In the planted bisection model a random graph G(n,p_+,p_-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to plus or minus 1). Any two vertices that belong to the same class are linked by an edge with probability p_+ and any two that belong to different classes with probability (p_-) c * sqrt((d_+)ln(d_+)) for a certain constant c>0.
Determining the minimum bisection width of a graph is NP-hard @cite_29 and there is evidence that the problem does not even admit a PTAS @cite_3 . On the positive side, it is possible to approximate the minimum bisection width within a factor of @math for graphs on @math vertices in polynomial time @cite_7 .
@cite_29 @cite_7 @cite_3
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We introduce new lower bounds for the minimum graph bisection problem. Within a branch-and-bound framework, they enable the solution of a wide variety of instances with tens of thousands of vertices to optimality. Our algorithm compares favorably with the best previous approaches, solving long-standing open instances in minutes. Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ‘‘cross edges’’ between the parts. We are interested in sparse random graphs Ž . G with edge probability c n. We show that, if c ln 4, then the bisection width is n n, c n with high probability; while if c ln 4, then it is equal to 0 with high probability. There are corresponding threshold results for partitioning into any fixed number of parts. 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 31 38, 2001 Graph Bisection is the problem of partitioning the vertices of a graph into two equal-size pieces so as to minimize the number of edges between the two pieces. This paper presents an algorithm that will, for almost all graphs in a certain class, output the minimum-size bisection. Furthermore the algorithm will yield, for almost all such graphs, a proof that the bisection is optimal. The algorithm is based on computing eigenvalues and eigenvectors of matrices associated with the graph. In a graph G=(V,E), a bisection (X,Y) is a partition of V into sets X and Y such that |X|=<|Y|=<|X|+1. The size of (X,Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G=(V,E) and are required to find a bisection of maximum size. It is not hard to see that @?|E| 2@? is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (Max-Bisec-ATLB): decide whether a graph G=(V,E) has a bisection of size at least @?|E| 2@?+k, where k is the parameter. We show that this parameterized problem has a kernel with O(k^2) vertices and O(k^3) edges, i.e., every instance of Max-Bisec-ATLB is equivalent to an instance of Max-Bisec-ATLB on a graph with at most O(k^2) vertices and O(k^3) edges.