Finite Domain Bounds Consistency Revisited

A widely adopted approach to solving constraint satisfaction problems combines systematic tree search with constraint propagation for pruning the search space. Constraint propagation is performed by propagators implementing a certain notion of consistency. Bounds consistency is the method of choice for building propagators for arithmetic constraints and several global constraints in the finite integer domain. However, there has been some confusion in the definition of bounds consistency. In this paper we clarify the differences and similarities among the three commonly used notions of bounds consistency.Comment: 12 page

Phase Transition Behavior of Cardinality and XOR Constraints

The runtime performance of modern SAT solvers is deeply connected to the phase transition behavior of CNF formulas. While CNF solving has witnessed significant runtime improvement over the past two decades, the same does not hold for several other classes such as the conjunction of cardinality and XOR constraints, denoted as CARD-XOR formulas. The problem of determining the satisfiability of CARD-XOR formulas is a fundamental problem with a wide variety of applications ranging from discrete integration in the field of artificial intelligence to maximum likelihood decoding in coding theory. The runtime behavior of random CARD-XOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a non-linear tradeoff between CARD and XOR constraints

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

Phase Transition Behavior of Cardinality and XOR Constraints

The runtime performance of modern SAT solvers is deeply connected to the phase transition behavior of CNF formulas. While CNF solving has witnessed significant runtime improvement over the past two decades, the same does not hold for several other classes such as the conjunction of cardinality and XOR constraints, denoted as CARD-XOR formulas. The problem of determining satisfiability of CARDXOR formulas is a fundamental problem with wide variety of applications ranging from discrete integration in the field of artificial intelligence to maximum likelihood decoding in coding theory. The runtime behavior of random CARD-XOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a non-linear tradeoff between CARD and XOR constraints

Arc Consistency on n-ary Monotonic and Linear Constraints

Many problems and applications can be naturally modelled and solved using constraints with more than two variables. Such n-ary constraints, in particular, arithmetic constraints are provided by many finite domain constraint programming systems. The best known worst case time complexity of existing algorithms(3l C-schema) for enforcing arc consistency on general CSPs isO( ) where d is the size of domain, e is the number of constraints and n is the maximum number of variables in a single constraint. We address the question of efficient consistency enforcing for n-ary constraints. An observation here is that even with a restriction of n-ary constraints to linear constraints, arc consistency enforcing is NP-complete. We identify a general class of monotonic n-ary constraints(3 h includes linear inequalities as a special case). Such monotonic constraints can be made arc consistent in time d). The special case of linear inequalities can be made arc consistent in time d) using bounds-consistency which exploits special properties of the projection function