Exact Matching and Top-k Perfect Matching Parameterized by Neighborhood Diversity or Bandwidth

TL;DR

This work investigates Exact Matching (EM) and Top‑k Perfect Matching (TkPM) in red/blue edge‑colored graphs, focusing on blown‑up (prototype) graphs. It delivers a family of results: (i) a parameterized, exact TkPM algorithm for graphs with bounded neighborhood diversity $\gamma$ running in time $\mathcal{O}\big(\binom{2k+\gamma-1}{\gamma-1} f(n)\big)$ by reducing to a Type‑Constrained Maximum Weight Matching, (ii) a $(1-\varepsilon)$‑approximation scheme with runtime $\mathcal{O}\left(\left(\frac{\log_2 k}{\log_2(1/(1-\varepsilon))}\right)^{\gamma^2} f(n)\right)$, and (iii) a recursive, separator‑based subexponential algorithm for TkPM on blowups of bounded‑bandwidth prototypes, achieving $2^{\mathcal{O}(\phi^2 \sqrt{n} \log^2 n)}$; the same framework adapts to EM with a refined balance parameter, yielding $2^{\mathcal{O}(\phi^2 n^{12/13} \log^2 n)}$ under a bandwidth bound. These results leverage many disjoint separators to guide a DP‑style decomposition and provide concrete progress toward understanding P vs RP for these problems in structured graph classes.

Abstract

The Exact Matching (EM) problem asks whether there exists a perfect matching which uses a prescribed number of red edges in a red/blue edge-colored graph. While there exists a randomized polynomial-time algorithm for the problem, only some special cases admit a deterministic one so far, making it a natural candidate for testing the P=RP hypothesis. A polynomial-time equivalent problem, Top-k Perfect Matching (TkPM), asks for a perfect matching maximizing the weight of the $k$ heaviest edges. We study the above problems, mainly the latter, in the scenario where the input is a blown-up graph, meaning a graph which had its vertices replaced by cliques or independent sets. We describe an FPT algorithm for TkPM parameterized by $k$ and the neighborhood diversity of the input graph, which is essentially the size of the graph before the blow-up; this graph is also called the prototype. We extend this algorithm into an approximation scheme with a much softer dependency on the aforementioned parameters, time-complexity wise. Moreover, for prototypes with bounded bandwidth but unbounded size, we develop a recursive algorithm that runs in subexponential time. Utilizing another algorithm for EM on bounded neighborhood diversity graphs, we adapt this recursive subexponential algorithm to EM. Our approach is similar to the use of dynamic programming on e.g. bounded treewidth instances for various problems. The main point is that the existence of many disjoint separators is utilized to avoid including in the separator any of a set of ``bad'' vertices during the split phase.

Figures (4)

• Figure 1: The graph used for the reduction of TC-MWM to MWPM.
• Figure 2: A case where a small separator (here $\{V_1\}$) in the prototype blows up into a big separator in the graph, both in terms of vertices and adjacent edges.
• Figure 3: An illustration of loose/tight bands. Loose bands (blue) contain less than $\sqrt{n}$ edges of the solution PM $M$ and tight ones (red) contain at least that many. The set $\{V_1\}$ is a loose separator (see definition \ref{['def:loose_separator']}), since it splits the prototype in a balanced way, contains at most $\phi$ blobs and is only adjacent to loose bands.
• Figure 4: An illustration of the recursive solution to the problem. The complete edge set $E'$ (in yellow) leaving blobs of the loose separator $\mathcal{S}$ is guessed. Then, the problem decomposes into the matchings $M_1, M_2$ of the smaller subproblems (in orange) as well as the matching $M_{blobs}$ which is comprised of edges which stay within a single blob of $\mathcal{S}$ (in pink).

Theorems & Definitions (27)

• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• definition thmcounterdefinition: Neighborhood diversity
• definition thmcounterdefinition
• theorem thmcountertheorem