problem (csp), namely, naive backtracking (BT), backjumping (BJ), conflict-directed backjumping

... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400-variable 3-SAT problems in about 2 hours on the average. In general, it can solve hard n-variable random 3-SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NP-complete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.

Scheduling deals with the allocation of resources over time to perform a collection of tasks. Scheduling problems arise in domains as diverse as manufacturing, computer processing, transportation, health care, space exploration, education, etc. Scheduling problems are conveniently formulated as Constraint Satisfaction Problems (CSPs) or Constrained Optimization Problems (COPs). A general paradigm for solving CSPs and COPs relies on the use of backtrack search. Within this paradigm, the scheduling problem is solved through the iterative selection of a subproblem and the tentative assignment of a solution to that subproblem. Because most scheduling problems are NP-complete, even finding a solution that simply satisfies the problem constraints could require exponential time in the worst case. This dissertation demonstrates that the granularity of the subproblems selected by the backtrack search procedure critically affects both the efficiency of the procedure and the quality of the result...

The Constraint Satisfaction Problem is a type of combinatorial search problem of much interest in Artificial Intelligence and Operations Research. The simplest algorithm for solving such a problem is chronological backtracking, but this method suffers from a malady known as "thrashing," in which essentially the same subproblems end up being solved repeatedly. Intelligent backtracking algorithms, such as backjumping and dependency-directed backtracking, were designed to address this difficulty, but the exact utility and range of applicability of these techniques have not been fully explored. This dissertation describes an experimental and theoretical investigation into the power of these intelligent backtracking algorithms. We compare the empirical performance of several such algorithms on a range of problem distributions. We show that the more sophisticated algorithms are especially useful on those problems with a small number of constraints that happen to be difficult for chronologica...

Many practical problems in Artificial Intelligence have search trees that are too large to search exhaustively in the amount of time allowed. Systematic techniques such as chronological backtracking can be applied to these problems, but the order in which they examine nodes makes them unlikely to find a solution in the explored fraction of the space. Nonsystematic techniques have been proposed to alleviate the problem by searching nodes in a random order. A technique known as iterative sampling follows random paths from the root of the tree to the fringe, stopping if a path ends at a goal node. Although the nonsystematic techniques do not suffer from the problem of exploring nodes in a bad order, they do reconsider nodes they have already ruled out, a problem that is serious when the density of solutions in the tree is low. Unfortunately, for many practical problems the order of examing nodes matters and the density of solutions is low. Consequently, neither chronological backtracking...

Practical Constraint Satisfaction Problems (CSPs) such as design of integrated circuits or scheduling generally entail large search spaces with hundreds or even thousands of variables, each with hundreds or thousands of possible values. Often, only a very tiny fraction of all these possible assignments participates in a satisfactory solution. This article discusses techniques that aim at reducing the effective size of the search space to be explored in order to find a satisfactory solution by judiciously selecting the order in which variables are instantiated and the sequence in which possible values are tried for each variable. In the CSP literature, these techniques are commonly referred to as variable and value ordering heuristics. Our investigation is conducted in the job shop scheduling domain. We show that, in contrast with problems studied earlier in the CSP literature, generic variable and value heuristics do not perform well in this domain. This is attributed to the difficulty of these heuristics to properly account for the tightness of constraints and/or the connectivity of the constraint graphs induced by job shop scheduling CSPs. A new probabilistic framework is introduced that better captures these key aspects of the job shop scheduling search space. Empirical results show that variable and value ordering heuristics

This paper studies a version of the job shop scheduling problem in which some operations have to be scheduled within non-relaxable time windows (i.e. earliest/latest possible start time windows). This problem is a wellknown NP-complete Constraint Satisfaction Problem (CSP). A popular method for solving these types of problems consists in using depth-first backtrack search. Our earlier work focused on developing efficient consistency enforcing techniques and efficient variable /value ordering heuristics to improve the efficiency of this procedure. In this paper, we combine these techniques with new lookback schemes that help the search procedure recover from so-called deadend search states (i.e. partial solutions that cannot be completed without violating some constraints). More specifically, we successively describe three intelligent backtracking schemes: Dynamic Consistency Enforcement dynamically enforces higher levels of consistency in selected critical subproblems, Learning From Fa...

Over the years a large number of algorithms has been discovered to solve instances of CSP problems. In a recent paper Prosser [9] proposed a new approach to these algorithms by splitting them up in groups with identical forward (Backtracking, Backjumping, Conflict-Directed Backjumping) and backward (Backtracking, Backmarking, Forward Checking) moves. By combining the forward move of an algorithm from the first group and the backward move of an algorithm from the second group he was able to develop four new hybrid algorithms: Backmarking with Backjumping (BMJ), Backmarking with Conflict-Directed Backjumping (BMCBJ) , Forward Checking with Backjumping (FC-BJ) and Forward Checking with Conflict-Directed Backjumping (FC-CBJ). Variable reordering heuristics have been suggested by, among others, by Haralick [6] and Purdom [11, 14] to improve the standard CSP algorithms. They obtained both analytical and empiral results about the performance of these heuristics in their research. In this thes...

We describe various computing techniques for tackling chessboard domination problems and apply these to the determination of domination and irredundance numbers for queens ’ and kings ’ graphs. In particular we show that γ(Q15) = γ(Q16) = 9, confirm that γ(Q17) =γ(Q18) = 9, show that γ(Q19) = 10, show that i(Q18) = 10, improve the bound for i(Q19) to10 ≤ i(Q19) ≤ 11, show that ir(Qn) =γ(Qn) for 1 ≤ n ≤ 13, show that IR(Q9) =Γ(Q9) = 13 and that IR(Q10) =Γ(Q10) = 15, show that γ(Q4k+1) =2k +1for16 ≤ k ≤ 21, improve the bound for i(Q22) toi(Q22) ≤ 12, and show that IR(K8) =17,IR(K9) =25,IR(K10) = 27, and IR(K11) = 36. 1

One of the difficulties in using nondeterministic algorithms for the solution of combinatorial problems is that most programming languages do not include features capable of easily repre-senting backtracking processes. This paper describes a procedure mechanism that uses co-routines as a means for the description and realization of nondeterministic algorithms. A solution to the eight queens problem is given to illustrate the application of the procedure mechanism to backtracking problems. I.