2,884 research outputs found
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
The Geometric Maximum Traveling Salesman Problem
We consider the traveling salesman problem when the cities are points in R^d
for some fixed d and distances are computed according to geometric distances,
determined by some norm. We show that for any polyhedral norm, the problem of
finding a tour of maximum length can be solved in polynomial time. If
arithmetic operations are assumed to take unit time, our algorithms run in time
O(n^{f-2} log n), where f is the number of facets of the polyhedron determining
the polyhedral norm. Thus for example we have O(n^2 log n) algorithms for the
cases of points in the plane under the Rectilinear and Sup norms. This is in
contrast to the fact that finding a minimum length tour in each case is
NP-hard. Our approach can be extended to the more general case of quasi-norms
with not necessarily symmetric unit ball, where we get a complexity of
O(n^{2f-2} log n).
For the special case of two-dimensional metrics with f=4 (which includes the
Rectilinear and Sup norms), we present a simple algorithm with O(n) running
time. The algorithm does not use any indirect addressing, so its running time
remains valid even in comparison based models in which sorting requires Omega(n
\log n) time. The basic mechanism of the algorithm provides some intuition on
why polyhedral norms allow fast algorithms.
Complementing the results on simplicity for polyhedral norms, we prove that
for the case of Euclidean distances in R^d for d>2, the Maximum TSP is NP-hard.
This sheds new light on the well-studied difficulties of Euclidean distances.Comment: 24 pages, 6 figures; revised to appear in Journal of the ACM.
(clarified some minor points, fixed typos
Exact value for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions
We show that the average cost for the traveling-salesman problem in two
dimensions, which is the archetypal problem in combinatorial optimization, in
the bipartite case, is simply related to the average cost of the assignment
problem with the same Euclidean, increasing, convex weights. In this way we
extend a result already known in one dimension where exact solutions are
avalaible. The recently determined average cost for the assignment when the
cost function is the square of the distance between the points provides
therefore an exact prediction for
large number of points . As a byproduct of our analysis also the loop
covering problem has the same optimal average cost. We also explain why this
result cannot be extended at higher dimensions. We numerically check the exact
predictions.Comment: 5 pages, 3 figure
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