Approximating TSP Solution by MST based Graph Pyramid

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Approximating TSP Solution by MST based Graph Pyramid Y. Haxhimusa 1,2, W. G. Kropatsch 2, Z.

Lehrbaum 2 1 Department of Psychological Sciences, Purdue University 2 PRIP,

by Austrian Science Founds P18716-N13 and S9103-N04 Proceedings of GbRPR

1 Approximating TSP Solution by MST based Graph Pyramid Y. Haxhimusa 1,2, W. G. Kropatsch 2, Z. Pizlo 1, A. Ion 2 and A. Lehrbaum 2 1 Department of Psychological Sciences, Purdue University 2 PRIP, Vienna University of Technology Air Force of Scientific Research Supported by Austrian Science Founds P18716-N13 and S9103-N04 Proceedings of GbRPR 2007 on Pages Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 1 / 24

2 Motivation Traveling Salesman Problem TSP Given a set of cities, find the route that visits all the cities and has the smallest cost. Humans solve TSP ( 50 cities) close to optimal and near linear [MacGregor et al., 2000] Evidence that suggests the existence of the hierarchical representations in the human vision system [Tsotsos, 1990] Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 2 / 24

3 Motivation Multiresolution Pyramid Representation Approach Approximate the TSP Solution NOT an attempt to solve TSP optimally (review of existing methods [Gutin and Punnen, 2002]) NOT an attempt to find the best approximation method The emphasis of this talk is on emulating human performance (time and accuracy) hierarchical representation Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 3 / 24

4 Motivation Multiresolution Pyramid Representation Approach Local to global: bottom-up reduction of resolution Global to local: top-down solution refinement Approximate the TSP solution using pyramid representation The idea 1 partition the input space 2 reduce number of cities 3 repeat until solution becomes trivial 4 refine solution top-down to the base level Algorithm Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 4 / 24

5 Motivation Multiresolution Pyramid Representation Approach Approximate the TSP Solution using pyramid representation Gaussian pyramid [Graham et al., 1995] Quad-tree pyramid [Pizlo et al., 2005] shift variant MST-based pyramid - this talk shift in-variant Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 5 / 24

6 Outline of the Talk Agenda 1 Motivation 2 Borůvka s Minimum Spanning Tree Algorithm 3 Approximate Solution of the Traveling Salesman Problem 4 Psychophysical Experiments and Evaluations Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 6 / 24

7 Partition Input Space How to Organize/Partition City-Space? Graph G = (V, E): city = vertex v V ; edges e E? complete graph: E = V V Delaunay triangulation E V V and Voronoi Diagram attribute edges and vertices Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 7 / 24

8 TSP Problem Specification Traveling Salesman Problem (TSP) Fully connected graph G = (V, E, w) and attributed by weight w costs w : e E R + Goal : Find the tour τ with the smallest overall cost w(e) min e τ If the weights are (2D) Euclidean distances (2D) E-TSP TSP (E-TSP) is hard optimization problem solution: approximation algorithms Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 8 / 24

9 Borůvka s Minimum Spanning Tree Algorithm TSP with triangle inequality Fakts: E-TSP satisfies triangle inequality MST is a natural lower bound for the length of the optimal tour In TSP with triangle inequality, it is possible to prove upper bounds in terms of the minimum spanning tree Christofides Heuristics [Christofides, 1976] Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 9 / 24

10 Borůvka s Minimum Spanning Tree Algorithm Minimum Spanning Tree (MST) Graph G = (V, E, w) connected weights w w : e E R + Goal : Find the spanning tree T with the smallest weight w(e) min e T Easy optimization problem solution: greedy algorithms Kruskal s [Kruskal, 1956] and Prim s algorithms [Prim, 1957] Borůvka s algorithm [Borůvka, 1926] (O( E log V )) Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 10 / 24

$e with the minimum weight which connects T to G \ T and add edge e to MST 5: using edge e merge pairs of trees in L 6: end while$

11 Boru vka s Minimum Spanning Tree Algorithm MST Algorithm Boru vka s MST Algorithm [Boru vka, 1926] Input: graph G = (V, E, w) 1: MST := empty edge list 2: v V make a list of trees L 3: while there is more than one tree in L do 4: each tree T L finds the edge e with the minimum weight which connects T to G \ T and add edge e to MST 5: using edge e merge pairs of trees in L 6: end while Output: minimum spanning tree of G Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 11 / 24

12 Borůvka s Minimum Spanning Tree Algorithm Borůvka s Algorithm and Graph Pyramid Graph contraction merges all trees T L in step 5. Step 4: called Borůvka s step G 1 graph contraction G 0 Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 12 / 24

13 Approximate Solution of the Traveling Salesman Problem Pyramid Solution MST based approximate Algorithm for TSP Input: graph G = (V, E, w) 1: while there is more than s cities do 2: merge no more than r cites using Borůvka s step 3: end while 4: Find the trivial tour τ 5: repeat 6: refine τ : τ τ 7: until the bottom of the pyramid Output: (approximated) tour τ r, s N and 2 r 10 and s = {3, 4} Idea Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 13 / 24

14 Approximate Solution of the Traveling Salesman Problem Top-down flow trivial tour τ super vertices Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 14 / 24

15 Approximate Solution of the Traveling Salesman Problem Top-down flow trivial tour τ approximated tour τ Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 14 / 24

16 Psychophysical Experiments and Evaluations Solution Errors on Random Instances Compared Human subjects MST pyramid Test set Random graphs of different sizes: 6, 10, 20, and 50 with 25 instances each MST Pyramid implementation issues s = 3 and r = 7 Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 15 / 24

BSL OSK MST pyramid model ZL ZP Yll Haxhimusa (Purdue

17 Psychophysical Experiments and Evaluations Solution Errors on Random Instances 10 city TSP solutions by Humans and MST Model BSL OSK MST pyramid model ZL ZP Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 15 / 24

