The metric Menger problem

Abstract

We study a generalization of the well-known disjoint paths problem which we call the metric Menger problem, denoted MM(r,k), where one is given two subsets of a graph and must decide whether they can be connected by $k$ paths of pairwise distance at least $r$. We prove that this problem is NP-complete for every $r\geq 3$ and $k\geq 2$ by giving a reduction from 3SAT. This resolves a conjecture recently stated by Georgakopoulos and Papasoglu. On the other hand, we show that the problem is in XP when parameterised by treewidth and maximum degree by observing that it is `locally checkable'. In the case $r\leq 3$, we prove that it suffices to parameterise by treewidth. We also state some open questions relating to this work.

Figures (2)

• Figure 1: An example of the reduction from $3\SAT$ to $\MM(3,2)$. The graph constructed above encodes the formula $(x_1 \vee \neg x_2 \vee x_3) \wedge (\neg x_1 \vee \neg x_2 \vee x_4) \wedge (x_2 \vee \neg x_3 \vee \neg x_4)$. For better overview, the dummy paths beginning in vertices $a_4, \dots, a_9, b_4, \dots, b_9$ are omitted.
• Figure 2: An example of the transformation from $3\SAT$ to $\MM(4,2)$ with an input graph of maximum degree three.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Lemma 1.1
• proof
• Theorem 2
• proof
• Definition 2.1: Simple $m$-locally checkable problem
• Theorem 2.2: bonomo2022new
• Lemma 2.3
• proof