Discrepancy-Based Sliced Neighborhood Search

Limited Discrepancy Search (LDS) is one of the most widely used search strategies in Constraint Programming; given a solution suggested by a search heuristics, LDS explores the search space at increasing discrepancy with respect to such solution. In optimization problems, LDS is used as a way to explore the k-distance neighborhood of an incumbent solution using constraint propagation and tree search. However, for large problems, the size of the resulting neighborhoods limits the k-distance (i.e. the number of discrepancies) that can be efficiently explored. If the first solution is far from the optimal one, exploring limited neighborhood leads to small improvements.Therefore, we propose a variant of LDS that samples the LDS space by exploring slices of (possibly very large) discrepancy based neighborhoods. Instead of deciding only the number of variables that can change (the k-distance) we decide which n-k variables should be fixed to the value they have in the incumbent solution. We present results on hard Asymmetric Traveling Salesman Problem with Time Windows (ATSPTW) instances to show the effectiveness of the approach.