Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP

Transcription

1 Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP Markus Reuther Zuse Institute Berlin January 19, 2012 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10

2 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Outline 1 The Traveling Salesman Problem 2 The Lin-Kernighan heuristic

3 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 The Traveling Salesman Problem Given Complete undirected graph G = (V, E) Metric edge costs c e 0 for all e E. Problem Find a hamiltionian cycle with minimal cost.

4 Lin-Kernighan (LK) Most famous and best local search approach for the sym. TSP. Developed by Shen Lin and Brian Kernighan in Best exact solver for the TSP is Concorde (Applegate, Bixby, Chva tal, Cook). Best LK-Code today: Keld Helsgaun. Concorde + Code of Helsgaun, 2006: pla85900 solved (world record). (Applegate, Bixby, Chva tal, Cook, Espinoza, Goycoolea, Helsgaun) Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10

5 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 k-opts k-opt neighborhood for tour x: N k (x) consists of all tours, which can be constructed from x by deleting and adding k edges.

6 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 k-opts k-opt neighborhood for tour x: N k (x) consists of all tours, which can be constructed from x by deleting and adding k edges. Observations Two hamiltionian cycles only differ in k edges (2 k n), i.e.: x opt N k (x) for every x. Problem 1: k-optimality in N k can only be tested in O(n k ). Problem 2: k is unknown. Approach: Choose an efficient searchable neighborhood such that k can be choosen dynamically.

7 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 k-opts k-opt neighborhood for tour x: N k (x) consists of all tours, which can be constructed from x by deleting and adding k edges. Observations Two hamiltionian cycles only differ in k edges (2 k n), i.e.: x opt N k (x) for every x. Problem 1: k-optimality in N k can only be tested in O(n k ). Problem 2: k is unknown. Approach: Choose an efficient searchable neighborhood such that k can be choosen dynamically. Sequential k-opt moves.

8 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 k-opts k-opt neighborhood for tour x: N k (x) consists of all tours, which can be constructed from x by deleting and adding k edges. Observations Two hamiltionian cycles only differ in k edges (2 k n), i.e.: x opt N k (x) for every x. Problem 1: k-optimality in N k can only be tested in O(n k ). Problem 2: k is unknown. Approach: Choose an efficient searchable neighborhood such that k can be choosen dynamically. Sequential k-opt moves. Definition: Sequential k-opt move A k-opt move is called sequential if it can be described by a path alternating between deleted and added edges.

9 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example Sequential 6-opt Double-Bridge-Move (4-opt)

10 Lin-Kernighan Flip operations b a b a Operation: flip(next(a), prev(b)) Gain g t of flip t: g t = c(a, next(a)) + c(prev(b), b) c(next(a), b) c(a, prev(b)) Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10

11 Lin-Kernighan Flip operations b a b a Operation: flip(next(a), prev(b)) Gain g t of flip t: Lin-Kernighan g t = c(a, next(a)) + c(prev(b), b) c(next(a), b) c(a, prev(b)) Choose a fix starting node a and construct the alternating path by a sequence of flip operations of the form flip(next(a), prev(b)). The goal within this construction is to obtain k g ti > 0. i=1 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10

12 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 7 v 8 v 10 v 9 v 12 v 2 v 11 v 1 v 5 v 6 v 1 v 2 v 8 v 7 v 4 v 3 v 10 v 9 v 12 v 11 v 6 v 5 flip(v 2, v 4 )

13 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 2 v 1 v 1 v 4 v 7 v 8 v 2 v 3 v 10 v 9 v 12 v 11 v 6 v 5 flip(v 4, v 6 )

14 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 2 v 1 v 5 v 6 v 1 v 6 v 11 v 12 v 9 v 10 v 3 v 2 v 8 v 7 v 4 v 5 flip(v 6, v 8 )

15 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 7 v 8 v 2 v 1 v 5 v 6 v 1 v 8 v 2 v 3 v 10 v 9 v 12 v 11 v 6 v 7 v 4 v 5 flip(v 8, v 10 )

16 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 7 v 8 v 10 v 9 v 2 v 1 v 5 v 6 v 1 v 10 v 3 v 2 v 8 v 9 v 12 v 11 v 6 v 7 v 4 v 5 flip(v 10, v 12 )

17 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Example v 4 v 3 v 7 v 8 v 10 v 9 v 12 v 2 v 11 v 1 v 5 v 6 v 1 v 12 v 9 v 8 v 2 v 3 v 10 v 11 v 6 v 7 v 4 v 5

18 Example Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10

19 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Implementation details Backtracking Neighborhoodgraph k-nearest graph α-nearest graph Delaunay triangulation Mak-Morton-Moves alternatestep() Swaps Bentley-Marking-Scheme Kicking-Strategic Double-Bridge-Moves...

20 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Neighborhood graphs for the choice of the edge {a, next(a)} Figure: Neighborhood graph: 15 nearest neighbors

21 Markus Reuther (Zuse Institute Berlin) Exercise 12: Implementing the Lin-Kernighan heuristic for the TSP January 19, / 10 Neighborhood graphs for the choice of the edge {a, next(a)} Figure: Neighborhood graph: Delaunay triangulation