Given a graph $G=(V,E)$ with costs on its edges, the minimum-cost edge cover problem consists of finding a subset of $E$ covering all vertices in $V$ at minimum cost. If $G$ is bipartite, this problem can be solved in time $O(|V|^3)$ via a well-known reduction to a maximum-cost matching problem on $G$. If in addition $V$ is a set of points on the Euclidean line, Collanino et al. showed that the problem can be solved in time $O(|V| \log |V|)$ and asked whether it can be solved in time $o(|V|^3)$ if $V$ is a set of points on the Euclidean plane. We answer this in the affirmative, giving an $O(|V|^{2.5} \log |V|)$ algorithm based on the Hungarian method using weighted Voronoi diagrams. We also propose some 2-approximation algorithms and give experimental results of our implementations.
翻译：鉴于GG=(V,E)的图表,其边缘费用最低成本边缘问题包括找到以最低成本以美元支付所有脊椎的费用的E$子集。如果G$是双方美元,这个问题可以及时解决。如果美元是双方美元,那么这个问题可以通过众所周知的将美元减到美元的最大成本匹配问题来及时解决。如果加上V$,这是Euclidean线上一套点数,Colanino等人则表明,问题可以及时解决,美元( ⁇ V ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ V ⁇ ⁇ ),并询问,如果美元是Euclidean飞机上的一组点数,这个问题能否及时解决。我们肯定地回答这个问题,用加权的Voronoioi图表给出以匈牙利方法为基础的一个$( ⁇ V ⁇ 2.5} log ⁇ V ⁇ ⁇ )的算法。我们还提出一些2 套套式算算算法,并给出我们执行的实验结果。
相关内容
微信扫码咨询专知VIP会员