NP-Completeness of the Combinatorial Distance Matrix Realisation Problem
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929393418371072 |
|---|---|
| author | Fairbairn, David L. Mertzios, George B. Peyerimhoff, Norbert |
| author_facet | Fairbairn, David L. Mertzios, George B. Peyerimhoff, Norbert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14729 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | NP-Completeness of the Combinatorial Distance Matrix Realisation Problem Fairbairn, David L. Mertzios, George B. Peyerimhoff, Norbert Data Structures and Algorithms Computational Complexity Discrete Mathematics F.2.2; G.2.2 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. |
| title | NP-Completeness of the Combinatorial Distance Matrix Realisation Problem |
| topic | Data Structures and Algorithms Computational Complexity Discrete Mathematics F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2406.14729 |