18 Psychophysical Experiments and Evaluations Solution Errors on Random Instances Subjects vs. MST-based Pyramid Subject: BSL Subject: OSK Subject: ZL Subject: ZP Model: MST Pyramid Subjects vs. MST-Pyramid Model 6 Solution Error (%) Number of Cities Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 15 / 24

19 Psychophysical Experiments and Evaluations Comparing Solutions of Algorithms Concorde vs Pyramid based methods Concorde algorithm 1 [Applegate et al., 2001] Adaptive binary pyramid 2 [Pizlo et al., 2006] Quad-tree pyramid 2 [Pizlo et al., 2005] MST-based pyramid 3 Test set Random graphs of different sizes 200, 400, 600, 800, pizlo/ 3 yll/ Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 16 / 24

20 Psychophysical Experiments and Evaluations Comparison of the Algorithms Solution Error Solution Error On Large Problems Solution Error (%) Adaptive Binary Pyramid Regular Quad Pyramid MST Pyramid Concorde Number of Cities Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 17 / 24

21 Psychophysical Experiments and Evaluations Comparison of the Algorithms Running Time MST vs Concorde Running Normalized Time Concorde MST Pyramid Number of Cities Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 17 / 24

22 Summary Summary Pyramid Solution of the Euclidean TSP Bottom-up reduction of the size of the problem by MST based technique Solving the trivial problem Top-down refining of the found problem MST-based pyramid solution shows similar results as the other pyramid models on random instances but shift invariant Performance of the graph pyramid model is comparable to human performance Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 18 / 24

23 Summary Summary Partition 2D space using Delaunay Triangulation to better capture proximity Psychophysical experiments on 3D TSP Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 18 / 24

24 Thanks you Summary Approximating TSP Solution by MST based Graph Pyramid Graph-based Representation for Pattern Recognition Workshop Alicante, Spain, June 2007 Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 19 / 24

25 Thanks you Summary Approximating TSP Solution by MST based Graph Pyramid Graph-based Representation for Pattern Recognition Workshop Alicante, Spain, June Motivation 2 Borůvka s Minimum Spanning Tree Algorithm 3 Approximate Solution of the Traveling Salesman Problem 4 Psychophysical Experiments and Evaluations Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 19 / 24

26 References I Selected References Jolion, J.-M. and Rosenfeld, A. (1994). A Pyramid Framework for Early Vision. Kluwer. Gutin, G., Punnen, A.P. (2002) The traveling salesman problem and its variations. Kluwer Bister, M., Cornelis, J., and Rosenfeld, A. (1990). A critical view of pyramid segmentation algorithms. Pattern Recognition Letters, 11(9): Kropatsch, W. G. (1995a). Building irregular pyramids by dual graph contraction. IEE-Proc. Vision, Image and Signal Processing, 142(6): Felzenszwalb, P. F. and Huttenlocher, D. P. (2004). Efficient graph-based image segmentation. International Journal of Computer Vision, 59(2): MacGregor, J.N., Ormerod, T.C., Chronicle, E.P.: A model of human performance on the traveling salesperson problem. Memory and Cognition 28 (2000) Tsotsos, J. K. (1990). Analyzing vision at the complexity level. Behavioral and Brain Sciences, 13(3): Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 20 / 24

27 References II Selected References S. M. Graham, Z. Pizlo and A. Joshi. Problem Solving in Human Beings and Computers. Annual Meeting of the Society for Mathematical Psychology, Ivine, CA, Z. Pizlo, E. Stefanov, J. Saalweachter, Z. Li, Y. Haxhimusa and W. G. Kropatsch. Traveling Salesman Problem: a Foveating Pyramid Model. Journal of Problem Solving,in press, Borůvka, O. (1926). O jistém problému minimálnim (about a certain minimal problem). Práce Moravské Přírodvědecké Společnosti v Brně (Acta Societ. Scienc. Natur. Moravicae), 3(3): Kruskal, J. B. J. (1956). On the shortest spanning subtree of a graph and the travelling salesman problem. In Proc. Am. Math. Soc., volume 7, pages Prim, R. C. (1957). Shortest connection networks and some generalizations. The Bell System Technical Journal, 36: A. Aggrawal, L. J. Guibas, J. Saxe, and P.W. Shor. A Linear-Time Algorithm for Computing the Voronoi Diagram of a Convex Polygon. Discrete & Comput. Geometry, 4(6): , C. H. Papadimitriou and S. Vempala. On the approximability of the traveling salesman problem). Proceedings of STOC 2000, extended abstract, Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 21 / 24

28 References III Selected References N. Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Graduate School of Industrial Administration, Carnegie-Mellon University, 1976, 388. D. Applegate, R. Bixby, V. Chvatal, and W. Cook TSP cuts which do not conform to the template paradigm. Computational combinatorial optimization: optimal or provably near optimal solutions, Lecture Notes in Computer Science, 2241, M. Jünger and D. Naddef, eds., Springer, 2001, pp Z. Pizlo, E. Stefanov, J. Saalweachter, Z. Li, Y. Haxhimusa and W. G. Kropatsch. Adaptive Pyramid Model for the Traveling Salesman Problem. Workshop on Human Problem Solving, June Yll Haxhimusa (Purdue University) Approximating TSP Solution by MST based Graph Pyramid 22 / 24

Approximating TSP Solution by MST based Graph Pyramid

Approximating TSP Solution by MST based Graph Pyramid Y. Haxhimusa 1,2, W. G. Kropatsch 1, Z. Pizlo 2, A. Ion 1, and A. Lehrbaum 1 1 Vienna University of Technology, Faculty of Informatics, Pattern Recognition