Average Sensitivity of Dynamic Programming

TL;DR

This paper investigates the average sensitivity of dynamic programming algorithms, proposing new algorithms with low average sensitivity for several classical problems, enhancing stability analysis in uncertain data processing.

Contribution

It introduces a framework for analyzing and reducing the average sensitivity of dynamic programming algorithms, including novel algorithms for multiple problems with provably low sensitivity.

Findings

Developed a $(1- ext{delta})$-approximation algorithm for maximum weight chain with low average sensitivity.

Extended the approach to problems like LIS, interval scheduling, LCS, and RNA folding via problem reductions.

Achieved low average sensitivity bounds for these problems, improving stability in uncertain data scenarios.

Abstract

When processing data with uncertainty, it is desirable that the output of the algorithm is stable against small perturbations in the input. Varma and Yoshida [SODA'21] recently formalized this idea and proposed the notion of average sensitivity of algorithms, which is roughly speaking, the average Hamming distance between solutions for the original input and that obtained by deleting one element from the input, where the average is taken over the deleted element. In this work, we consider average sensitivity of algorithms for problems that can be solved by dynamic programming. We first present a $(1-\delta)$-approximation algorithm for finding a maximum weight chain (MWC) in a transitive directed acyclic graph with average sensitivity $O(\delta^{-1}\log^3 n)$, where $n$ is the number of vertices in the graph. We then show algorithms with small average sensitivity for various dynamic programming problems by reducing them to the MWC problem while preserving average sensitivity, including the longest increasing subsequence problem, the interval scheduling problem, the longest common subsequence problem, the longest palindromic subsequence problem, the knapsack problem with integral weight, and the RNA folding problem. For the RNA folding problem, our reduction is highly nontrivial because a naive reduction generates an exponentially large graph, which only provides a trivial average sensitivity bound.