# An FPTAS for Stochastic Unbounded Min-Knapsack Problem

## TL;DR

This paper introduces a Fully Polynomial-Time Approximation Scheme (FPTAS) for the stochastic unbounded min-knapsack problem, enabling near-optimal solutions efficiently despite the problem's stochastic and unbounded nature.

## Contribution

The paper presents the first FPTAS for the stochastic unbounded min-knapsack problem, extending classical knapsack algorithms to handle stochastic weights with provable approximation guarantees.

## Key findings

- Provides an FPTAS with polynomial runtime in 1/ε, n, and log W.
- Achieves solutions within (1+ε) factor of the optimal.
- Handles stochastic weight distributions for multiple item types.

## Abstract

In this paper, we study the stochastic unbounded min-knapsack problem ($\textbf{Min-SUKP}$). The ordinary unbounded min-knapsack problem states that: There are $n$ types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least $W$ and the total cost is minimized. The \prob~generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In \prob, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for $\textbf{Min-SUKP}$, i.e., the approximate value our algorithm computes is at most $(1+\epsilon)$ times the optimum, and our algorithm runs in $poly(1/\epsilon,n,\log W)$ time.