Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let W be the total weight of all nodes in a planar graph G.
For any constant ε>0, our algorithm outputs a bipartition of the
nodes such that each part weighs at most W/2+ε and the total cost
of edges crossing the partition is at most (1+ε) times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was O(logn). Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201