Balanced connected partitions of graphs: approximation, parameterization and lower bounds

Abstract

A connected k-partition of a graph is a partition of its vertex set into k classes such that each class induces a connected subgraph. Finding a connected k-partition in which the classes have similar size is a classical problem that has been investigated since late seventies. We consider a more general setting in which the input graph $G=(V,E)$ has a nonnegative weight assigned to each vertex, and the aim is to find a connected k-partition in which every class has roughly the same weight. In this case, we may either maximize the weight of a lightest class (max–min BCP$_k$) or minimize the weight of a heaviest class (min–max BCP$_k$). Both problems are $\text {\textsc {NP}}$-hard for any fixed $k\ge 2$, and equivalent only when $k=2$. In this work, we propose a simple pseudo-polynomial $\frac{3}{2}$-approximation algorithm for min–max BCP$_3$, which is an $\mathcal {O}(|V ||E |)$ time $\frac{3}{2}$-approximation for the unweighted version of the problem. We show that, using a scaling technique, this algorithm can be turned into a polynomial-time $(\frac{3}{2} +{\varepsilon })$-approximation for the weighted version of the problem with running-time $\mathcal {O}(|V |^3 |E |/ {\varepsilon })$, for any fixed ${\varepsilon }>0$. This algorithm is then used to obtain, for min–max BCP$_k$, $k\ge 4$, analogous results with approximation ratio $(\frac{k}{2}+{\varepsilon })$. For $k\in \{4,5\}$, we are not aware of algorithms with approximation ratios better than those. We also consider fractional bipartitions that lead to a unified approach to design simpler approximations for both min–max and max–min versions. Additionally, we propose a fixed-parameter tractable algorithm based on integer linear programming for the unweighted max–min BCP parameterized by the size of a vertex cover. Assuming the Exponential-Time Hypothesis, we show that there is no subexponential-time algorithm to solve the max–min and min–max versions of the problem.