New Algorithms and Hardness Results for Connected Clustering

TL;DR

This work provides exact algorithms that run in polynomial time if the treewidth $w$ is a constant and obtains constant approximation algorithms that run in FPT time with respect to the parameter $\max(w,k)$.

Abstract

Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$ vertices into a given number $k$ of clusters such that every cluster forms a connected subgraph of $G$ and a given clustering objective gets minimized. The constraint that the clusters are connected has applications in many different fields, like for example community detection and geodesy. So far, $k$-center and $k$-median have been studied in this setting. It has been shown that connected $k$-median is $Ω(n^{1- ε})$-hard to approximate which also carries over to the connected $k$-means problem, while for connected $k$-center it remained an open question whether one can find a constant approximation in polynomial time. We answer this question by providing an $Ω(\log^*(k))$-hardness result for the problem. Given these hardness results, we study the problems on graphs with bounded treewidth. We provide exact algorithms that run in polynomial time if the treewidth $w$ is a constant. Furthermore, we obtain constant approximation algorithms that run in FPT time with respect to the parameter $\max(w,k)$. Additionally, we consider the min-sum-radii (MSR) and min-sum-diameter (MSD) objective. We prove that on general graphs connected MSR can be approximated with an approximation factor of $(3 + ε)$ and connected MSD with an approximation factor of $(4 + ε)$. The latter also directly improves the best known approximation guarantee for unconstrained MSD from $(6 + ε)$ to $(4 + ε)$.

Figures (7)

• Figure 1: A solution for the graph $G_t$ fulfilling the specification $(a,\{U_1,U_2,U_3\})$ where $a(U_1) = a(U_3) = c_1$ and $a(U_2) = c_2$ with $c_2 \not\in V_t$. The solution requires $3$ clusters, finished ones are circled in black, the unfinished ones in the color corresponding to their assignment.
• Figure 2: A sketch of a $2$-layer subinstance included in a $3$-layer instance. Note that the number of in- and outputs is not accurate and there are further gadgets, centers and subinstances in the 3-layer instance.
• Figure 3: A depictions how a solution for $G_t$ (corresponding to a node $t$ joining two children $s_1$ and $s_2$) can be split up into two solutions for $G_{s_1}$ and $G_{s_2}$. The green ellipsoids correspond to clusters in $G_{s_1}$, the blue ones to clusters in $G_{s_2}$ and the black ones correspond to the unfinished clusters of the solution for $G_t$.
• Figure 4: The connectivity graph corresponding to the 3-SAT formula $(x_1 \lor x_2 \lor x_3) \land (\overline{x}_2 \lor \overline{x}_3)$. Green vertices are placed at position $1$, blue ones at $2$ and black ones at $\perp$.
• Figure 5: A subgraph of the connectivity graph with two layers corresponding to the 3-SAT formula $(\overline{x}_1 \lor x_2) \land (\overline{x}_2)$. Blue edges correspond to the clause $(\overline{x}_1 \lor x_2)$, green ones to the clause $(\overline{x}_2)$, magenta ones to the variable $x_1$ and red ones to the variable $x_2$. The vertices and edges in the subgraph $I(p,\{1,2\})$ (with $p \in [2]$) are for the most part not depicted. The complete connectivity graph also contains the vertices $T_3$, $F_3$, $x_{1,3}$, $\overline{x}_{1,3}$, $x_{2,3}$ and $\overline{x}_{2,3}$, as well as $8$ copies of one-layer instances connecting them with the vertices with first coordinate $1$ and $8$ copies connecting them with the vertices with first coordinate $2$.

Theorems & Definitions (67)

• Definition 1
• Definition 2: Tree Decomposition
• Definition 3: Nice Tree Decomposition
• Lemma 4: Lemma 7.4 in cygan2015parameterized
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 7
• Definition 7
• Definition 7