NP-Completeness of the Combinatorial Distance Matrix Realisation Problem

TL;DR

The paper studies the problem $k$-CombDMR: given an $n\times n$ distance matrix $D$, can one realise it with an unweighted graph on at most $n+k$ vertices? It provides polynomial-time algorithms for $k\in\{0,1,2\}$ using structured 2-SAT formulations and constructive realisations, and proves NP-completeness for every fixed $k\ge3$ via a gadget-based reduction from $k$-colourability. It also connects tree realisations to the minimum weighted tree realisation problem, yielding an $O(n^2)$ method to decide and construct tree realisations when possible. Together, these results delineate a clear boundary between tractable and intractable instances and offer practical procedures for small $k$ and tree cases.

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.

Figures (5)

• Figure 1: Two graph realisations of the above matrix $D$, where $\Phi(i) = v_i$ for $i \in [3]$, while $k=3$ and $k=1$ in the left and the right realisation, respectively. The right realisation is a minimum.
• Figure 2: Minimum graph realisations of $D$ as in \ref{['eq:drpcomp1']}, for $k$-CombDMR (Left) and for the weighted graph realisation problem (Right).
• Figure 3: Optimum graph realisations of $D$ as in \ref{['eq:drpcomp2']}, with $\Phi(i)=v_i, i \in [8]$ for the weighted graph realisation problem with integer weights $w(e) = 1$ (Left and Right). Only the right graph realisation is optimum for the combinatorial distance realisation problem.
• Figure 4: Example of construction of a gadget graph $G_g$ from an input graph $G_c$ as in Algorithm \ref{['alg:k3redution']} with old vertices $v_1,\dots,v_4$ (i.e., $n_c = 4$) and new vertices $v_5,\dots,v_{15}$ (i.e., $n_g = 15$).
• Figure 5: A graph realisation of $D$ constructed from the example in Figure \ref{['fig:k3reduction']} with $k=3$. In accordance with the proof of Proposition \ref{['prop:k3redutionforward']}, the vertex $v_{1}$ inherits the colour of vertex $v_{n+2}$, the vertices $v_{2}$ and $v_{4}$ inherit the colour of vertex $v_{n+2}$, and the vertex $v_{3}$ inherits the colour of vertex $v_{n+3}$.

Theorems & Definitions (60)

• Example 2
• Lemma 3
• proof
• Definition 4: Distance matrix
• Proposition 5
• proof
• Proposition 6
• Definition 7: $q$-skeleton
• Lemma 8
• proof