Doubly Balanced Connected Graph Partitioning

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 )= ∑ j∈V p(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=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 )6=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. ∗Research partially supported by DTRA grant HDTRA1-13-1-0021 and CIAN NSF ERC under grant EEC0812072. †Research partially supported by NSF grant CCF-1320654. ar X iv :1 60 7. 06 50 9v 1 [ m at h. C O ] 2 1 Ju l 2 01 6