A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
نویسندگان
چکیده
منابع مشابه
A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
Given n elements with nonnegative integer weights w1, . . . , wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error 1±ε in time polynomial in n and 1/ε (fully polynomial...
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Given n elements with nonnegative integer weights w = (w1, . . . , wn), an integer capacity C and positive integer ranges u = (u1, . . . , un), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error in time ...
متن کاملA Polynomial Time Approximation Scheme for the Multiple Knapsack Problem
The multiple knapsack problem (MKP) is a natural and well-known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is...
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In previous chapters we have seen the definition of a constant factor approximation algorithm. In this chapter, we will introduce the notion of a polynomial time approximation scheme (PTAS), which allows approximability to any required degree. To illustrate how PTAS works, we will study two examples, including the knapsack problem and the bin packing problem. The dynamic programming technique w...
متن کاملA New Fully Polynomial Approximation Scheme for the Knapsack Problem
A new fully polynomial approximation scheme (FPTAS) is presented for the classical 0{1 knapsack problem. It considerably improves the space requirements. The two best previously known approaches need O(n+1=" 3) and O(n1=") space, respectively. Our new approximation scheme requires only O(n + 1=" 2) space while also reducing the running time.
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ژورنال
عنوان ژورنال: SIAM Journal on Computing
سال: 2012
ISSN: 0097-5397,1095-7111
DOI: 10.1137/11083976x