A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

TL;DR

This paper presents a polynomial-time bicriteria approximation scheme for the planar graph bisection problem, achieving near-optimal partitions with controlled imbalance and crossing edge costs, improving significantly over previous logarithmic approximations.

Contribution

It introduces the first polynomial-time bicriteria approximation scheme for planar graph bisection, allowing for a small imbalance and near-optimal edge cut cost.

Findings

Achieves (1+ε)-approximation for crossing edges

Allows small imbalance of W/2 + ε in partition weights

Improves approximation ratio from O(log n) to near-optimal

Abstract

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 $\varepsilon > 0$, our algorithm outputs a bipartition of the nodes such that each part weighs at most $W/2 + \varepsilon$ and the total cost of edges crossing the partition is at most $(1+\varepsilon)$ times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was $O(\log n)$. Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.