Hierarchical b-Matching

TL;DR

This paper introduces a polynomial-time algorithm for Hierarchical b-Matching, an extension of b-matching where vertices are organized in multiple hierarchical levels with bounds at each level, enabling complex structured matchings.

Contribution

It presents the first polynomial-time algorithm for Hierarchical b-Matching, handling any number of hierarchical levels in the problem.

Findings

Algorithm works efficiently for any number of hierarchy levels.

Successfully extends b-matching to hierarchical structures.

Provides a practical solution for complex structured matching problems.

Abstract

A matching of a graph is a subset of edges no two of which share a common vertex, and a maximum matching is a matching of maximum cardinality. In a $b$-matching every vertex $v$ has an associated bound $b_v$, and a maximum $b$-matching is a maximum set of edges, such that every vertex $v$ appears in at most $b_v$ of them. We study an extension of this problem, termed {\em Hierarchical b-Matching}. In this extension, the vertices are arranged in a hierarchical manner. At the first level the vertices are partitioned into disjoint subsets, with a given bound for each subset. At the second level the set of these subsets is again partitioned into disjoint subsets, with a given bound for each subset, and so on. In an {\em Hierarchical b-matching} we look for a maximum set of edges, that will obey all bounds (that is, no vertex $v$ participates in more than $b_v$ edges, then all the vertices in one subset do not participate in more that that subset's bound of edges, and so on hierarchically). We propose a polynomial-time algorithm for this new problem, that works for any number of levels of this hierarchical structure.