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在有向图中近似最长循环

[英]Approximating longest cycle in a directed graph

原文 2020-02-06 15:39:13 5 1 algorithm/ complexity-theory/ graph-theory/ computation-theory/ operations-research

Finding the longest cycle (By cycle I mean cycle with no node repitition) in a directed graph is an NP-hard problem, otherwise we can tell if the graph is Hamiltonian or not.在有向图中找到最长的循环（循环我的意思是没有节点重复的循环）是一个 NP-hard 问题，否则我们可以判断该图是否是哈密顿量。 My question is: Is there any alpha-approxiamtion polynomial algorithm for this problem ?我的问题是：是否有针对此问题的 alpha 近似多项式算法？

1 个解决方案

Since the longest directed path problem in a directed graph cannot be approximated in polynomial time within a factor of n^(1-epsilon) for any epsilon > 0 , we can quickly deduce that it is also the case for the longest cycle in a directed graph unless P=NP ( source ).由于对于任何epsilon > 0 ，有向图中最长有向路径问题不能在多项式时间内逼近n^(1-epsilon)因子，我们可以很快推断出有向图中最长循环也是这种情况除非 P=NP（来源）。

You can make the reduction as follows:您可以按以下方式进行减少：
Choose a vertex v , duplicate v into v1 and v2 , duplicate all concerned arcs as well.选择一个顶点v ，将v复制到v1和v2 ，同时复制所有相关的弧。 Now find the longest directed path from v1 to v2 .现在找到从v1到v2的最长有向路径。
Do that for all vertices in the graph.对图中的所有顶点执行此操作。 That gives you the longest directed cycle in the graph.这为您提供了图中最长的有向循环。

Conclusion: there is no alpha -approximation in polynomial time for the longest cycle problem in directed graphs (unless P=NP of course).结论：有向图中最长循环问题的多项式时间没有alpha近似（当然，除非 P=NP）。