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
1574334008 2963986064 2103363383 2890196309
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. We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + ln([EQUATION]), where amin is the minimum covering constraint and Δ1 is the maximum e1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearly-matching inapproximability and integrality-gap lower bounds. Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ1 which is always - and can be much - smaller than Δ0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + e) of some given upper-bounds, and show nearly-matching inapproximability. We consider the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph G = (V ,E ) and k request pairs R 1 , ..., R k , where each R i consists of a source-destination pair (s i ,t i ), a demand d i and a weight w i . The goal is to find a maximum weight subset of requests that can be routed unsplittably in G . Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of . [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O (logn ) approximation for UFP on trees when all weights are identical; this yields an O (log2 n ) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O (log2 n ); previously there was no relaxation with o (n ) gap. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results. Several algorithms with an approximation guarantee of @math are known for the Set Cover problem, where @math is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into @math , and we are required to cover at least @math elements from each color class @math , using the minimum number of sets. We give a randomized LP-rounding algorithm that is an @math approximation for the Partition Set Cover problem. Here @math denotes the approximation guarantee for a related Set Cover instance obtained by rounding the standard LP. As a corollary, we obtain improved approximation guarantees for various set systems for which @math is known to be sublogarithmic in @math . We also extend the LP rounding algorithm to obtain @math approximations for similar generalizations of the Facility Location type problems. Finally, we show that many of these results are essentially tight, by showing that it is NP-hard to obtain an @math -approximation for any of these problems.