Approximately Counting Knapsack Solutions in Subquadratic Time

TL;DR

This paper presents a novel randomized approximation algorithm for counting knapsack solutions that operates in subquadratic time, significantly improving efficiency over previous methods and employing advanced techniques from combinatorics and algorithms.

Contribution

It introduces the first sub-quadratic time randomized approximation scheme for #Knapsack, refining Dyer's framework with new structural and algorithmic techniques.

Findings

Achieves (n^{1.5} \u00b7 ^{-2}}) time complexity for approximate counting.

Reduces sample complexity using structural lemmas and hitting-set arguments.

Employs advanced subset sum algorithms and recent convolution techniques.

Abstract

We revisit the classic #Knapsack problem, which asks to count the Boolean points $(x_1,\dots,x_n)\in\{0,1\}^n$ in a given half-space $\sum_{i=1}^nW_ix_i\le T$. This #P-complete problem admits $(1\pm\epsilon)$-approximation. Before this work, [Dyer, STOC 2003]'s $\tilde{O}(n^{2.5}+n^2{\epsilon^{-2}})$-time randomized approximation scheme remains the fastest known in the natural regime of $\epsilon\ge 1/polylog(n)$. In this paper, we give a randomized $(1\pm\epsilon)$-approximation algorithm in $\tilde{O}(n^{1.5}{\epsilon^{-2}})$ time (in the standard word-RAM model), achieving the first sub-quadratic dependence on $n$. Such sub-quadratic running time is rare in the approximate counting literature in general, as a large class of algorithms naturally faces a quadratic-time barrier. Our algorithm follows Dyer's framework, which reduces #Knapsack to the task of sampling (and approximately counting) solutions in a randomly rounded instance with poly(n)-bounded integer weights. We refine Dyer's framework using the following ideas: - We decrease the sample complexity of Dyer's Monte Carlo method, by proving some structural lemmas for typical points near the input hyperplane via hitting-set arguments, and appropriately setting the rounding scale. - Instead of running a vanilla dynamic program on the rounded instance, we employ techniques from the growing field of pseudopolynomial-time Subset Sum algorithms, such as FFT, divide-and-conquer, and balls-into-bins hashing of [Bringmann, SODA 2017]. We also need other ingredients, including a surprising application of the recent Bounded Monotone (max,+)-Convolution algorithm by [Chi-Duan-Xie-Zhang, STOC 2022] (adapted by [Bringmann-D\"urr-Polak, ESA 2024]), the notion of sum-approximation from [Gawrychowski-Markin-Weimann, ICALP 2018]'s #Knapsack approximation scheme, and a two-phase extension of Dyer's framework for handling tiny weights.