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COMPLEXITY&APPROXIMABILITY OF QUANTIFIED&STOCHASTIC CONSTRAINT SATISFACTION PROBLEMS

Description: Let D be an arbitrary (not necessarily finite) nonempty set, let C be a finite set of constant symbols denoting arbitrary elements of D, and let S and T be an arbitrary finite set of finite-arity relations on D. We denote the problem of determining the satisfiability of finite conjunctions of relations in S applied to variables (to variables and symbols in C) by SAT(S) (by SATc(S).) Here, we study simultaneously the complexity of decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. We present simple yet general techniques to characterize simultaneously, the complexity or efficient approximability of a number of versions/variants of the problems SAT(S), Q-SAT(S), S-SAT(S),MAX-Q-SAT(S) etc., for many different such D,C ,S, T. These versions/variants include decision, counting, maximization and approximate maximization problems, for unquantified, quantified and stochastically quantified formulas. Our unified approach is based on the following two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic represent ability. Some of the results extend the earlier results in [Pa85,LMP99,CF+93,CF+94O]u r techniques and results reported here also provide significant steps towards obtaining dichotomy theorems, for a number of the problems above, including the problems MAX-&-SAT( S), and MAX-S-SAT(S). The discovery of such dichotomy theorems, for unquantified formulas, has received significant recent attention in the literature [CF+93,CF+94,Cr95,KSW97]
Date: January 1, 2001
Creator: Hunt, H. B. (Harry B.); Marathe, M. V. (Madhav V.) & Stearns, R. E. (Richard E.)
Partner: UNT Libraries Government Documents Department

Sequential dynamical systems with threshold functions.

Description: A sequential dynamical system (SDS) (see [BH+01] and the references therein) consists of an undirected graph G(V,E) where each node {nu} {epsilon} V is associated with a Boolean state (s{sub {nu}}) and a symmetric Boolean function f{sub {nu}} (called the local transition function at {nu}). The inputs to f{sub {nu}} are s{sub {nu}} and the states of all the nodes adjacent to {nu}. In each step of the SDS, the nodes update their state values using their local transition functions in the order specified by a given permutation {pi} of the nodes. A configuration of the SDS is an n-tuple (b{sub 1}, b{sub 2}...,b{sub n}) where n = |V| and b{sub i} {epsilon} {l_brace}0,1{r_brace} is the state value of node {nu}{sub i}. The system starts in a specified initial configuration and each step of the SDS produces a (possibly new) configuration.
Date: January 1, 2001
Creator: Barrett, C. L. (Christopher L.); Hunt, H. B.; Marathe, M. V. (Madhav V.); Ravi, S. S.; Rosenkrantz, D. J. (Daniel J.) & Stearns, R. E. (Richard E.)
Partner: UNT Libraries Government Documents Department

Predecessor and permutation existence problems for sequential dynamical systems

Description: A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in BMR99, BR991 as a formal model for analyzing simulation systems. An SDS S is a triple (G, F,n ),w here (i) G(V,E ) is an undirected graph with n nodes with each node having a state, (ii) F = (fi, fi, . . ., fn), with fi denoting a function associated with node ui E V and (iii) A is a permutation of (or total order on) the nodes in V, A configuration of an SDS is an n-vector ( b l, bz, . . ., bn), where bi is the value of the state of node vi. A single SDS transition from one configuration to another is obtained by updating the states of the nodes by evaluating the function associated with each of them in the order given by n. Here, we address the complexity of two basic problems and their generalizations for SDSs. Given an SDS S and a configuration C, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration C' such that S has a transition from C' to C. (If C has no predecessor, C is known as a garden of Eden configuration.) Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded treewidth. In contrast, we show that allowing just one non-symmetric node function renders the problem NP-complete even when the underlying graph is a tree (which has a treewidth of 1). We also show that the PRE problem is efficiently solvable for SDSs whose state values are from ...
Date: January 1, 2002
Creator: Barrett, C. L. (Christopher L.); Hunt, H. B. (Harry B.); Marathe, M. V. (Madhav V.); Rosenkrantz, D. J. (Daniel J.) & Stearns, R. E. (Richard E.)
Partner: UNT Libraries Government Documents Department