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Travelling Salesman Problem
Algorithms and Networks 2015/2016
Hans L. Bodlaender
Johan M. M. van Rooij
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Contents
TSP and its applications
Heuristics and approximation algorithms
Construction heuristics, a.o.: Christofides, insertion heuristics
Improvement heuristics, a.o.: 2-opt, 3-opt, Lin-Kernighan
Exact algorithms
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PROBLEM DEFINITION APPLICATIONS
Travelling Salesman Problem – Algorithms and Networks
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Problem
Instance: n vertices (cities), distance between every pair of vertices.
Question: Find shortest (simple) cycle that visits every city?
1 2
3 4
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2 3
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1 2
3 4
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2 3
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1 2
3 4
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2 3
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13 11
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Applications
Pickup and delivery problems
Robotics
Board drilling / chip manufacturing
Lift scheduling
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Assumptions / Problem Variants
Complete graph vs underlying graph structure.
Complete graph: a given distance between all pairs of cities.
Directed edges vs undirected arcs.
Undirected: symmetric: w(u,v) = w(v,u).
Directed: not symmetric.
Asymmetric examples: painting/chip machine application.
Lengths:
Lengths are non-negative (or positive).
Triangle inequality: for all x, y, z:
w(x,y) + w(y,z) w(x,z)
Different assumptions lead to different problems.
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NP-complete
Instance: cities, distances, k.
Question: is there a TSP-tour of length at most k?
Is an NP-complete problem.
Relation with Hamiltonian Circuit problem.
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If triangle inequality does not hold
Theorem: If PNP, then there is no polynomial time algorithm for TSP without triangle inequality that approximates within a ratio c, for any constant c.
Proof: Suppose there is such an algorithm A. We build a polynomial time algorithm for Hamiltonian Circuit (giving a contradiction): Take instance G=(V,E) of Hamiltonian Circuit.
Build instance of TSP: • A city for each v V.
• If (v,w) E, then d(v,w) = 1, otherwise d(v,w) = nc+1.
A finds a tour with distance at most nc, if and only if, G has a Hamiltonian circuit.
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CONSTRUCTION HEURISTICS
Travelling Salesman Problem – Algorithms and Networks
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Heuristics and approximations
Construction heuristics:
A tour is built from nothing.
Improvement heuristics:
Start with `some’ tour, and continue to change it into a better one as long as possible
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1st Construction Heuristic: Nearest neighbor
Start at some vertex s; v=s;
While not all vertices visited
Select closest unvisited neighbor w of v
Go from v to w; v=w
Go from v to s.
Without triangle inequality:
Approximation ratio arbitrarily bad.
With triangle inequality:
Approximation ratio O(log n).
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2nd Construction Heuristic: Heuristic with ratio 2
Find a minimum spanning tree
Report vertices of tree in preorder
Preorder: visit a node before its children
Approximation ratio 2 (with triangle inequality):
OPT ≥ MST
2 MST ≥ Result
Result / OPT ≤ 2MST / MST = 2
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3rd Construction Heuristic: Christofides
1. Make a Minimum Spanning Tree T.
2. Set W = {v | v has odd degree in tree T}.
3. Compute a minimum weight matching M in the graph G[W].
4. Look at the graph T+M.
Note that T+M is Eulerian!
5. Compute an Euler tour C’ in T+M.
6. Add shortcuts to C’ to get a TSP-tour.
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Ratio 1.5
Total length edges in T: at most OPT
Total length edges in matching M: at most OPT/2.
T+M has length at most 3/2 OPT.
Use D-inequality.
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4th Construction Heuristic: Closest insertion heuristic
Build tour by starting with one vertex, and inserting vertices one by one.
Always insert vertex that is closest to a vertex already in tour.
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Closest insertion heuristic has performance ratio 2
Build tree T: if v is added to tour, add to T edge from v to closest vertex on tour.
T is a Minimum Spanning Tree (Prim’s algorithm).
Total length of T OPT.
Length of tour 2* length of T.
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Many variants
Closest insertion: insert vertex closest to vertex in the tour.
