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Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond

Yasuaki Kobayashi, Kazuhiro Kurita, Kunihiro Wasa

TL;DR

This work addresses the problem of enumerating large maximal common independent sets in two matroids (and, more generally, matroid matching) with a cardinality threshold $\tau$. It develops a polynomial-delay, polynomial-space enumeration framework built on reverse search and an augmenting-path characterization for matroid intersection, complemented by a flashlight-search extension approach to obtain maximum sets. The authors extend these techniques to maximal matroid matchings in tractable pairs, provide a framework for ranked enumeration, and demonstrate wide applicability to combinatorial structures such as $b$-matchings, colorful forests, and degree-constrained subgraphs, including a reduction yielding efficient enumeration of minimal connected vertex covers in subcubic graphs. Overall, the paper unifies optimization and enumeration in matroid theory, delivering practical, scalable algorithms with broad implications for related combinatorial problems.

Abstract

Finding a maximum cardinality common independent set in two matroids (also known as \textsc{Matroid Intersection}) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an ``intersection'' of these problems: Given two matroids and a threshold $τ$, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least $τ$. We show that this problem can be solved in polynomial delay and polynomial space. Moreover, our technique can be extended to a more general problem, which is relevant to Matroid Matching. We give a polynomial-delay and polynomial-space algorithm for enumerating all maximal ``matchings'' with cardinality at least $τ$, assuming that the optimization counterpart is ``tractable'' in a certain sense. This extension allows us to enumerate small minimal connected vertex covers in subcubic graphs. We also discuss a framework to convert enumeration with cardinality constraints into ranked enumeration.

Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond

TL;DR

This work addresses the problem of enumerating large maximal common independent sets in two matroids (and, more generally, matroid matching) with a cardinality threshold . It develops a polynomial-delay, polynomial-space enumeration framework built on reverse search and an augmenting-path characterization for matroid intersection, complemented by a flashlight-search extension approach to obtain maximum sets. The authors extend these techniques to maximal matroid matchings in tractable pairs, provide a framework for ranked enumeration, and demonstrate wide applicability to combinatorial structures such as -matchings, colorful forests, and degree-constrained subgraphs, including a reduction yielding efficient enumeration of minimal connected vertex covers in subcubic graphs. Overall, the paper unifies optimization and enumeration in matroid theory, delivering practical, scalable algorithms with broad implications for related combinatorial problems.

Abstract

Finding a maximum cardinality common independent set in two matroids (also known as \textsc{Matroid Intersection}) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an ``intersection'' of these problems: Given two matroids and a threshold , the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least . We show that this problem can be solved in polynomial delay and polynomial space. Moreover, our technique can be extended to a more general problem, which is relevant to Matroid Matching. We give a polynomial-delay and polynomial-space algorithm for enumerating all maximal ``matchings'' with cardinality at least , assuming that the optimization counterpart is ``tractable'' in a certain sense. This extension allows us to enumerate small minimal connected vertex covers in subcubic graphs. We also discuss a framework to convert enumeration with cardinality constraints into ranked enumeration.
Paper Structure (11 sections, 25 theorems, 21 equations, 2 figures, 2 algorithms)

This paper contains 11 sections, 25 theorems, 21 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of an algorithm for finding a maximum common independent set
  4. Enumeration of maximum common independent sets
  5. Enumeration of large maximal common independent sets
  6. Proof of Lemma \ref{['lem:+1']}
  7. Beyond common independent sets in two matroids: matroid matching
  8. Enumerating maximum matroid matchings
  9. Enumerating of large maximal matroid matchings
  10. A framework to convert an enumeration algorithm with cardinality constraints into a ranked enumeration algorithm
  11. Applications of our algorithms

Key Result

Proposition 1

Let $I_1$ and $I_2$ be independent sets of $\mathbf M$ and let $\mathbf M' = (M \mathbin{\mid} (I_1 \cup I_2)) \mathbin{\slash} (I_1 \cap I_2)$. $I \subseteq I_1 \mathbin{\triangle} I_2$ is independent in $\mathbf M'$ if and only if $I \cup (I_1 \cap I_2)$ is independent in $\mathbf M$.

Figures (2)

  • Figure 1: This figure depicts an example of the auxiliary graph $D_{\mathbf M_1, \mathbf M_2}(\{1,2,3\})$. Let $\mathbf M_1$ and $\mathbf M_2$ be matroids with the same ground set $\{1, \dots, 7\}$ that defined by five bases $\{1,2,3,4\}$, $\{1,2,3,5\}$, $\{1,3,5,6\}$, $\{1,2,5,6\}$, $\{1,2,5,7\}$ and six bases $\{1,2,3,6\}$, $\{1,2,3,7\}$, $\{1,2,5,6\}$, $\{1,3,5,6\}$, $\{1,2,5,7\}$, $\{2,3,4,6\}$, respectively. In this example, $D_{\mathbf M_1, \mathbf M_2}(\{1,2,3\})$ has a directed $s$-$t$ path $P = (s, 5, 2, 6, t)$ without shortcuts and $\{1,3,5,6\}$ is a common independent set of $\mathbf M_1$ and $\mathbf M_2$.
  • Figure 2: The construction of a subcubic graph $G'$ from a subcubic graph $G$. The set of non-separating independent sets of $G$ corresponds to the set of maximal cographic matroid matching of $G'$.

Theorems & Definitions (43)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7: Corollary 3.2 in Lawler1975
  • Definition 8
  • ...and 33 more