Discrete Mathematics & Theoretical Computer Science 
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 qpartition 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 qPartition and is denoted by BCP_q. We show that for q≥q 2 the problem BCP_q is NPhard in the strong sense, even on qconnected 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 qconnected graphs, for q=2 the best result is a \frac43approximation algorithm obtained by Chleb\'ıková; for q=3 and q=4 we present 2approximation 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.
Source : ScholeXplorer
IsRelatedTo DOI 10.2298/bg20130708matic Source : ScholeXplorer IsRelatedTo HANDLE 21.15107/rcub_nardus_2842
