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)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} 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 BCPk\documentclass[12pt]{minimal}
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\begin{document}$$_k$$\end{document}) or minimize the weight of a heaviest class (min–max BCPk\documentclass[12pt]{minimal}
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\begin{document}$$_k$$\end{document}). Both problems are NP\documentclass[12pt]{minimal}
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\begin{document}$$\text {\textsc {NP}}$$\end{document}-hard for any fixed k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}, and equivalent only when k=2\documentclass[12pt]{minimal}
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\begin{document}$$k=2$$\end{document}. In this work, we propose a simple pseudo-polynomial 32\documentclass[12pt]{minimal}
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\begin{document}$$\frac{3}{2}$$\end{document}-approximation algorithm for min–max BCP3\documentclass[12pt]{minimal}
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\begin{document}$$_3$$\end{document}, which is an O(|V||E|)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(|V ||E |)$$\end{document} time 32\documentclass[12pt]{minimal}
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\begin{document}$$\frac{3}{2}$$\end{document}-approximation for the unweighted version of the problem. We show that, using a scaling technique, this algorithm can be turned into a polynomial-time (32+ε)\documentclass[12pt]{minimal}
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\begin{document}$$(\frac{3}{2} +{\varepsilon })$$\end{document}-approximation for the weighted version of the problem with running-time O(|V|3|E|/ε)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}(|V |^3 |E |/ {\varepsilon })$$\end{document}, for any fixed ε>0\documentclass[12pt]{minimal}
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\begin{document}$${\varepsilon }>0$$\end{document}. This algorithm is then used to obtain, for min–max BCPk\documentclass[12pt]{minimal}
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\begin{document}$$_k$$\end{document}, k≥4\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 4$$\end{document}, analogous results with approximation ratio (k2+ε)\documentclass[12pt]{minimal}
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\begin{document}$$(\frac{k}{2}+{\varepsilon })$$\end{document}. For k∈{4,5}\documentclass[12pt]{minimal}
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\begin{document}$$k\in \{4,5\}$$\end{document}, 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.
