## Frédéric Chataigner ; Liliane R. B. Salgado ; Yoshiko Wakabayashi - Approximation and inapproximability results on balanced connected partitions of graphs

dmtcs:384 - Discrete Mathematics & Theoretical Computer Science, January 1, 2007, Vol. 9 no. 1 - https://doi.org/10.46298/dmtcs.384
Approximation and inapproximability results on balanced connected partitions of graphs

Authors: Frédéric Chataigner ; Liliane R. B. Salgado ; Yoshiko Wakabayashi

Let G=(V,E) be a connected graph with a weight function w: V \to \mathbbZ₊, and let q ≥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, \ldots, V_q) of V such that G[V_i] is connected (1≤q i≤q q) and P maximizes \rm min\w(V_i): 1≤q i≤q q\. This problem is called \textitMax Balanced Connected q-Partition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NP-hard in the strong sense, even on q-connected graphs, and therefore does not admit a FPTAS, unless \rm P=\rm NP. We also show another inapproximability result for BCP₂ on arbitrary graphs. On q-connected graphs, for q=2 the best result is a \frac43-approximation algorithm obtained by Chleb\'ıková; 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 \textitMax Balanced Connected Partition, and denoted as BCP. We show that BCP does not admit an approximation algorithm with ratio smaller than 6/5, unless \rm P=\rm NP.

Volume: Vol. 9 no. 1
Section: Graph and Algorithms
Published on: January 1, 2007
Imported on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]