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 most relevant work to this paper is that of Kolliopoulos @cite_22 . The author studies CCIPs which satisfy a rather strong assumption, called the no bottleneck assumption , that the supply of any column is smaller than the demand of any row. Kolliopoulos @cite_22 shows that if one is allowed to violate the upper bounds by a multiplicative constant, then the integrality gap of the CCIP is within a constant factor of that of the original 0,1-CIP Such a result is implicit in the paper; the author only states a @math integrality gap. . As the author notes such a violation is necessary; otherwise the CCIP has unbounded integrality gap. If one is not allowed to violated upper bounds, nothing better than the result of @cite_19 is known for the special case of CCIPs.
@cite_19 @cite_22
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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 requests for capacity in a given tree network T e (V, E) where each edge e of the tree has some integer capacity ue. Each request f is a node pair with an integer demand df and a profit wf which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b-matching problems. When all demands are 1, we have the integer multicommodity flow problem. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand dmax is at most the minimum edge capacity umin, we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed. In a covering integer program (CIP), we seek an n-vector x of nonnegative integers, which minimizes cT ċ x, subject to Ax ≥ b, where all entries of A, b, c are nonnegative. In their most general form, CIPs include also multiplicity constraints of the type x ≤ d, i.e., arbitrarily large integers are not acceptable in the solution. The multiplicity constraints incur a dichotomy with respect to approximation between (0,1)-CIPs whose matrix A contains only zeros and ones and the general case. Let m denote the number of rows of A. The well known O(log m) cost approximation with respect to the optimum of the linear relaxation is valid for general CIPs, but multiplicity constraints can be dealt with effectively only in the (0,1) case. In the general case, existing algorithms that match the integrality gap for the cost objective violate the multiplicity constraints by a multiplicative O(log m) factor. We make progress by defining column-restricted CIPs, a strict superclass of (0,1)-CIPs, and showing how to find for them integral solutions of cost O(log m) times the LP optimum while violating the multiplicity constraints by a multiplicative O(1) factor. 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.