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 .
We note that PIPs are equivalent to the multi-dmensional knapsack problem. When @math we have the classical knapsack problem which admits a very efficient FPTAS (see @cite_11 ). There is a PTAS for any fixed @math @cite_0 but unless @math an FPTAS does not exist for @math .
@cite_0 @cite_11
2031666601 1989921461 1524700731 2060645964
Abstract We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting-plane approaches to integer linear programs. We focus our attention on approximation algorithms for the problem, describing a linear-storage Polynomial Time Approximation Scheme (PTAS) and a dynamic-programming based Fully Polynomial Time Approximation Scheme (FPTAS). The main ideas contained in our PTAS are used to derive PTAS's for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTAS's. We finally illustrate better PTAS's and FPTAS's for the subset sum case of the problem in which profits and weights coincide. 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. A fully polynomial time approximation scheme (FPTAS) is presented for the classical 0-1 knapsack problem. The new approach considerably improves the necessary space requirements. The two best previously known approaches need O(n+ 1 e 3 ) and O(n.1 e) space, respectively. Our new approximation scheme requires only O(n + 1 e 2 ) space while also reducing the running time. For multiobjective optimization problems, it is meaningful to compute a set of solutions covering all possible trade-offs between the different objectives. The multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values. For this problem, efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions, the Pareto frontier, are studied.For the multiobjective one-dimensional knapsack problem, a practical fully polynomial-time approximation scheme (FPTAS) is derived. It is based on a new approach to the single-objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length. For the multiobjectivem-dimensional knapsack problem, the first known polynomial-time approximation scheme (PTAS), based on linear programming, is presented.