Bisection Problem on Bipartite Graphs

Bisection Problem is to patition a given graph G into subgraphs G1 and G2 of equal size with minimum number of cut edges, i.e., edges between G1 and G2. This problem has been studied extensively by many researchers [1, 2, 4], and several algorithms have been proposed and analyzed both mathematically and experimentally. Here we propose yet another algorithm for this problem. In this paper, we consider only bipartite graphs G = (V1, V2, E) such that |V1| = |V2| = n for some size parameter n. That is, E is a subset of {(i, j) : i ∈ V1, j ∈ V2}. We will use i to denote vertices in V1 and j to denote vertices in V2. An equal size partition of V1 ∪ V2 is a partition (C, C−) such that V1 = C− ∩ V1 = C− ∩ V1 = C ∩ V2 = C− ∩ V2 = n/2. (We assume that n is even.) The Bisection Problem for general graphs is NP-hard, and it is easy to modify the proof to show the NP-hardness of our problem. Thus, one may not be able to hope for an efficient algorithm solving the problem for all instances. On the other hand, the problem seems not so hard on average under some reasonable probability model on input graphs. In fact, under “planted models”, which we explain below, several algorithms have been proposed and some of them are shown to solve the problem [1, 2, 3, 4] in polynomial-time on average. Let us define a planted model, a probability model for input graph distribution. Consider any even n, and let V1 and V2 be respectively a set of n vertices, and let (C, C−) be its equal size partition. For given parameters p and r, we consider the following random generation of a bipartite graph (V1, V2, E): For any vertex i ∈ V1 and j ∈ V2, add an edge (i, j) to E, with probability p if both i and j are in C (or, respectively, in C−) and with probability r otherwise. For any size parameter n, and parameters p and r, this model defines a probability distribution on bipartite of size 2n. We discuss the average performace of algorithms under this probability distribution. It has been shown [1] that if p− r is large engough (more specifically, p − r = Ω(n−2), then a planted solution, the partition used to generate an instance to the problem, is a unique solution for the generated instance with high probability. Thus, such parameters p and r, the goal of bisection algorithms is to find a planted solution. There is another natural way to formalize a problem for panted models. For any graph G = (V1, V2, E) and its partition (C, C−), and for given parameters p and r, the following is the probability that G is generated from (C, C−) as above.