Fixed-parameter algorithms for rectilinear Steiner tree and rectilinear traveling salesman problem in the plane - Cambazard, Hadrien; Catusse, Nicolas (European journal of operational research, 2018)
•A fixed-parameter algorithm for the rectilinear travelling salesman problem.•A fixed-parameter algorithm for the rectilinear Steiner tree problem.•A refined complexity analysis.•Experimental results showing the scalability of this approach. Given a set P of n points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the l1 distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in O(nh7h) where h ≤ n denotes the number of horizontal lines containing the points of P. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of P providing a O(nh5h) time complexity. Both bounds improve over the best time bounds known for these problems.