Basis sequence reconfiguration in the union of matroids

TL;DR

This work extends Spanning Tree Reconfiguration to Basis Sequence Reconfiguration, addressing the problem for sequences of matroids and two feasible basis sequences. It develops a polynomial-time algorithm, grounded in the exchangeability graph and a coloops-based criterion, to decide reconfigurability and to construct a reconfiguration sequence when feasible, under a basis-oracle input model. The authors also prove that finding the shortest reconfiguration sequence is NP-hard to approximate within a factor $c\log n$ via a Set Cover reduction using two partition matroids, highlighting a clear separation between decision and optimization variants. The study links reconfiguration problems with matroid theory and demonstrates foundational tools, such as tadpole-walks, with potential implications for multisolution reconfiguration in graphs and beyond.

Abstract

Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from $T$ to $T'$ such that all intermediates are also spanning trees of $G$, by exchanging an edge in $T$ with an edge outside $T$ at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of $T$ and $T'$. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of $c \log n$ for some constant $c > 0$, where $n$ is the total size of the ground sets of the input matroids.

Figures (5)

• Figure 1: The figure illustrates an instance in which a pair of edge-disjoint spanning trees (a) cannot be transformed into the other pair (b), where the spanning trees are indicated by dashed blue lines and solid red lines.
• Figure 2: A tadpole-walk starting from $x_0$ and ending at $x_n$.
• Figure 3: The figure depicts tadpole-walks (with shortcuts) and their updated tadpole-walks.
• Figure 4: The bold red walk in the upper digraph represents $W_p$ for $p = n-1$, and that in the lower digraph for $p = n$.
• Figure 5: The figure depicts (hypergraph representations of) two partition matroids $M_1$ and $M_2$. A set $S \in \mathcal{S}$ contains three elements $u, v, w \in U$ with $f({u},{S}) = 3$, $f({v},{S})= 2$, and $f({w},{S}) = 4$. Solid black circles represent elements in ${B}^\mathtt{s}_1$, and solid red circles represent elements in ${B}^\mathtt{s}_2$.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4: e.g., Murota2010-ar
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 8
• proof