An Invitation to "Fine-grained Complexity of NP-Complete Problems"
Jesper Nederlof
TL;DR
This paper surveys the fine-grained complexity of NP-complete problems, focusing on the fastest possible worst-case runtimes and when simple brute-force baselines may already be near-optimal under $P\neq NP$ and related hypotheses. It synthesizes a range of techniques—from sparsification lemmas and random restrictions to matrix-factorization methods, Yates’ transforms, and container-based set-cover strategies—to achieve nontrivial speedups for classic problems like CNF-SAT, Set Cover, Hamiltonicity, and Subset Sum. Key contributions include explicit runtime improvements such as $O^*(1.5^n)$ for 3-coloring, $O^*(2^{(1-1/O(k))n})$ for $k$-CNF-SAT, and $O^*(3^{n/2})$ for undirected Hamiltonicity, along with structural tools (sparsification, representations, containers) that connect to ETH/SETH and tensor-rank frameworks. The survey also highlights ongoing directions, including parameterized and coarser-grained analyses, as well as quantum speedups, illustrating how these ideas influence both practical algorithm design and foundational complexity theory.
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.