### Recording and Minimizing Nogoods from Restarts

#### Abstract

In this paper, nogood recording is investigated for CSP within the randomization and restart framework. Our goal is to avoid the same situations to occur from one run to the next ones. More precisely, nogoods are recorded when the current cutoff value is reached, i.e. before restarting the search algorithm. Such a set of nogoods is extracted from the last branch of the current search tree and exploited using the structure of watched literals originally proposed for SAT. We prove that the worst-case time complexity of extracting such nogoods at the end of each run is only O(n^2 d) where n is the number of variables of the constraint network and d the size of the greatest domain, whereas for any node of the search tree, the worst-case time complexity of exploiting these nogoods to enforce Generalized Arc Consistency (GAC) is O(n|B|) where |B| denotes the number of recorded nogoods. As the number of nogoods recorded before each new run is bounded by the length of the last branch, the total number of recorded nogoods is polynomial in the number of restarts. Interestingly, we show that when the minimization of the nogoods is envisioned with respect to an inference operator φ, it is possible to directly identify some nogoods that cannot be minimized. For φ = AC (i.e. for MAC), the worst-case time complexity of extracting minimal nogoods is slightly increased to O(en2 d3 ) where e is the number of constraints of the network. Experimentation over a wide range of CSP instances using a generic state-of-the-art CSP solver demonstrates the effectiveness of this approach. Recording nogoods (and in particular, minimal nogoods) from restarts significantly improves the robustness of the solver.

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