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

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 BCPk) or minimize the weight of a heaviest class (MIN-MAX BCPk). Both problems are NP-hard for any fixed k = 2, and equivalent only when k = 2. In this work, we propose a simple pseudo-polynomial 3/2-approximation algorithm for MIN-MAX BCP3, which is an O(vertical bar V vertical bar vertical bar E vertical bar) time 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 (3/2 + epsilon)-approximation for the weighted version of the problem with running-time O(vertical bar V vertical bar vertical bar(3)vertical bar E vertical bar/epsilon), for any fixed epsilon > 0. This algorithm is then used to obtain, for MIN-MAX BCPk, k >= 4, analogous results with approximation ratio (k/2 + epsilon). For k epsilon {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.