TSP Heuristics

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TSP Heuristics
January 2002
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Heuristics for the Travelling Salesman Problem

• Faced with the likely impossibility of an
  efficient algorithm for finding the optimum
  solution of problems like TSP, we resort to
  heuristics instead. A heuristic is a rule of
  thumb which should give a reasonable approximate
  solution, in polynomial time.
• E.g. a heuristic we might use to find a well-off
  person in the street is to look for smart clothes
  and neat hair.
• For example, consider the following TSP

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Example
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1. Nearest neighbour heuristic

• Heuristic 1.
• always move to the nearest unvisited city

• E.g.
• starting from A
• A F G B H D J E A - but misses C!
• Starting from G
• G F J H D E A B C G ? A B C G F J H D E A
• length 67

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2. Christofides algorithm

• Heuristic 2.
• Christofides algorithm
• three steps

• ? Find a minimum length spanning tree
• start with the shortest link (road), consider
  links of progressively greater length, add a link
  to the tree if it connects either a new node to
  the tree, or two subtrees

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Christofides algorithm contd

• Thus
• i. length 3 DH
• ii. length 4 HJ, FG, GB
• iii. length 5 EF, FJ, but not DJ
• iv. length 6 AF, but not GJ, GH
• v. length 7 not AG, EJ
• vi. length 8 CH, but not BH
• Tree complete since all nodes linked, and there
  are no circuits

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Christofides algorithm contd

• ? There must be an even number of odd order
  nodes (i.e. with an odd number of links) A B
  C D E H in this case
• Find the minimum length matching of disjoint
  pairs of these.
• Thus
• AB CD EH wont work (no link EH)
• AB CH DE 28
• AE BH CD 28
• AE BC DH 23 minimum length

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Christofides algorithm contd

• Add these new links (AE BC DH in this case) to
  the tree (dashed lines). Every node now has even
  order.

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Christofides algorithm contd

• ? An Euler tour traverses each link once (since
  each node now has even order)
• e.g.
• A F G B C H D H J F E A - total length 62
• Can use shortcuts to skip over cities already
  visited.
• A F G B C H D J E A - total length 57
• Shortcuts DJ and JE

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Christofides algorithm contd

• Shortcuts DJ JE
• A F G B C H D J E A - total length 57

• Shortcuts CD AG
• A E F J H D C B G A - total length 58

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Christofides algorithm contd

• The Christofides algorithm generates a good tour
  (but not necessarily the best!) in O(n3) time.
• E.g. shortcuts AG JE JD give route A G B C H D J
  E F A - length 55
• Cant guarantee to find best solution because
• then general problem of finding the best set of
  shortcuts around an Euler tour is NP-Complete

These shortcuts have not been found by bypassing
already-visited cities
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3. Branch and bound

• BB is A search strategy rather than a heuristic
• Algorithm
• Find one route R1 (e.g. at random) this has
  length L1
• Now search for another route R2
• build it city by city e.g. R2 a,b,c,, noting
  the total length L2 so far.
• if L2 gt L1 then stop, discard R2 (whether
  complete or not) and try another route.
• Never start a new route with a discarded route.

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Branch and bound contd

• 1. Pick a route R1
• R1 AFCEDBGDCA
• L1 24
• 2. Build new route R2
• R2 AF L2 3
• R2 AFE L2 13
• R2 AFEB L2 28
• L2 gt L1 therefore discard L2
• and try 2. again
• The best route cannot possibly start AFEB

e.g.
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Branch and bound contd

• Branch and bound prunes the tree of choices as we
  search.

• BB can simplify the search tree dramatically but
  in the worst case will not reduce problem time at
  all.

?
Prune tree here
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Summary

• TSP problem is NP-Complete
• But we can find reasonable solution in polynomial
  time.
• Use heuristics
• nearest neighbour
• christofides algorithm
• or search strategy
• branch and bound

Cant guarantee to find best solution
Cant guarantee any speed up
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Exercises

• 1. What does the following Turing Machine do
• Q q0, q1, q2, T 0, 1, F q2
• g(q0,0) (q0,0,R), g(q0,1) (q1,1,R),
• g(q1,0) (q2,1,R), g(q1,1) (q1,1,R)
• 2. Modify the machine in question 1 so that it
  always leaves an even number of ones on the tape

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Exercises

• 3. Apply the nearest neighbour heuristic and
  Christofides algorithm to the following
  travelling salesman problem, with starting point
  at A

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Exercises

• 4. Apply Christofides algorithm to the following
  travelling salesman problem, with starting point
  at A

B
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A
8
C
5
7
5
6
D
4
F
E
6
5
8
9
4
7
H
8
G
6
K
6
8
J
9
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