Polynomial-Time Approximation Schemes for Knapsack and Related Counting Problems using Branching Programs

TL;DR

This paper presents a deterministic polynomial-time approximation scheme for counting solutions to knapsack and related problems, improving over prior randomized methods and introducing branching programs for efficient approximation.

Contribution

It introduces a new deterministic approximation algorithm for knapsack and related problems using small-width branching programs, with polynomial dependence on accuracy.

Findings

Provides a (1+eps)-approximate counting algorithm with polynomial runtime

Extends to multidimensional knapsack and contingency tables with constant constraints

Introduces new query algorithms for learning functions of k halfspaces

Abstract

We give a deterministic, polynomial-time algorithm for approximately counting the number of {0,1}-solutions to any instance of the knapsack problem. On an instance of length n with total weight W and accuracy parameter eps, our algorithm produces a (1 + eps)-multiplicative approximation in time poly(n,log W,1/eps). We also give algorithms with identical guarantees for general integer knapsack, the multidimensional knapsack problem (with a constant number of constraints) and for contingency tables (with a constant number of rows). Previously, only randomized approximation schemes were known for these problems due to work by Morris and Sinclair and work by Dyer. Our algorithms work by constructing small-width, read-once branching programs for approximating the underlying solution space under a carefully chosen distribution. As a byproduct of this approach, we obtain new query algorithms for learning functions of k halfspaces with respect to the uniform distribution on {0,1}^n. The running time of our algorithm is polynomial in the accuracy parameter eps. Previously even for the case of k=2, only algorithms with an exponential dependence on eps were known.