Effective Tour Searching for Large TSP Instances. Gerold Jäger

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1 Effective Tour Searching for Large TSP Instances Gerold Jäger Martin-Luther-University Halle-Wittenberg (Germany) joint work with Changxing Dong, Paul Molitor, Dirk Richter German Research Foundation Grant Tolerance Based Algorithms for Solving the Traveling Salesman Problem

2 Overview 1 Traveling Salesman Problem Definition Importance Helsgaun s Heuristic

3 Overview 2 Description of the Algorithm Experimental Results Work on World-TSP Instance

4 Traveling Salesman Problem Definition Given: undirected graph G = (V, E), V = n, cost function c : E R. Traveling Salesman Problem (TSP): Find: tour (v 1, v 2,..., v n, v 1 ) with minimum costs c(v n, v 1 ) + n 1 i=1 c(v i, v i+1 ).

5 Traveling Salesman Problem Importance Easy to understand. Hard to solve: NP-Hard. Gap between few performance guarantees good practical results Many important applications: public transport tour planning design of microchips genome sequencing

6 Traveling Salesman Problem Helsgaun s Heuristic 1 Start with an arbitrary vertex. 2 In each step go to the nearest non-visited vertex. 3 If all vertices are visited, return to the starting point. 4 Use the resulted tour as starting tour for the next steps. 5 For k n apply a k-opt step, i.e.: Replace tour edges by non-tour edges, such that the edges are still a tour the tour is better than the original one 6 Repeat step 5 as long as improving steps can be found.

7 Traveling Salesman Problem Helsgaun s Heuristic Example of a 2-OPT step

8 Traveling Salesman Problem Helsgaun s Heuristic Best TSP heuristic: [Helsgaun, 1998, improved: 2007] Main ideas: [Lin, Kernighan, 1971] Optimizations: 1 Choose k small. 2 For each vertex consider only the s best neighboring edges, the so-called candidate system. Helsgaun s main improvement: For each vertex do not consider the s shortest neighboring edges, but the s neighboring edges with a criterion based on so-called tolerances of the minimum spanning tree. 3 Apply t (nearly) independent runs of the algorithm. The larger the algorithm parameters k, s and t are, the slower, but more effective is Helsgaun s Heuristic.

9 Description of the Algorithm 1 Using known heuristics, e.g., Helsgaun s Heuristic, find good starting tours. a b c

10 Description of the Algorithm 2 Find all common edges in these starting tours. a b c d Such edges are called pseudo backbone edges.

11 Description of the Algorithm 3 Contract all edges of paths of pseudo backbone edges to one edge. d eee

12 Description of the Algorithm 4 Create a new (reduced) instance by omitting the vertices, which lie on a path of pseudo backbone edges: d e f Fix the contracted edges, i.e., force them to be in the final tour.

13 Description of the Algorithm 5 Apply Helsgaun s Heuristic to the new instance. e f g

14 Description of the Algorithm 6 Re-contract the tour of the new instance to a tour of the original instance. g h The last tour is the optimum one. a b c

15 Description of the Algorithm Two advantages: 1 Reduction of the set of vertices. 2 Fixing of a part of the edges. Helsgaun s Heuristic can be applied with larger algorithm parameters k, s and t than for the original instance. The algorithm works rather good, if the starting tours are 1 good ones 2 not too similar (as otherwise the search space is restricted too strongly)

16 Experimental Results Competition: TSP homepage ( Large TSP datasets from practice: for comparison of exact algorithms and heuristics. 74 unsolved example instances: VLSI and national instances For 18 of 74 instances we have set a new record. 9 of 18 records are still up to date.

17 Experimental Results Our new records Date # Vertices Date # Vertices , , , , , , , , , , , , , , , , , ,608

18 Work on World-TSP Instance joint work also with Christian Ernst Most difficult instance of TSP homepage: World-TSP instance with 1, 904, 711 cities. World-TSP tour (Ambitious) aim: Computation of a new record World-TSP tour.

19 Work on World-TSP Instance Problem: Finding good starting tours is too difficult for the World-TSP instance. Idea: Combine with Partitioning.

20 Work on World-TSP Instance 1 Compute good tours in overlapping windows.

21 Work on World-TSP Instance The overlap size is chosen half the width (height) of the window frame. Each vertex is contained in exactly 4 windows, unless it is located near the boundary.

22 Work on World-TSP Instance 2 Use edges contained in all tours of 4 overlapping windows as pseudo backbone edges.

23 Work on World-TSP Instance 3 Apply the recursively. Our current tour: % over the current best tour 2 Found in 13 hours

24 Work on World-TSP Instance Thanks for your attention!

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