A Nested Family of kk-total Effective Rewards for Positional Games

We consider Gillette's two-person zero-sum stochastic games with perfect information. For each k \in \ZZ_+ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as {\it mean payoff} and {\it total reward}, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games

A pseudo-polynomial algorithm for mean payoff stochastic games with perfect information and few random positions

We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph G = (V;E), with local rewards r : E Z, and three types of positions: black VB, white VW, and random VR forming a partition of V . It is a long- standing open question whether a polynomial time algorithm for BWR-games exists, or not, even when |VR| = 0. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo-polynomial time. More precisely, in any BWR-game with |VR| = O(1), a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in |VW| + |VB|, the maximum absolute local reward, and the common denominator of the transition probabilities

Constraint Satisfaction Problems over Numeric Domains

We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra