Parameterized Algorithms for Colored Clustering

TL;DR

This paper studies the parameterized complexity of the Colored Clustering problem, providing algorithms, kernelization results, and complexity classifications for various parameters, advancing understanding of its computational boundaries.

Contribution

It introduces a fixed-parameter algorithm, resolves an open problem by establishing a problem kernel, and classifies the problem's complexity for multiple structural parameters.

Findings

Developed a $2^{O(k)} imes n^{O(1)}$-time algorithm.

Proved the existence of an $O(k^{5/2})$-sized problem kernel.

Established W[1]-hardness for vertex cover number and tree-cut width, but fixed-parameter tractability for slim tree-cut width.

Abstract

In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a $2^{\mathcal O(k)} \cdot n^{\mathcal O(1)}$-time algorithm where $k$ is the number of edges to be selected and $n$ the number of vertices. We also prove the existence of a problem kernel of size $\mathcal O(k^{5/2} )$, resolving an open problem posed in the literature. We consider parameters that are smaller than $k$, the number of edges to be selected, and $r$, the number of edges that can be deleted. Such smaller parameters are obtained by considering the difference between $k$ or $r$ and some lower bound on these values. We give both algorithms and lower bounds for Colored Clustering with such parameterizations. Finally, we settle the parameterized complexity of Colored Clustering with respect to structural graph parameters by showing that it is $W[1]$-hard with respect to both vertex cover number and tree-cut width, but fixed-parameter tractable with respect to slim tree-cut width.