Approximation 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 = ( V 1 , V 2 , … , V q ) of V such that G [ V i ] is connected ( 1 ≤ i ≤ q ) and P maximizes min { w ( V i ) : 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. This result implies that this problem does not admit a FPTAS, unless P = NP . For these graphs, for q = 2 the best result is a 4 3 -approximation algorithm obtained by Chlebíková [J. Chlebíková. “Approximating the maximally balanced connected partition problem in graphs”, Information Processing Letters, 60:225–230, 1996]; for q = 3 we present a 2-approximation algorithm. We also show a non-approximability result for BCP2.