Basis sequence reconfiguration in the union of matroids
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 ste...
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| creator | Hanaka, Tesshu Iwamasa, Yuni Kobayashi, Yasuaki Okada, Yuto Saito, Rin |
| description | 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. |
| doi_str_mv | 10.48550/arxiv.2409.07848 |
| format | Article |
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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
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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.</description><subject>Computer Science - Discrete Mathematics</subject><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMjGw1DMwtzCx4GQwd0oszixWKE4tLE3NS05VKEpNzs9Ly0wvLUosyczPU8jMUyjJSFUozQNx8tMUchNLivIzU4p5GFjTEnOKU3mhNDeDvJtriLOHLtiK-IKizNzEosp4kFXxYKuMCasAAApjM9w</recordid><startdate>20240912</startdate><enddate>20240912</enddate><creator>Hanaka, Tesshu</creator><creator>Iwamasa, Yuni</creator><creator>Kobayashi, Yasuaki</creator><creator>Okada, Yuto</creator><creator>Saito, Rin</creator><scope>AKY</scope><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20240912</creationdate><title>Basis sequence reconfiguration in the union of matroids</title><author>Hanaka, Tesshu ; Iwamasa, Yuni ; Kobayashi, Yasuaki ; Okada, Yuto ; Saito, Rin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2409_078483</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Computer Science - Discrete Mathematics</topic><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Hanaka, Tesshu</creatorcontrib><creatorcontrib>Iwamasa, Yuni</creatorcontrib><creatorcontrib>Kobayashi, Yasuaki</creatorcontrib><creatorcontrib>Okada, Yuto</creatorcontrib><creatorcontrib>Saito, Rin</creatorcontrib><collection>arXiv Computer Science</collection><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Hanaka, Tesshu</au><au>Iwamasa, Yuni</au><au>Kobayashi, Yasuaki</au><au>Okada, Yuto</au><au>Saito, Rin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Basis sequence reconfiguration in the union of matroids</atitle><date>2024-09-12</date><risdate>2024</risdate><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.</abstract><doi>10.48550/arxiv.2409.07848</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Computer Science - Discrete Mathematics Mathematics - Combinatorics |
| title | Basis sequence reconfiguration in the union of matroids |
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