On forward checking for non-binary constraint satisfaction

Abstract. We define Realtime Online solving of Quantified Constraint Satisfaction Problems (QCSPs) as a model for realtime online CSP solving. We use a combination of propagation, lookahead and heuristics and show how all three improve performance. For adversarial opponents we show that we can achieve promising results through good lookahead and heuristics and that a version of alpha beta pruning performs strongly. For random opponents, we show that we can frequently achieve solutions even on problems which lack a winning strategy and that we can improve our success rate by using Existential Quantified Generalised Arc Consistency, a lower level of consistency than SQGAC, to maximise pruning without removing solutions. We also consider the power of the universal opponent and show that through good heuristic selection we can generate a significantly stronger player than a static heuristic provides.

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“…To test the impact of this processing cost, we also implement solvers using forward checking. For non-binary QCSPs we extend nFC0 [23] …”

Section: Definition 2 (Eqgac) a Qcsp Is Existential Quantified Genermentioning

confidence: 99%

Realtime Online Solving of Quantified CSPs

Stynes

Brown

2009

Principles and Practice of Constraint Programming - CP 2009

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“…Figure 5 shows the initial CSP and the possible solutions (2,4,1,3) and (3,1,4,2) obtained using BT and BT+COH. COH checks how many tuples (from a given sample: s(n, d, e) = 16 tuples (see equation (2)) (4,3), (4, 4)}) satisfy each constraint and classifies them afterwards. We can observe that some constraints are tighter than others (c 1 , c 4 , c 6 satisfy less valid tuples).…”

Section: Example 2: (The 4-queens Problem)mentioning

confidence: 99%

“…As current techniques manage non-binary constraints in a natural way [4], new heuristics can be applied to these constraints to reduce the search space. As we pointed out in introduction some of these heuristics classify the non-binary constraints by means of the arity.…”

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A non-binary constraint ordering heuristic for constraint satisfaction problems

Salido

2008

Applied Mathematics and Computation

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Nowadays many real problems can be modelled as Constraint Satisfaction Problems (CSPs). A search algorithm for constraint programming requires an order in which variables and values should to be considered. Choosing the right order of variables and values can noticeably improve the efficiency of constraint satisfaction. Furthermore, the order in which constraints are studied can improve efficiency, particularly in problems with non-binary constraints. In this paper, we present a preprocess heuristic called Constraint Ordering Heuristic (COH) that studies the constrainedness of the scheduling problem and mainly classifies the constraints so that the tightest ones are studied first. Thus, constrainedness can be known in advance and overall inconsistencies can be found earlier and the number of constraint checks can significantly be reduced.Keywords: Heuristic Search, Constraint ordering, Non-binary Constraints, Constrainedness.1 2 IntroductionNowadays, many real problems can be efficiently modelled as Constraint Satisfaction Problems (CSPs) and solved using constraint programming techniques. Some problems can be modelled naturally using non-binary constraints [26,19,8]. Although, researchers have traditionally focused on binary constraints [24], the need to address issues regarding non-binary constraints has recently started to be widely recognized in the constraint satisfaction literature. Thus, to define heuristics to solve non-binary CSPs becomes relevant.One approach to solving CSPs is to use a depth-first backtrack search algorithm [5]. While the worst-case complexity of backtrack search is exponential, several heuristics to reduce its average-case complexity have been proposed in the literature [6]. However, determining which algorithms are superior to others remains difficult. Theoretical analysis provides worst-case guarantees which often do not reflect average performance. For instance, a backtracking-based algorithm that incorporates features such as variable ordering heuristics will often in practice have substantially better performance than a simpler algorithm without this feature [17], and yet the two share the same worst-case complexity. Similarly, one algorithm may be better than another on problems with a certain characteristic, and worse on another category of problem. Ideally, we would be able to identify this characteristic in advance and use it to guide our choice of algorithm.Many of the heuristics that improve backtracking-based algorithms are based on variable ordering and value ordering [20, 2], due to the additivity of the variables and values. However, constraints are also considered to be additive, that is, the order of imposition of constraints does not matter, all that matters is that the conjunction of constraints be satisfied [2].Thus, a real problem can be modelled as a CSP, and using some of the current ordering heuristics, it can be solved in a more efficient way by some of the backtracking-based search algorithms (Figure 1).In spite of the additivity of constraints, only...

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“…It was initially defined on binary CSPs. Numerous extensions and generalizations of FC have been proposed in order to solve non-binary CSPs or to exploit more powerful filters [7,13,1]. In this paper, we consider one of these extensions called nFC2 [1] whose level of filtering seems to realize a good trade-off according to the presented empirical results obtained on non-binary CSPs.…”

Section: Complexity Expressed By the Size Of Domainsmentioning

confidence: 99%

A New Evaluation of Forward Checking and Its Consequences on Efficiency of Tools for Decomposition of CSPs

Jégou

Ndiaye

Terrioux

2008

2008 20th IEEE International Conference on Tools With Artificial Intelligence

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In this paper, a new evaluation of the complexity of Forward Checking for solving non-binary CSPs with finite domains is proposed. Unlike what is done usually, it does not consider the size of domains, but the size of the relations associated to the constraints. It may lead sometimes to define better complexity bounds. By using this first result, we show that the tractability hierarchy proposed in [6] which compares different methods based on a decomposition of constraint networks can be seen from a new viewpoint.

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On forward checking for non-binary constraint satisfaction

Cited by 50 publications

References 4 publications

Realtime Online Solving of Quantified CSPs

Realtime Online Solving of Quantified CSPs

A non-binary constraint ordering heuristic for constraint satisfaction problems

A New Evaluation of Forward Checking and Its Consequences on Efficiency of Tools for Decomposition of CSPs

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