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**1 - 3**of**3**### Grammar Constraints

"... With the introduction of the Regular Membership Constraint, a new line of research has opened where constraints are based on formal languages. This paper is taking the next step, namely to investigate constraints based on grammars higher up in the Chomsky hierarchy. We devise a time- and space-effic ..."

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With the introduction of the Regular Membership Constraint, a new line of research has opened where constraints are based on formal languages. This paper is taking the next step, namely to investigate constraints based on grammars higher up in the Chomsky hierarchy. We devise a time- and space-efficient incremental arc-consistency algorithm for context-free grammars, investigate when logic combinations of grammar constraints are tractable, show how to exploit non-constant size grammars and reorderings of languages, and study where the boundaries run between regular, context-free, and context-sensitive grammar filtering.

### Orbital Shrinking: Theory and Applications

"... Symmetry plays an important role in optimization. The usual ap-proach to cope with symmetry in discrete optimization is to try to eliminate it by introducing artificial symmetry-breaking conditions into the problem, and/or by using an ad-hoc search strategy. This is the common approach in both the ..."

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Symmetry plays an important role in optimization. The usual ap-proach to cope with symmetry in discrete optimization is to try to eliminate it by introducing artificial symmetry-breaking conditions into the problem, and/or by using an ad-hoc search strategy. This is the common approach in both the mixed-integer programming (MIP) and constraint programming (CP) communities. In this paper we argue that symmetry is instead a benefi-cial feature that we should preserve and exploit as much as possible, breaking it only as a last resort. To this end, we outline a new approach, that we call orbital shrinking, where additional integer variables expressing variable sums within each symmetry orbit are introduced and used to “encapsulate ” model symmetry. This leads to a discrete relaxation of the original problem, whose solution yields a bound on its optimal value. Then, we show that orbital shrink-ing can be turned into an exact method for solving arbitrary symmetric MIP instances. The proposed method naturally provides a new way for devising hy-brid MIP/CP decompositions. Finally, we report computational results on two specific applications of the method, namely the multi-activity shift scheduling and the multiple knapsack problem, showing that the resulting method can be orders of magnitude faster than pure MIP or CP approaches.