Doubly Balanced Connected Graph Partitioning

Saleh Soltan, Mihalis Yannakakis, Gil Zussman

Abstract: We introduce and study the doubly balanced connected graph partitioning problem: Let G=(V, E) be a connected graph with a weight (supply/demand) function p : V → {−1, +1} satisfying p(V)=∑ j&isin 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{ ∣V1 / V2∣,∣V2 / V1∣} ≤ cs, for some constants cp and cs. When G is 2-connected, we show that a solution with cp=1 and cs=2 always exists and can be found in randomized 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 randomized 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.