An Invitation to "Fine-grained Complexity of NP-Complete Problems"

TL;DR

This paper surveys the field of fine-grained complexity of NP-complete problems, exploring whether current algorithms are optimal and examining recent theoretical and algorithmic developments across multiple areas.

Contribution

It provides an overview of classic and recent results in fine-grained complexity, highlighting open questions and interdisciplinary approaches.

Findings

Naive algorithms may be optimal for some NP-complete problems

Recent developments connect algebra, combinatorics, cryptography, and algorithms

The survey identifies key open problems and research directions in the field

Abstract

Assuming that P is not equal to NP, the worst-case run time of any algorithm solving an NP-complete problem must be super-polynomial. But what is the fastest run time we can get? Before one can even hope to approach this question, a more provocative question presents itself: Since for many problems the naive brute-force baseline algorithms are still the fastest ones, maybe their run times are already optimal? The area that we call in this survey "fine-grained complexity of NP-complete problems" studies exactly this question. We invite the reader to catch up on selected classic results as well as delve into exciting recent developments in a riveting tour through the area passing by (among others) algebra, complexity theory, extremal and additive combinatorics, cryptography, and, of course, last but not least, algorithm design.