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.
To the best of our knowledge, the column-restricted or priority versions of the line and tree cover problem have not been studied. The best known approximation algorithm known for both is the @math factor implied by the results of @cite_19 stated above. However, upon completion of our work, Nitish Korula @cite_23 pointed out to us that a @math -approximation for column-restricted line cover is implicit in a result of Bar- @cite_5 . We remark that their algorithm is not LP-based, although our general result on CCIPs is.
@cite_19 @cite_5 @cite_23
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Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min c^Tx:[email protected]?Z"+^n,Ax>=a,Bx==a) and multiplicity constraints (x= We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the local-ratio technique. We note that they can equivalently be interpreted within the primal-dual schema.