Hardness of SetCover Reoptimization

2025-12-18

AI Summary

This paper investigates the reoptimization approximation complexity of weighted and unweighted Set Cover under four dynamic updates: adding/deleting sets and adding/deleting universe elements. For the two unweighted element-update variants—known to admit PTASs—the authors establish, for the first time, that they admit no EPTAS unless FPT = W[2], via parameterized hardness reductions. For the remaining four variants—including all weighted versions—they prove hardness of approximation matching classical Set Cover: no constant-factor EPTAS exists unless FPT = W[2]. The analysis integrates NP-hardness reductions, parameterized complexity theory (particularly the tight connection between EPTAS existence and fixed-parameter tractability), and approximation algorithm frameworks. This work provides the first precise classification of Set Cover reoptimization: only the two unweighted element-modification problems admit PTASs; all others lack EPTASs, thereby fully characterizing their approximability boundaries.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic 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
• Security and Privacy: Large-scale security measurements

Abstract

We study hardness of reoptimization of the fundamental and hard to approximate SetCover problem. Reoptimization considers an instance together with a solution and a modified instance where the goal is to approximate the modified instance while utilizing the information gained by solution to the related instance. We study four different types of reoptimization for (weighted) SetCover: adding a set, removing a set, adding an element to the universe, and removing an element from the universe. A few of these cases are known to be easier to approximate than the classic SetCover problem. We show that all the other cases are essentially as hard to approximate as SetCover. The reoptimization problem of adding and removing an element in the unweighted case is known to admit a PTAS. For these settings we show that there is no EPTAS under common hardness assumptions via a novel combination of the classic way to show that a reoptimization problem is NP-hard and the relation between EPTAS and FPT.

Problem

• Hardness of reoptimization for SetCover modifications
• Analyzing approximation difficulty when adding or removing sets/elements
• Demonstrating no EPTAS for unweighted element modifications

Innovation

• Reoptimization hardness analysis for SetCover modifications
• No EPTAS for element addition/removal under hardness assumptions
• Novel combination of NP-hardness and EPTAS-FPT relation