Parity and Streett Games with Costs

TL;DR

This paper introduces cost-based parity and Streett games on finite graphs, establishing complexity results and strategy types for the first player, and generalizing classical omega-regular conditions with cost constraints.

Contribution

It extends classical parity and Streett games by incorporating costs, providing complexity classifications, and characterizing the memory requirements for winning strategies.

Findings

First player has positional strategies in cost-parity games.

Winner determination is in NP ∩ coNP for cost-parity games.

Winner determination is EXPTIME-complete for cost-Streett games.

Abstract

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical omega-regular conditions and the corresponding finitary conditions. For parity games with costs we show that the first player has positional winning strategies and that determining the winner lies in NP and coNP. For Streett games with costs we show that the first player has finite-state winning strategies and that determining the winner is EXPTIME-complete. The second player might need infinite memory in both games. Both types of games with costs can be solved by solving linearly many instances of their classical variants.