A Deterministic Fully Polynomial Time Approximation Scheme For Counting Integer Knapsack Solutions Made Easy

Given n elements with nonnegative integer weights w=(w_1,...,w_n), an integer capacity C and positive integer ranges u=(u_1,...,u_n), 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 epsilon in time polynomial in n, log U and 1/epsilon, where U=max_i u_i. More precisely, our algorithm runs in O((n^3 log^2 U)/epsilon) log (n log U)/epsilon) time. This is an improvement of n^2 and 1/epsilon (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints

A Faster FPTAS for #Knapsack

Given a set W = {w_1,..., w_n} of non-negative integer weights and an integer C, the #Knapsack problem asks to count the number of distinct subsets of W whose total weight is at most C. In the more general integer version of the problem, the subsets are multisets. That is, we are also given a set {u_1,..., u_n} and we are allowed to take up to u_i items of weight w_i. We present a deterministic FPTAS for #Knapsack running in O(n^{2.5}epsilon^{-1.5}log(n epsilon^{-1})log (n epsilon)) time. The previous best deterministic algorithm [FOCS 2011] runs in O(n^3 epsilon^{-1} log(n epsilon^{-1})) time (see also [ESA 2014] for a logarithmic factor improvement). The previous best randomized algorithm [STOC 2003] runs in O(n^{2.5} sqrt{log (n epsilon^{-1})} + epsilon^{-2} n^2) time. Therefore, for the case of constant epsilon, we close the gap between the O~(n^{2.5}) randomized algorithm and the O~(n^3) deterministic algorithm. For the integer version with U = max_i {u_i}, we present a deterministic FPTAS running in O(n^{2.5}epsilon^{-1.5}log(n epsilon^{-1} log U)log (n epsilon) log^2 U) time. The previous best deterministic algorithm [TCS 2016] runs in O(n^3 epsilon^{-1}log(n epsilon^{-1} log U) log^2 U) time