Average sensitivity of the Knapsack Problem

TL;DR

The paper addresses the stability of knapsack solutions under input perturbations by formalizing average sensitivity via the earth mover's distance between output distributions. It develops a stable-on-average $(1-\epsilon)$-approximation algorithm with average sensitivity $O(\epsilon^{-1}\log \epsilon^{-1})$, and complements this with a near-matching lower bound of $\Omega(\epsilon^{-1})$, establishing tightness up to a logarithmic factor. Extending to practice, it provides an FPRAS variant, a deterministic stable algorithm for the simple knapsack, and a method to translate average-sensitivity guarantees into efficient incremental dynamic knapsack performance in the random-order setting, achieving amortized recourse $O(\epsilon^{-1}\log\epsilon^{-1})$ and update time $O(\log n+f_\epsilon)$. The results offer a principled way to attain stable allocations in resource-constrained settings and enable robust dynamic/ streaming implementations. Overall, the work links average-case stability to dynamic optimization, with implications for resilient resource management and privacy-conscious algorithm design.

Abstract

In resource allocation, we often require that the output allocation of an algorithm is stable against input perturbation because frequent reallocation is costly and untrustworthy. Varma and Yoshida (SODA'21) formalized this requirement for algorithms as the notion of average sensitivity. Here, the average sensitivity of an algorithm on an input instance is, roughly speaking, the average size of the symmetric difference of the output for the instance and that for the instance with one item deleted, where the average is taken over the deleted item. In this work, we consider the average sensitivity of the knapsack problem, a representative example of a resource allocation problem. We first show a $(1-ε)$-approximation algorithm for the knapsack problem with average sensitivity $O(ε^{-1}\log ε^{-1})$. Then, we complement this result by showing that any $(1-ε)$-approximation algorithm has average sensitivity $Ω(ε^{-1})$. As an application of our algorithm, we consider the incremental knapsack problem in the random-order setting, where the goal is to maintain a good solution while items arrive one by one in a random order. Specifically, we show that for any $ε> 0$, there exists a $(1-ε)$-approximation algorithm with amortized recourse $O(ε^{-1}\log ε^{-1})$ and amortized update time $O(\log n+f_ε)$, where $n$ is the total number of items and $f_ε>0$ is a value depending on $ε$.

Theorems & Definitions (61)

• theorem 1
• proposition 1
• theorem 2
• theorem 3
• theorem 4
• lemma 1: dantzig1957discrete
• lemma 2
• lemma 3
• proof
• lemma 4