关于限制圈覆盖的逼近
On Approximating Restricted Cycle Covers
摘要：
一个图的圈覆盖是一组圈，使得每个顶点恰好是一个圈的一部分。L-圈覆盖是指每个圈的长度都在集合L中的圈覆盖。一个边权图的圈覆盖的权是它的边的权之和。我们接近解决计算L-圈覆盖的复杂性和可逼近性。一方面，我们证明了对于几乎所有的L，计算有向图和无向图的最大权的L-圈覆盖是APX-困难和NP-困难的。即使边的权重被限制为0和1，我们的大多数硬度结果仍然成立。另一方面，我们表明，计算L-圈覆盖的最大重量的问题，可以近似在一个因素为2无向图和8/3的情况下，有向图。这对任意集合L成立。
英文摘要：
A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.
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