Several models based on constraint programming have been proposed to solve the traveling salesman problem (TSP). The most efficient ones, such as the weighted circuit constraint (WCC), mainly rely on the Lagrangian relaxation of the TSP, based on the search for spanning tree or more precisely "1-tree". The weakness of these approaches is that they do not include enough structural constraints and are based almost exclusively on edge costs. The purpose of this paper is to correct this drawback by introducing the Hamiltonian cycle constraint associated with propagators. We propose some properties preventing the existence of a Hamiltonian cycle in a graph or, conversely, properties requiring that certain edges be in the TSP solution set. Notably, we design a propagator based on the research of k-cutsets. The combination of this constraint with the WCC constraint allows us to obtain, for the resolution of the TSP, gains of an order of magnitude for the number of backtracks as well as a strong reduction of the computation time.
The currently best CP method for solving the Travelling Salesman Problem is the Weighted Circuit Constraint associated with the LCFirst search strategy. The use of Embarrassingly Parallel Search (EPS) for this model is problematic because EPS decomposition is a depth-bounded process unlike the LCFirst search strategy which is depth-first. We present Bound-Backtrack-and-Dive, a method which solves this issue. First, we run a sequential solving of the problem with a bounded number of backtracks in order to extract key information from LCFirst, then we decompose with EPS using that information rather than LCFirst. The experimental results show that we obtain almost a linear gain on the number of cores and that Bound-Backtrack-and-Dive may considerably reduce the number of backtracks performed for some problems.
We are interested in the consequences of imposing edges in T a minimum spanning tree. We prove that the sum of the replacement costs in T of the imposed edges is a lower bounds of the additional costs. More precisely if r-cost(T, e) is the replacement cost of the edge e, we prove that if we impose a set I of nontree edges of T then e∈I rcost(T, e) ≤ cost(Te∈I ), where I is the set of imposed edges and Te∈I a minimum spanning tree containing all the edges of I.
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