Hardness of SetCover Reoptimization

TL;DR

This paper investigates the computational hardness of reoptimizing the SetCover problem under various modifications, demonstrating that most reoptimization scenarios remain as hard as the original problem, with some cases proven to lack efficient approximation schemes.

Contribution

It establishes the hardness of multiple reoptimization variants of SetCover, including new results on the non-existence of EPTAS for certain cases under standard assumptions.

Findings

Most reoptimization variants are as hard as SetCover.

Adding/removing elements in unweighted case admit PTAS, but no EPTAS under common assumptions.

New techniques combine NP-hardness proofs with FPT and EPTAS relations.

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.