2,089 research outputs found
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of 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 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 where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
The 2-period balanced traveling salesman problem
In the 2-period Balanced Traveling Salesman Problem (2B-TSP), the customers must be visited over a period of two days: some must be visited daily, and the others on alternate days (even or odd days); moreover, the number of customers visited in every tour must be balancedĂą, i.e. it must be the same or, alternatively, the difference between the maximum and the minimum number of visited customers must be less than a given threshold. The salesman's objective is to minimize the total distance travelled over the two tours. Although this problem may be viewed as a particular case of the Period Traveling Salesman Problem, in the 2-period Balanced TSP the assumptions allow for emphasizing on routing aspects, more than on the assignment of the customers to the various days of the period. The paper proposes two heuristic algorithms particularly suited for the case of Euclidean distances between the customers. Computational experiences and a comparison between the two algorithms are also given.
Labeled Traveling Salesman Problems: Complexity and approximation
We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a -approximation algorithm based on local improvements and show that the problem is APX-hard. For the minimization version, we show that it is not approximable within for any fixed . When every color appears in the graph at most times and is an increasing function of , the problem is shown not to be approximable within factor . For fixed constant we analyze a polynomial-time approximation algorithm, where is the -th harmonic number, and prove APX-hardness for . For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances
A Parameterized Complexity Analysis of Bi-level Optimisation with Evolutionary Algorithms
Bi-level optimisation problems have gained increasing interest in the field
of combinatorial optimisation in recent years. With this paper, we start the
runtime analysis of evolutionary algorithms for bi-level optimisation problems.
We examine two NP-hard problems, the generalised minimum spanning tree problem
(GMST), and the generalised travelling salesman problem (GTSP) in the context
of parameterised complexity.
For the generalised minimum spanning tree problem, we analyse the two
approaches presented by Hu and Raidl (2012) with respect to the number of
clusters that distinguish each other by the chosen representation of possible
solutions. Our results show that a (1+1) EA working with the spanning nodes
representation is not a fixed-parameter evolutionary algorithm for the problem,
whereas the global structure representation enables to solve the problem in
fixed-parameter time. We present hard instances for each approach and show that
the two approaches are highly complementary by proving that they solve each
other's hard instances very efficiently.
For the generalised travelling salesman problem, we analyse the problem with
respect to the number of clusters in the problem instance. Our results show
that a (1+1) EA working with the global structure representation is a
fixed-parameter evolutionary algorithm for the problem
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