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Approximation and parameterized algorithms to find balanced connected partitions of graphs

by   Phablo F. S. Moura, et al.
University of Waterloo
Universidade de São Paulo
Universidade Federal de Minas Gerais

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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