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Hardness of SetCover Reoptimization

Klaus Jansen, Tobias Mömke, Björn Schumacher

TL;DR

The paper investigates the reoptimization hardness of SetCover and its weighted variant under four modification types: adding/removing a set and adding/removing an element. It establishes that, aside from a few unweighted cases with PTAS, most reoptimization problems inherit the same inapproximability as SetCover, and it rules out the existence of EPTAS under standard complexity assumptions for several settings. A key contribution is a framework linking reoptimization approximability to classical SetCover bounds and to fixed-parameter tractability, enabling conditional impossibility results for EPTAS and sharpening bounds in the weighted setting. The results delineate when reoptimization can yield near-optimal solutions and when the added information from related instances fails to surpass the fundamental hardness of SetCover, informing both theory and potential practical reoptimization strategies.

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.

Hardness of SetCover Reoptimization

TL;DR

The paper investigates the reoptimization hardness of SetCover and its weighted variant under four modification types: adding/removing a set and adding/removing an element. It establishes that, aside from a few unweighted cases with PTAS, most reoptimization problems inherit the same inapproximability as SetCover, and it rules out the existence of EPTAS under standard complexity assumptions for several settings. A key contribution is a framework linking reoptimization approximability to classical SetCover bounds and to fixed-parameter tractability, enabling conditional impossibility results for EPTAS and sharpening bounds in the weighted setting. The results delineate when reoptimization can yield near-optimal solutions and when the added information from related instances fails to surpass the fundamental hardness of SetCover, informing both theory and potential practical reoptimization strategies.

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.

Paper Structure

This paper contains 16 sections, 15 theorems, 10 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Results
  4. Outline of the paper
  5. Preliminaries
  6. Unweighted Set Cover
  7. $S^{+}$-SetCover
  8. $S^{-}$-SetCover
  9. $e^{+}$-SetCover
  10. $e^{-}$-SetCover
  11. Weighted Set Cover
  12. $S^{+}$-wSetCover
  13. $S^{-}$-wSetCover
  14. $e^{+}$-wSetCover
  15. $e^{-}$-wSetCover
  16. ...and 1 more sections

Key Result

lemma 1

Let $\rho \geq 1$. Let $f : \mathbb{N} \to \mathbb{Q}_{\geq 1}$. An $f(\abs{U})$-approximation algorithm for ($S^{+}$, $\rho$)-SetCover implies an $2f(2\abs{U})$-approximation algorithm for SetCover.

Figures (2)

  • Figure 1: Construction in \ref{['prop:setc:add-set:approx']}. The colorful center is the original instance, the outer dots are the duplicates, and the black sets are the sets that cover every element together with its duplicate. The gray set is added as the local modification to cover all duplicates.
  • Figure 2: Construction in \ref{['prop:wsetc:adding:inapprox']}. The black dots represent the original elements of the SetCover instance. The colorful sets are the sets of the original instance but each extended with a correspondingly colored element. The extra sets $G$ and $R$ are the brown and gray sets respectfully. The element that is added as the local modification is labeled $e_{\mathrm{new}}$.

Theorems & Definitions (27)

  • definition 1: wSetCover
  • lemma 1
  • proof
  • remark 2
  • corollary 3
  • lemma 4
  • proof
  • corollary 5
  • lemma 6: EPTAS implies Parameterized Algorithm
  • proof
  • ...and 17 more