TSP! Find a tour (hamiltonian circuit) that visits! every city exactly once and is of minimal cost.!

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Rafe Norris
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1 TSP! Find a tour (hamiltonian circuit) that visits! every city exactly once and is of minimal cost.!

2 Local Search! TSP! What should be the neighborhood?! 2-opt: Find two edges in the current solution and! swap them with two edges in the current solution! to obtain a tour.! Basic idea is the same as for graph partitioning! The details are more complicated

3 TSP! Finding an initial solution! Updating a solution! How to find an initial solution?! Nearest Neighbor! Insertion Methods

4 Nearest Neighbor! With the triangle inequality, we have that for a TSP with n nodes NS 0.5 ( logn + 1) OS where NS is the nearest neighbor heuristics! OS is the optimal solution Is this any good?! 500 cities! optimum = 10, NS ~50,000!

5 Insertion Methods! Basic Idea! Start with a tour between 2 cities! Add the remaining cities one by one in such!!a way that the tour is increased by the!!minimal amount! Main issues! which two cities are chosen to start?! which city is inserted at each stage (most!!important)?! Farthest Insertion! take the two cities that are far apart! for each inserted city v, compute the!!minimum distance between v and any!!city in the current tour.!

6 Insertion Methods! Nearest Insertion! Choose the city to insert as the one whose! distance to any city in the tour is minimal!

7 Insertion Methods! Cheapest Insertion! Choose the city to insert as the one which! increases the cost of the tour the least! Basic Results! I S OS logn + 1 NIS 2 OS! CIS 2 OS

8 Swapping! Back to Local Search! Need some bookkeeping

9 Local Search: 3-opt! Need to swap the direction between cities 2 and 3

10 Local Search: k-opt! Kerninghan & Lin! 2-opt! 3-opt: much better quality than 2-opt much!!!!more expensive! 4-opt: marginally better than 3-opt much!!!!more expensive! How to improve?! same idea as in graph partitioning! find k (in k-opt) dynamically!

11 TSP! t2 y1 x1 t1 First Step! choose a vertex t 1! choose an edge adjacent to t 1 :x 1! let t 2 be the end-vertex of x 1! choose an edge y 1 such that!!!! x 1 - y 1 > 0! if none exists, choose another vertex and!!start again!

12 TSP (2-opt)! t4 t3 y1 x2 t2 x1 t1 We have a solution by connecting t 4 with t 1! Compute its cost (how?)! but do not insert the edge.!

13 TSP (3-opt)! t4 t3 y1 y2 t2 x1 x2 t1 t4 t3 t2 t1 y1 x3 y2 t5 t6

14 TSP (4-opt)! t4 t3 t7 y3 y1 x3 y2 t5 t6 t2 t1

15 Branch and Bound! Two main steps! Bounding: relaxing the problem to find an!!optimistic evaluation on the cost! Branching: making a choice! Relaxation! Constraints of the problem: a hamiltonian!!tour (i.e. a tour which visits each city!!exactly once)! Can we think of a problem that could give!!a lower bound?!

16 Branch and Bound! for the TSP! Minimum Spanning Tree! We can do slightly better!

17 B&B for TSP: 1-tree! Intuition! Let T be a tour! remove two edges intersecting at one city! x1 We have a spanning tree through the other nodes! Relaxation: Take the minimum spanning tree Relaxation of the overall problem! mst(path) + cost of the two cheapest edges! from x 1 to the other vertices!

18 B&B for TSP: 1-tree! G = (V, E) with edge costs (c e : e is in G)! v 1 is in V!! A = min{ c e + c f ( e, f ) δ( v 1 ), e f } B is the cost of the MST of G[ V \ {v 1 } ]! A + B is a lower bound for the TSP on G! Of course, we can try several v 1 to improve! the bound!

19 Bounding! Branch and Bound! for the TSP! Use the 1-tree bound Pruning! getting rid of impossible edges At any stage, a 1-tree bound is computed given! a number of already selected edges

20 Branch and Bound! for the TSP! Let E be the edges selected so far.! Compute a 1-tree bound in such a way that! the MST includes all the selected edges and! do not consider impossible edges! Choose a city with no successor and! nondeterministically assign an edge!

