平面二分问题的多项式时间双准则逼近格式
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
摘要：
给定一个具有边成本和节点权重的无向图，最小二分法问题要求将节点划分为两个相等权重的部分，使得部分之间的边成本之和最小化。给出了平面图二分问题的一个多项式时间双准则逼近方案。具体地，令$W$为平面图$G$中所有节点的总权重。对于任何恒定的$\varepsilon > 0$，我们的算法输出的节点，使每个部分的重量最多$W/2 + \varepsilon$和总成本的边缘穿过分区是最多$(1+\varepsilon)$倍的最佳二分法的总成本的一个bipartition。以前最著名的平面最小二分法的近似，即使有单位节点权重，是$O(\log n)$。我们的算法实际上解决了一个更一般的问题，其中输入可能包括一个目标权重的较小的一面的bipartition。
英文摘要：
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.
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