具有一般目标函数的集合覆盖问题
Set Covering Problems with General Objective Functions
摘要：
我们引入了一个参数化版本的集合覆盖，概括了以前研究的几个问题。给定基集V和V的子集集合S_i，可行解是V的划分，使得划分的每个子集包含在S_i之一中。这个问题涉及到最大化分区的平均子集大小，其中平均值是参数p的广义平均值，取所有元素。当p=-1时，该问题等价于经典的最小集覆盖问题.对于p=0，它等价于Halperin和Karp提出的最小熵集合覆盖问题。对于p=1，该问题包括最大边团划分问题作为特殊情况。我们证明了对于R^+中的任意p，贪心算法同时在（p+1）^1/p的因子内逼近问题，并且这是最好的可能，除非P=NP。这些结果推广和简化了以前的特殊情况下的结果。我们还考虑了相应的图着色问题，证明了几个易处理性和不可逼近性结果。最后，我们考虑进一步推广的集合覆盖问题，我们的目标是最小化的部分大小的一些凹函数的总和。作为一个应用，我们得到了一个近似比租或买集覆盖问题。
英文摘要：
We introduce a parameterized version of set cover that generalizes several previously studied problems. Given a ground set V and a collection of subsets S_i of V, a feasible solution is a partition of V such that each subset of the partition is included in one of the S_i. The problem involves maximizing the mean subset size of the partition, where the mean is the generalized mean of parameter p, taken over the elements. For p=-1, the problem is equivalent to the classical minimum set cover problem. For p=0, it is equivalent to the minimum entropy set cover problem, introduced by Halperin and Karp. For p=1, the problem includes the maximum-edge clique partition problem as a special case. We prove that the greedy algorithm simultaneously approximates the problem within a factor of (p+1)^1/p for any p in R^+, and that this is the best possible unless P=NP. These results both generalize and simplify previous results for special cases. We also consider the corresponding graph coloring problem, and prove several tractability and inapproximability results. Finally, we consider a further generalization of the set cover problem in which we aim at minimizing the sum of some concave function of the part sizes. As an application, we derive an approximation ratio for a Rent-or-Buy set cover problem.
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