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(γ), and assuming that the integrality gap of the priority version of the 0,1-CIP is O(ω), we give a factor O(γ+ω) 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.
The only work on priority versions of covering problems that we are aware of is due to Charikar, Naor and Schieber @cite_8 who studied the priority Steiner tree and forest problems in the context of QoS management in a network multicasting application. present a @math -approximation algorithm for the problem, and @cite_2 later show that no efficient @math approximation algorithm can exist unless @math ( @math is the number of vertices).
@cite_2 @cite_8
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Consider the following classical network design problem: a set of terminals T e ti wishes to send traffic to a root r in an n-node graph G e (V, E). Each terminal ti sends di units of traffic and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue and hence provisioning k × ue bandwidth on edge e incurs a cost of ⌈k⌉ times the cost of that edge. The objective is a minimum-cost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation is governed by side constraints: edges can only allow a subset of cables to be purchased on them or certain quality-of-service requirements may have to be met. In this work, we show that this problem and, in fact, several basic problems in this general network design framework cannot be approximated better than Ω(log log n) unless NP ⊆ DTIME (nO(log log log n)), where vVv e n. In particular, we show that this inapproximability threshold holds for (i) the Priority-Steiner Tree problem, (ii) the (single-sink) Cost-Distance problem, and (iii) the single-sink version of an even more fundamental problem, Fixed Charge Network Flow. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first nonconstant hardness results known for all these problems. We consider a network design problem, where applications require various levels of Quality-of-Service (QoS) while connections have limited performance. Suppose that a source needs to send a message to a heterogeneous set of receivers. The objective is to design a low-cost multicast tree from the source that would provide the QoS levels (e.g., bandwidth) requested by the receivers. We assume that the QoS level required on a link is the maximum among the QoS levels of the receivers that are connected to the source through the link. In accordance, we define the cost of a link to be a function of the QoS level that it provides. This definition of cost makes this optimization problem more general than the classical Steiner tree problem. We consider several variants of this problem all of which are proved to be NP-Hard. For the variant where QoS levels of a link can vary arbitrarily and the cost function is linear in its QoS level, we give a heuristic that achieves a multicast tree with cost at most a constant times the cost of an optimal multicast tree. The constant depends on the best constant approximation ratio of the classical Steiner tree problem. For the more general variant, where each link has a given QoS level and cost we present a heuristic that generates a multicast tree with cost O(min log r, k ) times the cost of an optimal tree, where r denotes the number of receivers, and k denotes the number of different levels of QoS required. We generalize this result to hold for the case of many multicast groups.