Dynamic Set Cover with Worst-Case Recourse

2025-11-10

AI Summary

This paper studies the dynamic set cover problem, aiming to efficiently maintain an approximate optimal solution under frequent element insertions and deletions, while jointly optimizing worst-case update time and recourse—the number of set changes per update. We propose a generic black-box transformation framework that converts any static set cover algorithm into a low-recourse dynamic one; integrating greedy and primal-dual techniques, it supports weighted instances and bounds solution cost fluctuations. Theoretically, our approach achieves $O(alpha)$ worst-case recourse and $O(T + alpha C)$ update time for any constant $C$, where $alpha$ is the approximation ratio and $T$ the running time of the underlying static algorithm. To our knowledge, this is the first result to simultaneously guarantee both approximation quality and efficient updates while establishing a worst-case bound on recourse—thereby bridging a fundamental gap in the joint optimization of low adjustment cost and fast updates.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the dynamic set cover (SC) problem, the input is a dynamic universe of at most $n$ elements and a fixed collection of $m$ sets, where each element belongs to at most $f$ sets and each set has cost in $[1/C, 1]$. The objective is to efficiently maintain an approximate minimum SC under element updates; efficiency is primarily measured by the update time, but another important parameter is the recourse (number of changes to the solution per update). Ideally, one would like to achieve low worst-case bounds on both update time and recourse. One can achieve approximation $(1+epsilon)ln n$ (greedy-based) or $(1+epsilon)f$ (primal-dual-based) with worst-case update time $O(flog n)$ (ignoring $epsilon$ dependencies). However, despite a large body of work, no algorithm with low update time (even amortized) and nontrivial worst-case recourse is known, even for unweighted instances ($C = 1$)! We remedy this by providing a transformation that, given as a black-box a SC algorithm with approximation $alpha$ and update time $T$, returns a set cover algorithm with approximation $(2 + epsilon)alpha$, update time $O(T + alpha C)$, and worst-case recourse $O(alpha C)$. Our main results are obtained by leveraging this transformation for constant $C$:...

Problem

• Maintaining approximate minimum set cover under dynamic element updates
• Achieving low worst-case update time and bounded recourse simultaneously
• Transforming existing algorithms to guarantee worst-case recourse bounds

Innovation

• Transforms existing set cover algorithms with approximation guarantees
• Achieves constant worst-case recourse per update operation
• Maintains approximation ratio while bounding update time