Symmetry breaking in CSPs has attracted considerable attention in recent years. Various general schemes have been proposed to eliminate symmetries during search. In general, these schemes may take exponential space or time to eliminate all symmetries. This paper studies classes of CSPs for which symmetry breaking is tractable. It identifies several CSP classes which feature various forms of value interchangeability and shows that symmetry breaking can be performed in constant time and space during search using dedicated search procedures. Experimental results also show the benefits of symmetry breaking on these CSPs, which encompass many practical applications.

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A hybrid Mixed-Integer Linear Programming (MILP)/Constraint Programming (CP) decomposition algorithm is proposed for the short-term scheduling of batch plants that rely on the State Task Network representation. The decisions about the type and number of tasks performed, as well as the assignment of units to tasks are made by the MILP master problem. The CP

Abstract. This paper reconsiders the problems of discovering symmetries in constraint satisfaction problems (CSPs). It proposes a compositional approach which derives symmetries of the applications from primitive constraints. Its key insight is the recognition of the special role of global constraints in symmetry detection. Once the symmetries of global constraints are available, it often becomes much easier to derive symmetries compositionally and efficiently. The paper demonstrates the potential of this approach by studying several classes of value and variable symmetries and applying the resulting techniques to two non-trivial applications. The paper also discusses the potential of reformulations and high-level modelling abstractions to strengthen symmetry discovery. Compositional Derivation of Symmetries for Constraint Satisfaction This paper reconsiders the problems of discovering symmetries in constraint satisfaction problems (CSPs). It proposes a compositional approach which derives symmetries of the applications from primitive constraints. Its key insight is the recognition of the special role of global constraints in symmetry detection. Once the symmetries of global constraints are available, it often becomes much easier to derive symmetries compositionally and efficiently. The paper demonstrates the potential of this approach by studying several classes of value and variable symmetries and applying the resulting techniques to two non-trivial applications. The paper also discusses the potential of reformulations and high-level modeling abstractions to strengthen symmetry discovery.

Using constraint programming, we effectively model and efficiently solve the problem of balancing and minimising the traffic complexities of an airspace of adjacent sectors. The traffic complexity of a sector is here defined in terms of the numbers of flights within it, near its border, and on non-level segments within it. The allowed forms of complexity resolution are the changing of the take-off times of not yet airborne flights, the changing of the remaining approach times into the chosen airspace of already airborne flights by slowing down and speeding up within the two layers of feeder sectors around that airspace, as well as the changing of the levels of passage over way-points in that airspace. Experiments with actual European flight profiles obtained from the Central Flow Management Unit (CFMU) show that these forms of complexity resolution can lead to significant complexity reductions and rebalancing. 1

The availability of commodity multi-core and multi-processor machines and the inherent parallelism in constraint programming search offer significant opportunities for constraint programming. They also present a fundamental challenge: how to exploit parallelism transparently to speed up constraint programs. This paper shows how to parallelize constraint programs transparently without changes to the code. The main technical idea consists of automatically lifting a sequential exploration strategy into its parallel counterpart, allowing workers to share and steal subproblems. Experimental results show that the parallel implementation may produces significant speedups on multi-core machines. 1

Abstract. Invariants that incrementally maintain the value of expressions under assignments to their variables are a natural abstraction to build high-level local search algorithms. But their functionalities are not sufficient to allow arbitrary expressions as constraints or objective functions as in constraint programming. Differentiable invariants bridge this expressiveness gap. A differentiable invariant maintains the value of an expression and its variable gradients, it supports differentiation to evaluate the effect of local moves. The benefits of differentiable invariants are illustrated on a number of applications which feature complex, possibly reified, expressions and whose models are essentially similar to their CP counterparts. Experimental results demonstrate their practicability. 1

Abstract. In recent years, symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention. Various general schemes have been proposed to eliminate symmetries. In general, these schemes may take exponential space or time to eliminate all the symmetries. We identify several classes of CSPs that encompass many practical problems and for which symmetry breaking for various forms of value or variable interchangeability is tractable using dedicated search procedures. We also show the limits of efficient symmetry breaking for such dominance-detection schemes by proving intractability results for some classes of CSPs. 1

Meta-heuristics are methods that sit on top of local search algorithms. They perform the function of avoiding or escaping a local optimum and/or premature convergence. The aim of this paper is to survey, compare and contrast meta-heuristics for local search. First, we present the technique of local search (or hill climbing as it is sometimes known). We then present a table displaying the attributes of all the different meta-heuristics. After this, we give a short description and discussion of each meta-heuristic with pseudo code. Finally, we describe why, in general, these techniques work and present some ideas of what is needed from the next generation of meta-heuristics.

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