NP-Completeness of the Combinatorial Distance Matrix Realisation Problem

TL;DR

This paper investigates the computational complexity of the combinatorial distance matrix realization problem, showing polynomial solutions for small cases and NP-completeness for larger cases, with algorithms based on 2-SAT reductions.

Contribution

It establishes the NP-completeness of the problem for k ≥ 3 and provides polynomial algorithms for k=1 and k=2, including 2-SAT based construction methods.

Findings

Polynomial algorithms for k=1 and k=2 cases.

NP-completeness for k ≥ 3 via reduction from k-colourability.

Discussion on polynomial solvability of tree realizability.

Abstract

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.