Core Concepts
Primal Pathwidth-Strong Exponential Time Hypothesis (PP-SETH) unifies fine-grained questions in parameterized complexity, showing equivalence and sharpening lower bounds.
Abstract
Standalone Note here
Introduction:
Dynamic Programming (DP) importance in parameterized complexity.
Central role of DP in NP-hard graph problems with graph width parameters.
Interest in optimal algorithms and known results for Coloring problem.
Robustness:
PP-SETH equivalence to various SAT variants and CSPs with non-binary alphabets.
Importance of promise CSP for reductions.
Supporting Evidence:
PP-SETH implication by SETH, k-OVA, and Set Cover Conjecture.
Strengthening confidence in lower bounds under more plausible assumptions.
Single-exponential FPT problems:
Equivalence of various problems to falsifying the PP-SETH.
Implications for k-Coloring, Independent Set, Dominating Set, and Set Cover.
Super-exponential FPT problems:
Equivalence of Coloring and C4-Hitting Set to PP-SETH falsification.
XNLP-complete problems:
Equivalence of List Coloring, DFA Intersection, and Independent Set Reconfiguration to PP-SETH falsification.
Overview of techniques:
Importance of robustness in formulations for reductions.
Previous work:
Comparison with ETH-based bounds limitations due to one-way reductions.
Discussion and Directions for Further Work:
Potential investigation into treewidth variant of the PP-SETH hypothesis.
Stats
3-SAT cannot be solved in time (2 − ε)pwnO(1)
(k − ε)pwnO(1) algorithm for k-coloring is optimal under SETH
List Coloring can be solved in time n(1−ε)pw under PP_SETH