Farthest insertion: insert vertex whose minimum distance to a node on the cycle is maximum.
Cheapest insertion: insert the node that can be inserted with minimum increase in cost. Gives also ratio 2 .
Computationally expensive.
Random insertion: randomly select a vertex.
Each time: insert vertex at position that gives minimum increase of tour length.
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5th Construction Heuristic: Cycle merging heuristic
Start with n cycles of length 1.
Repeat:
Find two cycles with minimum distance.
Merge them into one cycle.
Until 1 cycle with n vertices.
This has ratio 2: compare with algorithm of Kruskal for MST.
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6th Construction Heuristic: Savings heuristic
Cycle merging heuristic where we merge tours that provide the largest “savings”:
Saving for a merge: merge with the smallest additional cost / largest savings.
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Some test results
In an overview paper, Junger et al. report on tests on set of instances (105 – 2392 vertices; city-generated TSP benchmarks) Nearest neighbor: 24% away from optimal in average Closest insertion: 20%; Farthest insertion: 10%; Cheapest insertion: 17%; Random Insertion: 11%; Preorder of min spanning trees: 38%; Christofides: 19% with improvement 11% / 10%; Savings method: 10% (and fast).
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IMPROVEMENT HEURISTICS
Travelling Salesman Problem – Algorithms and Networks
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Improvement heuristics
Start with a tour (e.g., from heuristic) and improve it stepwise 2-Opt
3-Opt
K-Opt
Lin-Kernighan
Iterated LK
Simulated annealing, …
Iterative improvement
Local search
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Scheme
Rule that modifies solution to different solution
While there is a Rule(sol, sol’) with sol’ a better solution than sol Take sol’ instead of sol
Cost decrease
Stuck in `local minimum’
Can use exponential time in theory…
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Very simple
Node insertion:
Take a vertex v and put it in a different spot in the tour.
Edge insertion:
Take two successive vertices v, w and put these as edge somewhere else in the tour.
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2-opt
Take two edges (v,w) and (x,y) and replace them by (v,x) and (w,y) if this improves the tour.
Costly: part of tour should be turned around.
On R2: get rid of crossings of tour.
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2-Opt improvements
Reversing shorter part of the tour.
We need a direction on the tour to decide what reconnect option leads to subtours.
Postpone correcting tour.
Clever search to improving moves.
Restrict distance function evaluations.
Prove that you do not need to consider some potential 2-opt moves, because these were not improving in earlier steps.
Look only to subset of candidate improvements.
Combine with node insertion.
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3-opt
Choose three edges from tour
Remove them, and combine the three parts to a tour in the cheapest way to link them
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3-opt
Costly to find 3-opt improvements: O(n3) candidates
k-opt: generalizes 3-opt
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Lin-Kernighan
Idea: modifications that are bad can lead to enable something good
Tour modification:
Collection of simple changes
Some increase length
Total set of changes decreases length
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Lin-Kernighan
One LK step:
Make sets of edges X = {x1, …, xr}, Y = {y1,…,yr}
• If we replace X by Y in tour then we have another tour
Sets are built stepwise
Repeated until …
Variants on scheme possible
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One Lin-Kernighan step
Choose vertex t1, and edge x1 = (t1,t2) from tour.
i=1.
Choose edge y1=(t2, t3) not in tour with g1 = w(x1)–w(y1) > 0 (or, as large as possible).
Repeat a number of times, or until … i++;
Choose edge xi = (t2i-1,t2i) from tour, such that: • xi not one of the edges yj.
• oldtour – X + (t2i,t1) +Y is also a tour.
if oldtour – X + (t2i,t1) +Y has shorter length than oldtour, then take this tour: done.
Choose edge yi = (t2i, t2i+1) such that: • gi = w(xi) – w(yi) > 0.
• yi is not one of the edges xj .
• yi not in the tour.
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Iterated Lin-Kernighan
Construct a start tour.
Repeat the following r times:
Improve the tour with Lin-Kernighan until not possible.
Do a random 4-opt move that does not increase the length with more than 10 percent.
