Core Concepts
The authors present improved complexity results and algorithms for approximating the limit value of stateful-discounted objectives and the parity value in concurrent stochastic games.
Abstract
The content discusses concurrent stochastic games, which are two-player zero-sum games played on finite-state graphs for an infinite number of steps. In each step, both players simultaneously and independently choose an action, and the next state is obtained according to a stochastic transition function.
The authors consider two types of objectives:
Stateful-discounted objectives: Similar to the classical discounted-sum objectives, but each state is associated with a different discount factor.
Parity objectives: A canonical representation for ω-regular objectives, where each state is associated with a priority, and the objective is to minimize the minimum priority visited infinitely often.
The main computational problems are the value-approximation problems, which aim to compute an approximation of the value within an arbitrary additive error.
The authors establish the following results:
The value-approximation problem for the limit value of stateful-discounted objectives and the parity value are in TFNP[NP], improving the previous EXPSPACE and PSPACE upper bounds, respectively.
The authors present algorithms that improve the dependency on the number of actions in the exponent from linear to logarithmic. In particular, if the number of states is constant, the algorithms run in polynomial time.
The technical contributions include:
A bound on the roots of multi-variate polynomials with integer coefficients, which is used to establish a connection between the stateful-discounted value and the limit value.
New characterizations of the stateful-discounted value and the limit value, leading to the improved complexity and algorithmic results.