Doubly balanced connected graph partitioning

Abstract

We introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let G=(V, E) be a connected graph with a weight (supply/demand) function p:V→{−1, +1} satisfying p(V)=Σj∈vp(j)=0. The objective is to partition G into (V1, V2) such that G[V1] and G[V2] are connected, |p(V1)|, |p(V2)|≤cp, and max [EQUATION], for some constants cp and cs. When G is 2-connected, we show that a solution with cp=1 and cs=3 always exists and can be found in polynomial time. Moreover, when G is 3-connected, we show that there is always a 'perfect' solution (a partition with p(V1)=p(V2)=0 and |V1|=|V2|, if |V|≡0(mod 4)), and it can be found in polynomial time. Our techniques can be extended, with similar results, to the case in which the weights are arbitrary (not necessarily ±1), and to the case that p(V)≠0 and the excess supply/demand should be split evenly. They also apply to the problem of partitioning a graph with two types of nodes into two large connected subgraphs that preserve approximately the proportion of the two types.