Publisher Summary This chapter discusses the concepts needed for defining the complexity classes. A complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form—the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R , where n is the size of the input. The simpler complexity classes are defined by various factors. The type of computational problem in which the most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, etc.

A catalog of complexity classes

Controlled stochastic systems occur in science engineering, manufacturing, social sciences, and many other cntexts. If the systems is modeled as a Markov decision process (MDP) and will run ad infinitum, the optimal control policy can be computed in polynomial time using linear programming. The problems considered here assume that the time that the process will run is finite, and based on the size of the input. There are mny factors that compound the complexity of computing the optimal policy. For instance, there are many factors that compound the complexity of this computation. For instance, if the controller does not have complete information about the state of the system, or if the system is represented in some very succint manner, the optimal policy is provably not computable in time polynomial in the size of the input. We analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a MDP, based on a number of confounding factors, including the observability of the system state; the succinctness of the representation; the type of policy; even the number of actions relative to the number of states. In almost every case, we show that the decision problem is complete for some known complexity class. Some of these results are familiar from work by Papadimitriou and Tsitsiklis and others, but some, such as our PL-completeness proofs, are surprising. We include proofs of completeness for natural problems in the as yet little-studied classes NPPP.

Complexity of finite-horizon Markov decision process problems

Definition de deux hierarchies des classes de complexite. Localisation de la hierarchie de differences et de la hierarchie du tableau de verite pour NP. Exemples d'ensembles complets dans les deux hierarchies et examen de leurs correlations

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The difference and truth-table hierarchies for NP

Short encodings of planar graphs and maps

We show that there exists a graph G with n /spl middot/ 2/sup O(log* n)/ nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies the existence of a graph with n/sup k/2/sup O(log* n)/ nodes that contains all n-node graphs of fixed arboricity k as node-induced subgraphs. We provide a lower bound of /spl Omega/(n/sup k/) for the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.

Small induced-universal graphs and compact implicit graph representations

This work presents a different approach to the continuing work done in the development of efficient algorithms for the solution of graph problems. While until now the main effort has been directed towards designing more efficient algorithms, we try to get a better solution by changing the graph representation. We suggest a new model for graph representation, which we named Graph Construction Representation.
The new model proved to have two significant advantages over the conventional graph representations: (1) For many graphs, this model will use less space, or in other words will have a "succinct" representation. (2) We developed several algorithms which accept the new model as input. The time complexity for those algorithms when applied to graphs which are represented "succinctly" is improved in comparison with the best known algorithm.
In this thesis the Graph Construction Representation is investigated. For many graph families it is shown how they are represented using this model. Bounds on the size of the representation are given. Several algorithms for checking graph properties are described, among them "Connectivity", "Has a Triangle?" and "Maximum Independent Set". We identify problems which are still open concerning the representation of graphs which are related to a graph already succinctly represented. We have only partial answers for these questions. We also present another graph representation, which was named the Small Circuit Representation, for which we prove that checking many graph properties is hard.

Hana Galperin

Papers

Succinct representation of graphs