21 Branch and Bound! for the TSP! What is the cost of the 1-tree bound?! What is the bound on the original tour?!

22 Branch and Bound! for the TSP! Can we improve the 1-tree bound?! v v u 0 10 e tree bound: 0! What is the minimum cost of a tour? Why is the 1-tree bound poor? We use three edges out of city u! Any tour can only use two of these and then it! is forced to use an edge of cost 10!

23 Branch and Bound! for the TSP! How to remedy this problem?! What is the cost of the tour when we increase! to 10 the cost of all the edges connected to! v compared to the previous tour?! NT = OT +! What is the cost of the 1-tree bound?! What is the bound on the original tour?!

24 B&B for TSP: Improving! the 1-tree Bound! Assign a number y v to each node v! The cost of an edge between v and w is! c e ~ = c e y v y w In the previous example, we assigned -10! to y v (and this added 10 to its adjacent! edges)! Held & Karp Bound! Let G = (V, E) be a graph with edge costs! v V let y v be a real number Let the new cost of (v, w)=e be! c~ e = c e y v y w and let c be the cost of an optimal 1-tree w.r.t.! new cost! ( ) c Then! 2 y v v V + is a lower bound for! the TSP on G!

25 Branch and Bound! for the TSP! How to find a set of good numbers? Iterative Scheme! Suppose we have a 1-tree T with the new cost! for each node v in V, δ T (v) denotes the number of! edges meeting at v.! if δ T (v) > 2 then! if δ T (v) = 1 then! Held & Karp idea! y v = y v + step * (2-δ T (v))! where step is the step size! Iterates this process to obtain a sequence of! H&K bounds!

26 Held and Karp! When to stop iterating?! H* = - ; For i = 1 to MAXCHANGES For k = 1 to NUMITERATIONS Let T be the 1-tree solution H its cost if H > H* then H* = H if T is a tour, stop let q = a(u-h)/sum(v in V)(2-d T (v)) 2 if q < LIMIT then STOP; Update the weights with q as step size replace a by a*b; take a = 2, b = 0.5

27 Lagrangian Relaxation! maximize w x subject to A 1 x b 1 [Hard] A 2 x b 2 [easy] x integral Upper bound (why?)! L(y) = max w x + y (b 1 - A 1 x) subject to A 2 x b 2 x integral when y 0 How to choose the y?! y(k+1) = y(k) t(k)(b 1 A 1 x(k)) Choose t(k)! to converge towards 0! not too fast (sum t(k) = )!

28 TSP!

29 TSP revisited! Formulate it as an integer program.! min! subject to! where! δ(i) are edges adjacent to node i!! other notations:! γ(s): edges whose vertices are in S.!

30 TSP revisited! Subtour constraints! :! What is the problem with subtour constraints?!

31 Relation with Held & Karp! How to express a 1-tree?! min! s.t.! (subtour)! The subtour problem is just! min! s.t.! (redundant)!

32 Relation with Held & Karp! min! s.t.! (A)! (B)! (C)! dual problem! a dual variable for each vertex! +! 1 dual variable for each constraint of the 1-tree! Suppose that we have an optimal solution! :! Assume that we fix the dual variables to their! values! The remaining variables are an optimal solution! to the dual! min! s.t. (A)(B)(C) x(δ(v1)) = 2!

33 TSP: more cuts! Obviously the subtour constraints are not!!sufficient! We need other cuts! How many edges do you cross?!

34 TSP: more cuts! How many edges do you cross?! COMB: basic idea is to reason about the number of! crossings! C: set of edges either in the handle or in a tooth! CUT: bound the number of edges in the COMB!

35 TSP: more cuts! number of edges from the COMB: H, T 1,,T k in! any tour is! H + ½(k-1)! in our example!

36 TSP: COMB cuts! Tell me what the cut is!!

Combinatorial Optimization - Lecture 14 - TSP EPFL

Combinatorial Optimization - Lecture 14 - TSP EPFL 2012 Plan Simple heuristics Alternative approaches Best heuristics: local search Lower bounds from LP Moats Simple Heuristics Nearest Neighbor (NN) Greedy