We consider approximation algorithms for packing integer programs (PIPs) of the form @math where @math , @math , and @math are nonnegative. We let @math denote the width of @math which is at least @math . Previous work by bansal-sparse obtained an @math -approximation ratio where @math is the maximum number of nonzeroes in any column of @math (in other words the @math -column sparsity of @math ). They raised the question of obtaining approximation ratios based on the @math -column sparsity of @math (denoted by @math ) which can be much smaller than @math . Motivated by recent work on covering integer programs (CIPs) cq,chs-16 we show that simple algorithms based on randomized rounding followed by alteration, similar to those of bansal-sparse (but with a twist), yield approximation ratios for PIPs based on @math . First, following an integrality gap example from bansal-sparse , we observe that the case of @math is as hard as maximum independent set even when @math . In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width @math where @math , we obtain an @math -approximation. In the large width regime, when @math , we obtain an @math -approximation. We also obtain a @math -approximation when @math .
There is a large literature on deterministic and randomized rounding algorithms for packing and covering integer programs and connections to several topics and applications including discrepancy theory. @math -sparsity guarantees for covering integer programs were first obtained by Chen, Harris and Srinivasan @cite_13 partly inspired by @cite_16 .
@cite_16 @cite_13
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Several important NP-hard combinatorial optimization problems can be posed as packing covering integer programs; the randomized rounding technique of Raghavan and Thompson is a powerful tool with which to approximate them well. We present one elementary unifying property of all these integer linear programs and use the FKG correlation inequality to derive an improved analysis of randomized rounding on them. This yields a pessimistic estimator, thus presenting deterministic polynomial-time algorithms for them with approximation guarantees that are significantly better than those known. We consider the problem of approximating an integer program by first solving its relaxation linear program and "rounding" the resulting solution. For several packing problems, we prove probabilistically that there exists an integer solution close to the optimum of the relaxation solution. We then develop a methodology for converting such a probabilistic existence proof to a deterministic approximation algorithm. The methodology mimics the existence proof in a very strong sense. The Lova´sz local lemma due to Erdods and Lova´sz (Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 11, 1975, pp. 609-627) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson (Combinatorica, 7 (1987), pp. 365-374) to derive good approximation algorithms for such problems. We use our extension of the local lemma to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan (J. Comput. System Sci., 37 (1988), pp. 130-143), to obtain constructive (algorithmic) versions of our results for covering integer programs. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.