Effective Tour Searching for Large TSP Instances. Gerold Jäger

Transcription

1 Effective Tour Searching for Large TSP Instances Gerold Jäger Martin-Luther-University Halle-Wittenberg joint work with Changxing Dong, Paul Molitor, Dirk Richter November 14, 2008

2 Overview 1 Introduction Definition Graph Model Applications Importance of TSP Exact Algorithms

3 Overview 2 Heuristics Nearest-Neighbor Algorithm k-opt Algorithm Helsgaun s Algorithm

4 Overview 3 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm Experimental Results Work on World-TSP

5 Introduction Definition A traveling salesman wants to visit a given number of cities. Finally he wants to return to the starting point. This is called a tour through all cities. Each pair of cities receives a cost value for the distance between these cities. The Traveling Salesman Problem (TSP) is to find a tour with minimum costs.

6 Introduction Graph Model Let G = (V, E) be a complete undirected graph. V is the set of vertices, where V = n. E is the set of edges. Let c : E R + 0 be a cost function. Find a tour (v 1, v 2,..., v n, v 1 ) such that the following term is minimized: n 1 c(v n, v 1 ) + c(v i, v i+1 ) i=1

7 Introduction Applications Public transport: vehicle stops at given stations. Warehouse logistics: single items of an order are searched in a warehouse. Tour planning: vehicle visits depots. Design of microchips: vertices are boreholes. Genome sequencing: vertices are DNA ranges.

8 Introduction Importance of TSP Easy to understand. NP-Hard. Theoretical and practical methods for the TSP have shown to be successful also for related optimization problems, e.g., integer programming, branch-and-cut. Large datasets from practice exist for comparison of 1 exact algorithms 2 heuristics

9 Introduction Exact Algorithms Largest solved example instances Year Research Team Number of Vertices 1954 Dantzig, Fulkerson, Johnson Held, Karp Camerini, Fratta, Maffioli Grötschel Crowder, Padberg Padberg, Rinaldi Grötschel, Holland Padberg, Rinaldi Applegate, Bixby, Chvátal, Cook Applegate, Bixby, Chvátal, Cook (USA tour) 2001 Applegate, Bixby, Chvátal, Cook (D tour) 2004 App., Bixby, Chvátal, Cook, Helsgaun (Swe tour) 2006 App., Bixby, Chvátal, Cook, Helsgaun The 5 largest instances were solved by Concorde, an exact TSP solver, based on branch-and-cut.

10 Introduction Exact Algorithms Shortest tour through cities in Germany

11 Heuristics Nearest-Neighbor Algorithm Simple idea: 1 In each step go to the nearest non-visited vertex. 2 If all vertices are visited, return to the starting point. Advantage: At the beginning of the tour the costs are rather small. Disadvantage: At the end of the tour the set of non-visited vertices is small, and the costs become very large. This algorithm needs O(n 2 ) steps.

12 Heuristics k-opt Algorithm By a variation of the Nearest-Neighbor Algorithm, find a good tour, the so-called starting tour. Replace tour edges by non-tour edges, such that 1 The edges are still a tour. 2 The tour is better than the original one.

13 Heuristics k-opt Algorithm For two edges: For k edges this step is called k-opt step. Apply k-opt steps, as long as they exist.

14 Heuristics Helsgaun s Algorithm Best TSP heuristic: [Helsgaun, 1998, improved: 2007]. It combines variations of the Nearest-Neighbor Algorithm and the k-opt Algorithm. Optimizations: 1 Choose k small (Standard k = 5) 2 For each vertex consider only the s shortest neighboring edges, the so-called candidate system. (Standard s = 5) 3 Apply t (nearly) independent runs of the algorithm. (Standard t = 10)

15 Heuristics Helsgaun s Algorithm The larger the algorithm parameters k, s and t are, the slower, but more effective is Helsgaun s Algorithm. Helsgaun s main improvement: For each vertex do not consider the t shortest neighboring edges, but the t neighboring edges with a criterion based on the minimum spanning tree.

16 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm 1 Using known algorithms, e.g., Helsgaun s Algorithm, find good starting tours. a b c

17 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm 2 Find all common edges in these starting tours. d Such edges are called pseudo backbone edges. (Edges, which are contained in all optimum tours, are called backbone edges.)

18 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm 3 Create a new instance by omitting the vertices, which lie on a path of pseudo backbone edges. 4 Contract all edges of paths of pseudo backbone edges to one edge. 5 Fix these edges, i.e., these edges are forced to be in the final tour.

19 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm 6 Apply Helsgaun s Algorithm to the new instance. d e

20 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm 7 Re-contract the tour of the new instance to a tour of the original instance. e f Indeed, the last tour is the optimum one.

21 Our Approach: Pseudo Backbone Contraction Algorithm Description of the Algorithm Two advantages: 1 Reduction of dimension. 2 Fixing of a part of the edges. Helsgaun s Algorithm can be applied with larger algorithm parameters k, s and t. It is much more effective than applied 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)

22 Our Approach: Pseudo Backbone Contraction Algorithm Experimental Results Competition: TSP homepage ( For example instances, for which the computation of an optimum tour is too difficult, find a tour as good as possible. 74 unsolved example instances from TSP homepage come from two sources: VLSI and national instances. For 19 instances we have set a new world record. 12 of 19 world records are still up to date.

23 Our Approach: Pseudo Backbone Contraction Algorithm Experimental Results Our world records, part 1 Date Research team Number of Vertices Richter, Goldengorin, Jäger, Molitor Richter, Goldengorin, Jäger, Molitor Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter

24 Our Approach: Pseudo Backbone Contraction Algorithm Experimental Results Our world records, part 2 Date Research team Number of Vertices Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Jäger, Molitor, Richter Dong, Molitor Dong, Molitor Dong, Molitor Dong, Jäger, Molitor, Richter

25 Our Approach: Pseudo Backbone Contraction Algorithm Work on World-TSP The most difficult (and interesting) instance of the TSP homepage is the World-TSP with cities. World-Tour (Ambitious) aim: computation of a world record tour.

26 Our Approach: Pseudo Backbone Contraction Algorithm Work on World-TSP Ideas 1 Hierarchical application of Pseudo Backbone Contraction Algorithm. 2 Compute good tours in overlapping windows. Compose an overall tour using only edges contained in 4 areas.

27 Our Approach: Pseudo Backbone Contraction Algorithm Work on World-TSP Our current tour: 1 0, 1% over the current best tour. 2 Found in a few days.

28 Our Approach: Pseudo Backbone Contraction Algorithm Work on World-TSP Thanks for your attention!