanov's property.

Theorems & Definitions (51)

• Definition 1
• Lemma 2
• Definition 3
• Theorem 4
• Definition 5: Karzanov's Property
• Definition 7: Odd Chord, Even Chord, Split of a Chord
• Definition 8: Adjacent Chords
• Definition 9: Chord Property
• Lemma 9: Chord Property is Sufficient
• Lemma 9: on Graphs with the Chord Property