Alternative Filtering for the Weighted Circuit Constraint: Comparing Lower Bounds for the TSP and Solving TSPTW

Many problems, and in particular routing problems, require to find one or many circuits in a weighted graph. The weights often express the distance or the travel time between vertices. We propose in this paper various filtering algorithms for the weighted circuit constraint which maintain a circuit in a weighted graph. The filtering algorithms are typical cost based filtering algorithms relying on relaxations of the Traveling Salesman Problem. We investigate three bounds and show that they are incomparable. In particular we design a filtering algorithm based on a lower bound introduced in 1981 by Christophides et al.. This bound can provide stronger filtering than the classical Held and Karp’s approach when additional information, such as the possible positions of the clients in the tour, is available. This is particularly suited for problems with side constraints such as time windows. Baseline CP for TSP/TSPTW Many problems, and in particular routing problems, require to find one or many circuits in a weighted graph. For example, the Traveling Salesman Problem (TSP) consists in finding a circuit of minimum total weight that visits all vertices of the graph. Time Windows are often added to the formulation to express the fact that a vertex or ‘a client’ can be visited only when available (TSPTW). To the best of our knowledge, the state-of-the-art to solve this problem is based on dynamic programming combined with column generation (Baldacci, Mingozzi, and Roberti 2012). Related work: In Constraint Programming (CP), two approaches have been developed to tackle the TSPTW. The first approach (Pesant et al. 1998) uses redundant constraints in order to reduce the search space. In particular, a constraint to reduce the domains of the time windows is added as well as an arc elimination constraint for filtering possible direct successors of a vertex. But the objective function is propagated independently of the circuit constraint which leads to poor global lower bounds. The second approach (Focacci, Lodi, and Milano 2002) adds a lower bound computed with an assignment relaxation. Additionally, an effective filtering for weighted circuit was proposed in (Benchimol et al. 2012) based on the Copyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 1-tree relaxation by Held and Karp (Held and Karp 1970; 1971) but has not been used for the TSPTW. Our paper is based on the work of (Benchimol et al. 2012). We propose different filtering algorithms, all based on a relaxation of the TSP taking into account different aspects that are relevant for asymmetric or time-constrained cases such as the TSPTW. A second contribution is to show that the three bounds are incomparable and in particular the two bounds based on Lagrangian relaxation. Problem definition: Let G = (N,E) be a complete directed graph without loops with vertex set N = {0, . . . , n+ 1}, the vertex 0 (resp. n + 1) corresponds to the start (resp. the end) of the route (N = {0, . . . , n} and N = {1, . . . , n + 1}). We denote by dij the distance (or cost) of arc (i, j). A salesman tour is defined as a path in G from vertex 0 to vertex n + 1, visiting each vertex 1, . . . , n exactly once. The Traveling Salesman Problem (TSP) consists in finding a salesman tour of minimum distance in G. In presence of time constraints, each vertex i is associated to a time windows [ai, bi], and each arc (i, j) to a traveling time tij . The service time of customer i is included in tij . The Traveling Salesman Problem with Time Windows (TSPTW) consists in identifying a shortest salesman tour visiting each vertex within its time window. Note that the problem has n + 1 cities: n clients plus one depot where the tour starts and ends. In the following, we denote by D(x), the domain of variable x and by x (resp. x) the upper (resp. lower) bound of x.