关于列限制优先覆盖规划
On Column-restricted and Priority Covering Integer Programs
摘要：
在列限制覆盖整数规划（CCIP）中，约束矩阵的任何一列的所有非零元素都是相等的。这样的程序捕获覆盖问题的能力限制版本。本文研究了CCIP的可逼近性，特别是它们与0，1-CIP的完整性间隙的关系。如果0，1-CIP的完整性间隔为O（gamma），并假设0，1-CIP的优先级版本的完整性间隔为O（omega），我们给出了CCIP的一个因子为O（gamma + omega）的近似算法。0，1-CIP（PCIP）的优先级版本自然地捕获覆盖问题中的服务质量类型约束。我们调查优先版本的线（PLC）和（根）树覆盖（PTC）的问题。除了作为自然的研究对象，这些问题属于一类基本的几何覆盖问题。我们通过一个常数来约束这个PCIP的某些类的完整性。在数学上，我们给出了PLC的一个多时间精确算法，证明了PTC问题是APX困难的，并给出了它的一个因子2-近似算法。
英文摘要：
In a column-restricted covering integer program (CCIP), all the non-zero entries of any column of the constraint matrix are equal. Such programs capture capacitated versions of covering problems. In this paper, we study the approximability of CCIPs, in particular, their relation to the integrality gaps of the underlying 0,1-CIP. If the underlying 0,1-CIP has an integrality gap O(gamma), and assuming that the integrality gap of the priority version of the 0,1-CIP is O(omega), we give a factor O(gamma + omega) approximation algorithm for the CCIP. Priority versions of 0,1-CIPs (PCIPs) naturally capture quality of service type constraints in a covering problem. We investigate priority versions of the line (PLC) and the (rooted) tree cover (PTC) problems. Apart from being natural objects to study, these problems fall in a class of fundamental geometric covering problems. We bound the integrality of certain classes of this PCIP by a constant. Algorithmically, we give a polytime exact algorithm for PLC, show that the PTC problem is APX-hard, and give a factor 2-approximation algorithm for it.
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