Approximation and Parameterized Algorithms for Balanced Connected Partition Problems.

For a given integer k ≥ 2 , partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or fixed) orders is a classical problem that has been intensively investigated since late seventies. A connected k -partition of a graph is a partition of its vertex set into classes such that each one induces a connected subgraph. Given a connected graph G = ( V , E ) and a weight function w : V → Q ≥ , the balanced connected k -partition problem looks for a connected k -partition of G into classes of roughly the same weight. To model this concept of balance, we seek connected k -partitions that either maximize the weight of a lightest class ( M A X - M I N BCP k ) or minimize the weight of a heaviest class ( M I N - M A X BCP k ) . These problems, known to be NP-hard, are equivalent only when k = 2 . We present a simple pseudo-polynomial k 2 -approximation algorithm for M I N - M A X BCP k that runs in time O ( W | V | | E | ) , where W = ∑ v ∈ V w ( v ) ; then, using a scaling technique, we obtain a (polynomial) ( k 2 + ε ) -approximation with running-time O ( | V | 3 | E | / ε ) , for any fixed ε > 0 . Additionally, we propose a fixed-parameter tractable algorithm for the unweighted M A X - M I N BCP (where k is part of the input) parameterized by the size of a vertex cover.