A Chronology of Set Cover Inapproximability Results

TL;DR

This paper presents a chronological overview of the development of inapproximability bounds for the Set Cover problem, highlighting key proof techniques and their evolution towards establishing the ln(n) bound as tight.

Contribution

It unifies and compares various proof methods for Set Cover inapproximability, clarifying the progression towards the optimal ln(n) bound.

Findings

Establishment of ln(n) as the tight inapproximability bound for Set Cover

Comparison of different proof techniques and their effectiveness

Unified framework for understanding Set Cover hardness results

Abstract

It is well-known that an algorithm exists which approximates the NP-complete problem of Set Cover within a factor of ln(n), and it was recently proven that this approximation ratio is optimal unless P = NP. This optimality result is the product of many advances in characterizations of NP, in terms of interactive proof systems and probabilistically checkable proofs (PCP), and improvements to the analyses thereof. However, as a result, it is difficult to extract the development of Set Cover approximation bounds from the greater scope of proof system analysis. This paper attempts to present a chronological progression of results on lower-bounding the approximation ratio of Set Cover. We analyze a series of proofs of progressively better bounds and unify the results under similar terminologies and frameworks to provide an accurate comparison of proof techniques and their results. We also treat many preliminary results as black-boxes to better focus our analysis on the core reductions to Set Cover instances. The result is alternative versions of several hardness proofs, beginning with initial inapproximability results and culminating in a version of the proof that ln(n) is a tight lower bound.