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Stefan Hougardy

Publications and source records attributed to Stefan Hougardy.

24 records · Page 2Linked to original sources
arXiv2014

Edge Elimination in TSP Instances

The Traveling Salesman Problem is one of the best studied NP-hard problems in combinatorial optimization. Powerful methods have been developed over the last 60 years to find optimum solutions to large TSP instances. The largest TSP instance so far that has been solved optimally has 85,900 vertices. Its solution required more than 136 years of total CPU time using the branch-and-cut based Concorde TSP code [1]. In this paper we present graph theoretic results that allow to prove that some edges of a TSP instance cannot occur in any optimum TSP tour. Based on these results we propose a combinatorial algorithm to identify such edges. The runtime of the main part of our algorithm is $O(n^2 \log n)$ for an n-vertex TSP instance. By combining our approach with the Concorde TSP solver we are able to solve a large TSPLIB instance more than 11 times faster than Concorde alone.

cs.DM↗
arXiv2014

On the Nearest Neighbor Rule for the Metric Traveling Salesman Problem

We present a very simple family of traveling salesman instances with $n$ cities where the nearest neighbor rule may produce a tour that is $Θ(\log n)$ times longer than an optimum solution. Our family works for the graphic, the euclidean, and the rectilinear traveling salesman problem at the same time. It improves the so far best known lower bound in the euclidean case and proves for the first time a lower bound in the rectilinear case.

cs.DS↗
arXiv2009

Surface realization with the intersection edge functional

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R^3 for all vertex-minimal triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12 vertices were obtained.

math.MG↗
arXiv2007

Polyhedral tori with minimal coordinates

We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9, and 10 vertices. For the unique 7-vertex triangulation of the torus we show that all corresponding 72 oriented matroids are realizable in the 6x6x6-cube. Moreover, we present polyhedral tori with 8 vertices in the 2x2x2-cube, general position realizations of triangulated tori with 8 vertices in the 2x2x3-cuboid as well as polyhedral tori with 9 and 10 vertices in the 1x2x2-cuboid.

math.MG↗