Kernelization, Proof Complexity and Social Choice
Abstract
We display an application of the notions of kernelization and data reduction from parameterized complexity to proof complexity: Specifically, we show that the existence of data reduction rules for a parameterized problem having (a). a smalllength reduction chain, and (b). smallsize (extended) Frege proofs certifying the soundness of reduction steps implies the existence of subexponential size (extended) Frege proofs for propositional formalizations of the given problem. We apply our result to infer the existence of subexponential Frege and extended Frege proofs for a variety of problems. Improving earlier results of Aisenberg et al. (ICALP 2015), we show that propositional formulas expressing (a stronger form of) the KneserLovász Theorem have polynomial size Frege proofs for each constant value of the parameter k. Previously only quasipolynomial bounds were known (and only for the ordinary KneserLovász Theorem). Another notable application of our framework is to impossibility results in computational social choice: we show that, for any fixed number of agents, propositional translations of the Arrow and GibbardSatterthwaite impossibility theorems have subexponential size Frege proofs.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.13681
 arXiv:
 arXiv:2104.13681
 Bibcode:
 2021arXiv210413681I
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Computer Science and Game Theory;
 Computer Science  Logic in Computer Science;
 Mathematics  Logic
 EPrint:
 Revised version will appear in the Proceedings of ICALP 2021