Report the best tour seen.
Cost much time. Gives excellent
results!
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Other methods
Simulated annealing and similar methods.
Problem specific approaches, special cases.
Iterated LK combined with treewidth/branchwidth approach:
Run ILK a few times (e.g., 5).
Take graph formed by union of the 5 tours.
Find minimum length Hamiltonian circuit in graph with clever dynamic programming algorithm.
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DYNAMIC PROGRAMMING
Travelling Salesman Problem – Algorithms and Networks
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Held-Karp algorithm for TSP
O(n22n) algorithm for TSP.
Uses dynamic programming.
Take some starting vertex s.
For set of vertices R (s R), vertex w R,
let B(R,w) = minimum length of a path, that
Starts in s.
Visits all vertices in R (and no other vertices).
Ends in w.
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TSP: Recursive formulation
B({s},s) = 0
B({s},v) = ∞ if v ≠ s
If |R| > 1, then
B(R,x) = minvR\{x}B( R\{x}, v ) + w(v,x)
If we have all B(V,v) then we can solve TSP.
Gives requested algorithm using DP-techniques.
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Space Improvement for Hamiltonian Cycle
Dynamic programming algorithm uses:
O(n22n) time.
O(n2n) space.
In practice space becomes a problem before time does.
Next, we give an algorithm for Hamiltonian Cycle that uses:
O(n32n) time.
O(n) space.
This algorithm counts solutions using ‘Inclusion/Exclusion’.
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Counting (Non-)Hamiltonian Cycles
Computing/counting tours is much easier if we do not care about which cities are visited.
This uses exponential space in the DP algorithm.
Define: Walks[ vi, k ] = the number of ways to travel from v1 to vi traversing k times an edge.
We do not care whether nodes/edges are visited (or twice).
Using Dynamic Programming:
Walks[ vi, 0 ] = 1 if i = 1
Walks[ vi, 0 ] = 0 otherwise
Walks[ vi, k ] = ∑vj ∈ N(vi) Walks[ vj, k – 1 ]
Walks[ v1, n ] counts all length n walks that return in v1. This requires O(n3) time and O(n) space.
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How Does Counting ‘Arbitrary’ Cycles Help?
A useful formula:
Number of cycles through vj =
total number of cycles (some go through vj some don’t)
- total number of cycles that do not go through vj.
Hamiltonian Cycle: cycle of length n that visits every city.
Keeping track of whether every vertex is visited is hard.
Counting cycles that do not go through some vertex is easy.
Just leave out the vertex in the graph.
Counting cycles that may go through some vertex is easy.
We do not care whether it is visited or not (or twice) anymore.
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Using the formula on every city
A useful formula:
Number of cycles through vj =
total number of cycles (some go through vj some don’t)
- total number of cycles that do not go through vj.
What if we apply the above formula to every city (except v1)?
We get 2n-1 subproblems, were we count walks from v1 to v1 where either the city is not allowed to be visited, or may be visited (not important whether it is visited, or twice, or not).
After combining the computed numbers from all 2n-1 subproblems using the above formula, we get:
The number of cycles of length n visiting every vertex.
I.e, the number of Hamiltonian cycles.
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Apply the formula to every city: the Inclusion/Exclusion formula
A useful formula:
Number of cycles through vj =
total number of cycles (some go through vj some don’t)
- total number of cycles that do not go through vj.
Let CycleWalks( V’ ) be the number of walks from v1 to v1 using only vertices in V’, then:
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CONCLUSIONS
Travelling Salesman Problem – Algorithms and Networks
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Conclusions
TSP has many applications.
Also many applications for variants of TSP.
Heuristics.
Construction heuristics
Improvement heuristics
Dynamic programming in O(n22n) time and O(n2n) space.
Inclusion/Exclusion for Hamiltonian Cycle in O(n32n).
Further reading:
M. Jünger, G. Reinelt, G. Rinaldi, The Traveling Salesman Problem, in: Handbooks in Operations Research and Management Science, volume 7: Network Models, North-Holland Elsevier, 1995.