Complexity of Local Search for Euclidean Clustering Problems

TL;DR

It is shown that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete, even when the edge weights are given by the Euclidean distances between the points in some set.

Abstract

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k = 2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $\mathcal{X} \subseteq \mathbb{R}^d$; a problem which is equivalent to Min Sum 2-Clustering.

Figures (4)

• Figure 1: Graph of the PLS-reductions used in this paper. Reductions represented by solid lines are tight, reductions represented by dashed lines are not.
• Figure 2: Schematic overview of the reduction used in the proof of \ref{['lemma:halfposnaesat']}. On the left we have a vertex $v \in V$ and its neighbors $\{u_1, u_2, u_3\}$ in a Max Cut instance, with weights on the edges between $v$ and its neighbors. The $\operatorname{NAE}$ clauses on the right are the clauses constructed from $v$. In the actual reduction, these clauses are added for all level 3 variables. The right-most column shows the weights assigned to the clauses. The middle column shows for the level 3 clauses which subset of $N(v)$ corresponds to which clause. The constants $L$ and $M$ are chosen so that $1 \ll L \ll M$.
• Figure 3: Schematic overview of the reduction used in the proof of \ref{['lemma:bisection_to_densestcut']}. On the left side we have an instance of Odd Max Bisection/Flip and on the right side we have the corresponding instance of Densest Cut/Flip. The edges inside of $G$ together with their weights are not depicted but are identical in both instances. Let $(A,B)$ be the partition corresponding to some locally optimal solution of the Densest Cut/Flip instance. Then, $A$ contains exactly one endpoint of each of the $n^4$ isolated edges and $|\, |A\cap V(G)|-|B\cap V(G)|\, |=1$.
• Figure 4: A simple Densest Cut instance. The matrix $D$ on the right is constructed from the graph on the left in the proof of \ref{['lemma:densestcut_to_hartigan']}. The rows of $D$ correspond to points in an instance of 2-Means/Flip. The actual instance of 2-Means does not use these points directly, as we only have to encode the weights, which are squared distances and integral in this reduction.

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Definition 3
• Lemma 4
• proof
• Lemma 5
• proof