Approximation and parameterized algorithms to find balanced connected partitions of graphs
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Partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph G=(V,E) and a weight function w : V →ℚ_≥, a connected k-partition of G is a partition of V such that each class induces a connected subgraph. The balanced connected k-partition problem consists in finding a connected k-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected k-partitions that either maximize the weight of a lightest class (max-min BCP_k) or minimize the weight of a heaviest class (min-max BCP_k). Such problems are equivalent when k=2, but they are different when k≥ 3. In this work, we propose a simple pseudo-polynomial k/2-approximation algorithm for min-max BCP_k which runs in time 𝒪(W|V||E|), where W = ∑_v ∈ V w(v). Based on this algorithm and using a scaling technique, we design a (polynomial) (k/2 +ε)-approximation for the same problem with running-time 𝒪(|V|^3|E|/ε), for any fixed ε>0. Additionally, we propose a fixed-parameter tractable algorithm based on integer linear programming for the unweighted max-min BCP_k parameterized by the size of a vertex cover.
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