Parameterized Complexity of Graph Partitioning into Connected Clusters

TL;DR

This paper investigates the computational complexity of partitioning graphs into connected clusters with specified sizes, providing hardness results and fixed parameter tractable algorithms for various graph classes and parameters.

Contribution

It establishes W[1]-hardness for balanced connected q-partition problems and develops FPT algorithms and kernels for specific cases and graph classes.

Findings

W[1]-hardness for BCP_q with q ≥ 2.

FPT algorithms for BCP_2 parameterized by treewidth and on planar graphs.

Polynomial kernel for BCP_2 on unit disk graphs.

Abstract

Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, balanced connected $q$-partition problem ($BCP_q$) asks whether there exists a partition of the vertex set $V$ of $G$ into $q$ parts $V_1,V_2,V_3,\cdots, V_q$ such that for all $i\in[1,q]$, $|V_i|=n_i$ and the graph induced on $V_i$ is connected. A related problem denoted as the balanced connected $q$-edge partition problem ($BCEP_q$) is defined as follows. Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, $BCEP_q$ asks whether there exists a partition of the edge set of $G$ into $q$ parts $E_1,E_2,E_3,\cdots, E_q$ such that for all $i\in[1,q]$, $|E_i|=n_i$ and the graph induced on the edge set $E_i$ is connected. Here we study both the problems for $q=2$ and prove that $BCP_q$ for $q\geq 2$ is $W[1]$-hard. We also show that $BCP_2$ is unlikely to have a polynomial kernel on the class of planar graphs. Coming to the positive results, we show that $BCP_2$ is fixed parameter tractable (FPT) parameterized by treewidth of the graph, which generalizes to FPT algorithm for planar graphs. We design another FPT algorithm and a polynomial kernel on the class of unit disk graphs parameterized by $\min(n_1,n_2)$. Finally, we prove that unlike $BCP_2$, $BCEP_2$ is FPT parameterized by $\min(n_1,n_2)$.