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

Alexander Kozachinskiy; Mikhail Vyalyi

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arXiv · 1902.07175

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

Alexander Kozachinskiy·Mikhail Vyalyi

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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.

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Alexander Kozachinskiy, Mikhail Vyalyi. 2019-02-19. An application of communication complexity, Kolmogorov complexity and extremal combinatorics to parity games. https://arxiv.org/abs/1902.07175

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