An application of communication complexity, Kolmogorov complexity and extremal combinatorics to parity games

TL;DR

This paper investigates the limitations of separation automata in solving parity games by establishing exponential lower bounds for a restricted class, using advanced techniques from communication complexity, Kolmogorov complexity, and extremal combinatorics.

Contribution

It introduces a novel lower bound for a class of separation automata, combining multiple theoretical techniques to advance understanding of parity game algorithms.

Findings

Proves exponential lower bounds for certain separation automata

Develops new bounds in extremal combinatorics related to set families

Integrates communication and Kolmogorov complexity methods in automata analysis

Abstract

So-called separation automata are in the core of several recently invented quasi-polynomial time algorithms for parity games. An explicit $q$-state separation automaton implies an algorithm for parity games with running time polynomial in $q$. It is open whether a polynomial-state separation automaton exists. A positive answer will lead to a polynomial-time algorithm for parity games, while a negative answer will at least demonstrate impossibility to construct such an algorithm using separation approach. In this work we prove exponential lower bound for a restricted class of separation automata. Our technique combines communication complexity and Kolmogorov complexity. One of our technical contributions belongs to extremal combinatorics. Namely, we prove a new upper bound on the product of sizes of two families of sets with small pairwise intersection.