Traveling salesman and hamiltonian cycle problems
Worst-case bounds for subadditive geometric graphs.
M. Bern and D. Eppstein.
9th ACM Symp. Comp. Geom., San Diego, 1993, pp. 183–188.For many geometric graph problems for points in the unit square, such as minimum spanning trees, matching, and traveling salesmen, the sum of edge lengths is O(sqrt n) and the sum of dth powers of edge lengths is O(log n). We provide a "gap theorem" showing that if these bounds do not hold for a class of graphs, both sums will instead be Omega(n). For traveling salesmen the O(log n) bound is tight but for some other graphs the sum of dth powers of edge lengths is O(1).
Approximation algorithms for geometric problems.
M. Bern and D. Eppstein.
Approximation Algorithms for NP-hard Problems, D. Hochbaum, ed., PWS Publishing, 1996, pp. 296–345.Considers problems for which no polynomial-time exact algorithms are known, and concentrates on bounds for worst-case approximation ratios, especially those depending intrinsically on geometry rather than on more general graph theoretic or metric space formulations. Includes sections on the traveling salesman problem, Steiner trees, minimum weight triangulation, clustering, and separation problems.
Fast hierarchical clustering and other applications of dynamic closest pairs.
D. Eppstein.
9th ACM-SIAM Symp. Discrete Algorithms, San Francisco, 1998, pp. 619–628.
arXiv:cs.DS/9912014.
J. Experimental Algorithmics 5 (1): 1–23, 2000.This paper shows how to use my dynamic closest pair data structure from "Dynamic Euclidean minimum spanning trees" for some non-geometric problems including hierarchical clustering, greedy matching, and TSP heuristics. Experiments show variants of my data structures to be faster than previously used heuristics.
The traveling salesman problem for cubic graphs.
D. Eppstein.
arXiv:cs.DS/0302030.
8th Worksh. Algorithms and Data Structures, Ottawa, 2003.
Springer, Lecture Notes in Comp. Sci. 2748, 2003, pp. 307–318.
J. Graph Algorithms and Applications 11 (1): 61–81, 2007.We find improved exponential-time algorithms for exact solution of the traveling salesman problem on graphs of maximum degree three and four. We also consider related problems including counting the number of Hamiltonian cycles in such graphs.
Single-strip triangulation of manifolds with arbitrary topology.
D. Eppstein and M. Gopi.
13th Video Review of Computational Geometry, 2004.
20th ACM Symp. Comp. Geom., Brooklyn, 2004, pp. 455–456 (abstract for video).
25th Conf. Eur. Assoc. for Computer Graphics (EuroGraphics '04), Grenoble, 2004 (2nd best paper award).
Eurographics Forum 23 (3): 371–379, 2004.
arXiv:cs.CG/0405036.We describe a new algorithm, based on graph matching, for subdividing a triangle mesh (without boundary) so that it has a Hamiltonian cycle of triangles, and prove that this subdivision process increases the total number of triangles in the mesh by at most a factor of 3/2. We also prove lower bounds on the increase needed for meshes with and without boundary.
Single triangle strip and loop on manifolds with boundaries.
A. Bushan, P. Diaz-Gutierrez, D. Eppstein, and M. Gopi.
Tech. Rep. 05-11, UC Irvine, School of Information and Computer Science, 2005.
Proc. 19th Brazilian Symp. Computer Graphics and Image Processing (SIBGRAPI 2006), pp. 221–228.This follows on to our previous paper on using graph matching to cover a triangulated polyhedral model with a single triangle strip by extending the algorithms to models with boundaries. We provide two methods: one is based on using an algorithm for the Chinese Postman problem to find a small set of triangles to split in order to find a perfect matching in the dual mesh, while the other augments the model with virtual triangles to remove the boundaries and merges the strips formed by our previous algorithm on this augmented model. We implement the algorithms and report some preliminary experimental results.