On Problems Related to Unbounded SubsetSum: A Unified Combinatorial Approach

TL;DR

This paper introduces a unified combinatorial approach to solve unbounded subset sum problems and their generalizations, achieving near-linear algorithms and establishing a connection with min-plus convolution complexity.

Contribution

It presents the first near-linear algorithms for CoinChange and Residue Table problems and links All-Target Unbounded Knapsack to min-plus convolution complexity.

Findings

Near-linear algorithms for CoinChange and Residue Table

First deterministic near-linear algorithms for these problems

Establishes equivalence between Unbounded Knapsack and min-plus convolution

Abstract

Unbounded SubsetSum is a classical textbook problem: given integers $w_1,w_2,\cdots,w_n\in [1,u],~c,u$, we need to find if there exists $m_1,m_2,\cdots,m_n\in \mathbb{N}$ satisfying $c=\sum_{i=1}^n w_im_i$. In its all-target version, $t\in \mathbb{Z}_+$ is given and answer for all integers $c\in[0,t]$ is required. In this paper, we study three generalizations of this simple problem: All-Target Unbounded Knapsack, All-Target CoinChange and Residue Table. By new combinatorial insights into the structures of solutions, we present a novel two-phase approach for such problems. As a result, we present the first near-linear algorithms for CoinChange and Residue Table, which runs in $\tilde{O}(u+t)$ and $\tilde{O}(u)$ time deterministically. We also show if we can compute $(\min,+)$ convolution for $n$-length arrays in $T(n)$ time, then All-Target Unbounded Knapsack can be solved in $\tilde{O}(T(u)+t)$ time, thus establishing sub-quadratic equivalence between All-Target Unbounded Knapsack and $(\min,+)$ convolution.