Approximation and Parameterized Algorithms for Balanced Connected Partition Problems

Abstract

For a given integer $k\ge 2$, partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or fixed) orders is a classical problem that has been intensively investigated since late seventies. A connected k-partition of a graph is a partition of its vertex set into classes such that each one induces a connected subgraph. Given a connected graph $G = (V, E)$ and a weight function $w : V \rightarrow \mathbb {Q}_\ge$, the balanced connected k-partition problem looks for a connected k-partition of G into classes of roughly the same weight. To model this concept of balance, we seek connected k-partitions that either maximize the weight of a lightest class $(\textsc {max}\hbox {-}\textsc {min\,\,BCP}_k)$ or minimize the weight of a heaviest class $(\textsc {min}\hbox {-}\textsc {max\,\,BCP}_k)$. These problems, known to be NP-hard, are equivalent only when $k=2$. We present a simple pseudo-polynomial $\frac{k}{2}$-approximation algorithm for $\textsc {min}\hbox {-}\textsc {max\,\,BCP}_k$ that runs in time $\mathcal {O}(W|V||E|)$, where $W = \sum _{v \in V} w(v)$; then, using a scaling technique, we obtain a (polynomial) $(\frac{k}{2} +{\varepsilon })$-approximation with running-time $\mathcal {O}(|V|^3|E|/{\varepsilon })$, for any fixed ${\varepsilon }>0$. Additionally, we propose a fixed-parameter tractable algorithm for the unweighted $\textsc {max}\hbox {-}\textsc {min\,\,BCP}$ (where k is part of the input) parameterized by the size of a vertex cover.

Research partially supported by grant #2015/11937-9, São Paulo Research Foundation (FAPESP). Moura is supported by FAPEMIG (Proc. APQ-01040-21) and Pró-Reitoria de Pesquisa da Universidade Federal de Minas Gerais. Wakabayashi is supported by CNPq (Proc. 306464/2016-0 and 423833/2018-9).