Approximation and Inapproximability Results on Balanced Connected Partitions of Graphs

Let G = (V, E) be a connected graph with a weight function w : V → Z+ and let q ≥ 2 be a positive integer. For X ⊆ V , let w(X) denote the sum of the weights of the vertices in X . We consider the following problem on G: find a q-partition P = (V1, V2, . . . , Vq) of V such that G[Vi] is connected (1 ≤ i ≤ q) and P maximizes min{w(Vi) : 1 ≤ i ≤ q}. This problem is called Max Balanced Connected q-Partition and is denoted by BCPq . We show that for q ≥ 2 the problem BCPq is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless P = NP. We also show another inapproximability result for BCP2. For the problem BCPq restricted to q-connected graphs, it is known that for q = 2 the best result is a 4 3 -approximation algorithm obtained by Chlebı́ková; for q = 3 and q = 4 we present 2-approximation algorithms. When q is not fixed (it is part of the instance), the corresponding problem is called Max Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless P = NP.