Unveiling the Primal Pathwidth SETH: Fine-Grained Complexity Questions Explored

Core Concepts

The author explores fundamental fine-grained questions in parameterized complexity, showing that they are all equivalent to the Primal Pathwidth-Strong Exponential Time Hypothesis (PP-SETH), indicating a meta-complexity property cutting across traditional complexity classes.

Abstract

The content delves into the equivalence of various problems under the PP-SETH, showcasing how different complexities converge to a single hypothesis. It highlights the importance of dynamic programming algorithms over linear structures and challenges existing assumptions like the SETH. The paper presents sharp lower bounds for problems previously known under the SETH, emphasizing a more plausible hypothesis with broader implications.

Stats

Dominating Set: time (3 − ε)pwnO(1) Coloring: time pw(1−ε)pwnO(1) Reconfiguration between size-k independent sets: time n(1−ε)k List Coloring: time n(1−ε)pw Short word accepted by k n-state DFAs: time n(1−ε)k

Quotes

"We achieve this by putting forth a natural complexity assumption which we call the Primal Pathwidth-Strong Exponential Time Hypothesis (PP-SETH)." "Our results indicate that PP-SETH-equivalence is a meta-complexity property that cuts across traditional complexity classes." "Improving upon this global algorithm for one problem should mean something for the others."

Key Insights Distilled From

The Primal Pathwidth SETH

by Michael Lamp... at arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07239.pdf

Deeper Inquiries

What implications does disproving the PP-SETH have on existing fine-grained lower bounds

Disproving the PP-SETH would have significant implications on existing fine-grained lower bounds. The PP-SETH serves as a unifying hypothesis that ties together various parameterized complexity problems, showing their equivalence to each other. If the PP-SETH were to be falsified, it would mean that all the problems currently known to have optimal algorithms based on dynamic programming over linear structures may not actually be at their optimal complexity level. This would necessitate a reevaluation of existing lower bounds and potentially lead to the development of new algorithms with improved time complexities for these problems.

How does the PP-SETH challenge traditional complexity assumptions and classifications

The PP-SETH challenges traditional complexity assumptions and classifications by introducing a more plausible and flexible hypothesis compared to existing assumptions like the SETH or ETH. Unlike one-way reductions commonly seen in complexity theory, the PP-SETH establishes an equivalence class among parameterized problems solvable using simple DP algorithms over linear structures. This meta-complexity property cuts across traditional complexity classes, highlighting how breaking any bound under the PP-SETH requires breaking all related bounds, old and new. By challenging established assumptions and classifications, the PP-SETH offers a fresh perspective on problem optimality in parameterized complexity.

Can advancements in DP algorithms over linear structures lead to breakthroughs in other complex problems beyond those discussed in this paper

Advancements in DP algorithms over linear structures can indeed lead to breakthroughs in other complex problems beyond those discussed in this paper. The unifying nature of the PP-SETH suggests that improvements in DP algorithms for specific parameterized problems could have ripple effects across different problem domains sharing similar characteristics. For instance, if a faster algorithm is developed for solving List Coloring or Independent Set Reconfiguration based on pathwidth considerations, it could inspire novel approaches or optimizations applicable to related XNLP-complete problems or even broader classes of NP-hard graph optimization challenges. These advancements may pave the way for enhanced algorithmic techniques with wider applicability and efficiency gains across various computational tasks beyond what is explicitly covered in this study.