David L. Fairbairn, George B. Mertzios, Norbert Peyerimhoff
{"title":"组合距离矩阵实现问题的 NP 完备性","authors":"David L. Fairbairn, George B. Mertzios, Norbert Peyerimhoff","doi":"arxiv-2406.14729","DOIUrl":null,"url":null,"abstract":"The $k$-CombDMR problem is that of determining whether an $n \\times n$\ndistance matrix can be realised by $n$ vertices in some undirected graph with\n$n + k$ vertices. This problem has a simple solution in the case $k=0$. In this\npaper we show that this problem is polynomial time solvable for $k=1$ and\n$k=2$. Moreover, we provide algorithms to construct such graph realisations by\nsolving appropriate 2-SAT instances. In the case where $k \\geq 3$, this problem\nis NP-complete. We show this by a reduction of the $k$-colourability problem to\nthe $k$-CombDMR problem. Finally, we discuss the simpler polynomial time\nsolvable problem of tree realisability for a given distance matrix.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"75 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"NP-Completeness of the Combinatorial Distance Matrix Realisation Problem\",\"authors\":\"David L. Fairbairn, George B. Mertzios, Norbert Peyerimhoff\",\"doi\":\"arxiv-2406.14729\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The $k$-CombDMR problem is that of determining whether an $n \\\\times n$\\ndistance matrix can be realised by $n$ vertices in some undirected graph with\\n$n + k$ vertices. This problem has a simple solution in the case $k=0$. In this\\npaper we show that this problem is polynomial time solvable for $k=1$ and\\n$k=2$. Moreover, we provide algorithms to construct such graph realisations by\\nsolving appropriate 2-SAT instances. In the case where $k \\\\geq 3$, this problem\\nis NP-complete. We show this by a reduction of the $k$-colourability problem to\\nthe $k$-CombDMR problem. Finally, we discuss the simpler polynomial time\\nsolvable problem of tree realisability for a given distance matrix.\",\"PeriodicalId\":501216,\"journal\":{\"name\":\"arXiv - CS - Discrete Mathematics\",\"volume\":\"75 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2406.14729\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.14729","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
NP-Completeness of the Combinatorial Distance Matrix Realisation Problem
The $k$-CombDMR problem is that of determining whether an $n \times n$
distance matrix can be realised by $n$ vertices in some undirected graph with
$n + k$ vertices. This problem has a simple solution in the case $k=0$. In this
paper we show that this problem is polynomial time solvable for $k=1$ and
$k=2$. Moreover, we provide algorithms to construct such graph realisations by
solving appropriate 2-SAT instances. In the case where $k \geq 3$, this problem
is NP-complete. We show this by a reduction of the $k$-colourability problem to
the $k$-CombDMR problem. Finally, we discuss the simpler polynomial time
solvable problem of tree realisability for a given distance matrix.