Average sensitivity of the Knapsack Problem

TL;DR

This paper analyzes the average sensitivity of algorithms for the knapsack problem, providing approximation algorithms with provable bounds and applying them to incremental settings with random item arrivals.

Contribution

It introduces a $(1- heta)$-approximation algorithm with optimal average sensitivity bounds and applies it to incremental knapsack in the random-order model.

Findings

Approximation algorithm with $O(rac{1}{ heta} ext{log} rac{1}{ heta})$ average sensitivity.

Lower bound showing any $(1- heta)$-approximation has $ ilde{ heta}^{-1}$ average sensitivity.

Efficient incremental algorithm with $O(rac{1}{ heta} ext{log} rac{1}{ heta})$ amortized recourse and update time.

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 $(1-\epsilon)$-approximation algorithm for the knapsack problem with average sensitivity $O(\epsilon^{-1}\log \epsilon^{-1})$. Then, we complement this result by showing that any $(1-\epsilon)$-approximation algorithm has average sensitivity $\Omega(\epsilon^{-1})$. 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 $\epsilon > 0$, there exists a $(1-\epsilon)$-approximation algorithm with amortized recourse $O(\epsilon^{-1}\log \epsilon^{-1})$ and amortized update time $O(\log n+f_\epsilon)$, where $n$ is the total number of items and $f_\epsilon>0$ is a value depending on $\epsilon$.