The Complexity of Planning Problems With Simple Causal Graphs

We present three new complexity results for classes of planning problems with simple causal graphs. First, we describe a polynomial-time algorithm that uses macros to generate plans for the class 3S of planning problems with binary state variables and acyclic causal graphs. This implies that plan generation may be tractable even when a planning problem has an exponentially long minimal solution. We also prove that the problem of plan existence for planning problems with multi-valued variables and chain causal graphs is NP-hard. Finally, we show that plan existence for planning problems with binary state variables and polytree causal graphs is NP-complete

Structure and Complexity in Planning with Unary Operators

Unary operator domains -- i.e., domains in which operators have a single effect -- arise naturally in many control problems. In its most general form, the problem of STRIPS planning in unary operator domains is known to be as hard as the general STRIPS planning problem -- both are PSPACE-complete. However, unary operator domains induce a natural structure, called the domain's causal graph. This graph relates between the preconditions and effect of each domain operator. Causal graphs were exploited by Williams and Nayak in order to analyze plan generation for one of the controllers in NASA's Deep-Space One spacecraft. There, they utilized the fact that when this graph is acyclic, a serialization ordering over any subgoal can be obtained quickly. In this paper we conduct a comprehensive study of the relationship between the structure of a domain's causal graph and the complexity of planning in this domain. On the positive side, we show that a non-trivial polynomial time plan generation algorithm exists for domains whose causal graph induces a polytree with a constant bound on its node indegree. On the negative side, we show that even plan existence is hard when the graph is a directed-path singly connected DAG. More generally, we show that the number of paths in the causal graph is closely related to the complexity of planning in the associated domain. Finally we relate our results to the question of complexity of planning with serializable subgoals

The Influence of k-Dependence on the Complexity of Planning

A planning problem is k-dependent if each action has at most k pre-conditions on variables unaffected by the action. This concept is well-founded since k is a constant for all but a few of the standard planning domains, and is known to have implications for tractability. In this paper, we present several new complexity results for P(k), the class of k-dependent planning problems with binary variables and polytree causal graphs. The problem of plan generation for P(k) is equivalent to determining how many times each variable can change. Using this fact, we present a polytime plan generation algorithm for P(2) and P(3). For constant k> 3, we introduce and use the notion of a cover to find conditions under which plan generation for P(k) is polynomial

Debt, deficits and finite horizons: the stochastic case

We introduce aggregate uncertainty and complete markets into Blanchard's (1985) perpetual youth model. We show how to construct a simple formula for the pricing kernel in terms of observable aggregate variables. We study a pure trade version of our model and we show it behaves much like the two-period overlapping generations model. Our methods are easily generalized to economies with production and they should prove useful to researchers who seek a tractable stochastic model in which fiscal policy has real effects on aggregate allocations.Overlapping generations ; indeterminacy ; sunspot equilibria ; aggregate uncertainty

An Extended Mean Field Game for Storage in Smart Grids

We consider a stylized model for a power network with distributed local power generation and storage. This system is modeled as network connection a large number of nodes, where each node is characterized by a local electricity consumption, has a local electricity production (e.g. photovoltaic panels), and manages a local storage device. Depending on its instantaneous consumption and production rates as well as its storage management decision, each node may either buy or sell electricity, impacting the electricity spot price. The objective at each node is to minimize energy and storage costs by optimally controlling the storage device. In a non-cooperative game setting, we are led to the analysis of a non-zero sum stochastic game with $N$ players where the interaction takes place through the spot price mechanism. For an infinite number of agents, our model corresponds to an Extended Mean-Field Game (EMFG). In a linear quadratic setting, we obtain and explicit solution to the EMFG, we show that it provides an approximate Nash-equilibrium for $N$-player game, and we compare this solution to the optimal strategy of a central planner.Comment: 27 pages, 5 figures. arXiv admin note: text overlap with arXiv:1607.02130 by other author