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The Euclidean $k$-Matching Problem is NP-hard

José-Miguel Díaz-Báñez, Ruy Fabila-Monroy, José-Manuel Higes-López, Nestaly Marín, Miguel-Angel Pérez-Cutiño, Pablo Pérez-Lantero

TL;DR

This work proves that the Euclidean $k$-matching problem is $NP$-hard for every fixed $k\ge 3$, resolving a long-standing open question. The authors build a layered reduction from Cubic Planar Monotone $1$-in-$3$ SAT, using grid-embedded SAT graphs to instantiate geometric gadgets that enforce SAT constraints via minimum-spanning-tree costs on $k$-subsets. The core innovation is a charge-propagation framework based on $k$-minos that generalizes the $3$-matching construction to arbitrary $k$, accompanied by a suite of gadgets (variable, wire, clause, switches, amplifiers, junctions, XOR filters/enforcers, and $\Delta$-networks) to enforce the desired truth assignments through geometric constraints. The results further extend to the path variant (E$k$-MPP) by a careful geometric modification that guarantees path-shaped trees, preserving the hardness and opening avenues for future approximation approaches in Euclidean clustering-like problems.

Abstract

Let $G$ be a complete edge-weighted graph on $n$ vertices. To each subset of vertices of $G$ assign the cost of the minimum spanning tree of the subset as its weight. Suppose that $n$ is a multiple of some fixed positive integer $k$. The $k$-matching problem is the problem of finding a partition of the vertices of $G$ into $k$-sets, that minimizes the sum of the weights of the $k$-sets. The case $k=3$ has been shown to be NP-hard [Johnsson et al.,1998]. In the Euclidean version, the vertices of $G$ are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean $k$-matching problem. We show that, for every fixed $k \ge 3$, the Euclidean $k$-matching is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case $k=3$. We also show that the problem remains NP-hard if the trees are required to be paths.

The Euclidean $k$-Matching Problem is NP-hard

TL;DR

This work proves that the Euclidean -matching problem is -hard for every fixed , resolving a long-standing open question. The authors build a layered reduction from Cubic Planar Monotone -in- SAT, using grid-embedded SAT graphs to instantiate geometric gadgets that enforce SAT constraints via minimum-spanning-tree costs on -subsets. The core innovation is a charge-propagation framework based on -minos that generalizes the -matching construction to arbitrary , accompanied by a suite of gadgets (variable, wire, clause, switches, amplifiers, junctions, XOR filters/enforcers, and -networks) to enforce the desired truth assignments through geometric constraints. The results further extend to the path variant (E-MPP) by a careful geometric modification that guarantees path-shaped trees, preserving the hardness and opening avenues for future approximation approaches in Euclidean clustering-like problems.

Abstract

Let be a complete edge-weighted graph on vertices. To each subset of vertices of assign the cost of the minimum spanning tree of the subset as its weight. Suppose that is a multiple of some fixed positive integer . The -matching problem is the problem of finding a partition of the vertices of into -sets, that minimizes the sum of the weights of the -sets. The case has been shown to be NP-hard [Johnsson et al.,1998]. In the Euclidean version, the vertices of are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean -matching problem. We show that, for every fixed , the Euclidean -matching is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case . We also show that the problem remains NP-hard if the trees are required to be paths.

Paper Structure

This paper contains 7 sections, 7 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Cubic Planar Monotone 1-in-3 SAT
  3. Reduction to the Euclidean 3-matching Problem
  4. The Euclidean $k$-Matching Problem
  5. Reduction to the Euclidean $k$-matching Problem
  6. The Euclidean $k$-path matching Problem
  7. Conclusions

Figures (18)

  • Figure 1: The graph $G_\psi$, and a grid embedding $D_\psi$, with respect to the formula $\psi:= \overbrace{(x_1 \lor x_2 \lor x_4)}^{C_1} \land ( \overbrace{\overline{x_1} \lor \overline{x_2} \lor \overline{x_3})}^{C_2} \land \overbrace{(x_2 \lor x_3 \lor x_4)}^{C_3} \land \overbrace{(\overline{x_1} \lor \overline{x_3} \lor \overline{x_4})}^{C_4}$
  • Figure 2: The variable gadget
  • Figure 3: Clause gadgets
  • Figure 4: The grid embedding $D_\psi$ associated to the formula $\psi:= \overbrace{(x_1 \lor x_2 \lor x_4)}^{C_1} \land ( \overbrace{\overline{x_1} \lor \overline{x_2} \lor \overline{x_3})}^{C_2} \land \overbrace{(x_2 \lor x_3 \lor x_4)}^{C_3} \land \overbrace{(\overline{x_1} \lor \overline{x_3} \lor \overline{x_4})}^{C_4}$
  • Figure 5: A wire, for $k=5$ carrying a charge of $3$ from left to right and a charge of $2$ from right to left.
  • ...and 13 more figures