1. CS 312: Algorithm Analysis
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CS 312: Algorithm Analysis
Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds
Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick

2. Announcements Homework #24 due now Homework #25 due Friday
Project #7: TSP
ASAP: Read the helpful “B&B for TSP Notes” linked from the schedule
Read Project Instructions
Today: We continue discussing main ideas
Next Wednesday: Early day
Week from Friday: due

3. Objectives Review the Traveling Salesman Problem (TSP)
Develop a good bound function for the TSP
Reason about Tight Bounds
Augment general B&B algorithm

4. Traveling Salesman (Optimization) Problem
Rudrata or Hamiltonian Cycle
Cycle in the graph that passes through each vertex exactly once +
Find the least cost or “shortest” cycle 1 2 3 4 5 8 6 7 9 10 12
Distinguish from the TSP search problem and the TSP decision problem

5. How to solve?
If with B&B, what do we need?

6. How to solve?
If with B&B, what do we need?

7. Initial BSSF How to compute? Should be quick.1 2 3 4 5
How to compute?
Should be quick.
What if you have a complete graph?
What if you don’t? 9 1 8 2 10 6 5 3 7 12 4
If not, think SCC algorithm

8. Simple-Minded Initial BSSF1 2 3 4 5
Cost of BSSF
= = 31 9 1 8 2 10 6 5 3 7 12 4

9. A Bound on Possible TSP Tours
We need a bound function. Lower or Upper?
How to compute? 1 2 3 4 5 8 6 7 9 10 12
One good idea not pursued here: MST

10. A Bound on Possible TSP Tours
We need a bound function. Lower or Upper?
How to compute? 1 2 3 4 5 8 6 7 9 10 12
One good idea not pursued here: MST

11. A Bound on Possible TSP Tours1 2 3 4 5 9 1 8 2 10 6 5 3 7 12 4
What’s the cheapest way to leave each vertex?

12. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound = = 20 9 1 8 2 10 6 5 3 7 12 4
Save the sum of those costs in the bound (as a rough draft).

13. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20
9-8=1 1
8-8=0 2 10 6 4 3 7 12 4
For a given vertex, subtract the least cost departure from each edge leaving that vertex.

14. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20 1 9 2 1 6 1
Repeat for the other vertices.
What do the numbers on the edges mean now?

15. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20 1 9 2 1 6 1
Now, can we find a tighter lower bound?

16. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20 1 9 2 1 6 1
Does that set of edges now having 0 residual cost arrive at every vertex?

17. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20 1 9 2 1 6 1
In this case, those edges never arrive at vertex #3.

18. Bound on Possible TSP Tours1 2 3 4 5
Rough draft bound
= 20 1 9 2 1 6 1
We have to take an edge to vertex 3 from somewhere.
Assume we take the cheapest.

19. Bound on Possible TSP Tours1 2 3 4 5
Bound = 21 1 9 1 6 1
Subtract its cost from other edges entering vertex 3 and add the cost to the bound.
We have just tightened the bound.

20. This Bound
It will cost at least this much to visit all the vertices in the graph.
There’s no cheaper way to get in and out of each vertex.
Each edge is now labeled with the extra cost of choosing that edge.
The bound is not a solution; it’s a bound!
Why are tight bounds desirable?

21. Bound on Possible TSP Tours
From: 1 2 3 4 5 1 2 3 4 5 8 6 7 9 10 12
Our algorithm can do this reasoning using a cost matrix.

22. Bound on Possible TSP Tours
From: 1 2 3 4 5 1 2 3 4 5 9 6
Reduce all rows.

23. Bound on Possible TSP Tours
From: 1 2 3 4 5 1 2 3 4 5 9 6
Then reduce column #3. Now we have a tighter bound.

24. Search Let’s start the search Arbitrarily start at vertex 1 Focus on:
Why is this OK?
Focus on:
the bound function and
the reduced cost matrix representation of states

25. Using this bound for TSP in B&B
Start at vertex 1 in graph (arbitrary)
bound = 21
BSSF=31
What should our state expansion strategy be?

26. Using this bound for TSP in B&B
Start at vertex 1 in graph (arbitrary)
bound = 21
BSSF=31
1-2
1-3
1-4
1-5
bound = 21+1
bound = 21

27. Add extra cost from 1 to 2, exclude edges from 1 or into 2.
Focus: going from 1 to 2
bound = 21
BSSF=31
Before
After 1 2 3 4 5 1 2 3 4 5 1 1
1-2 9 9 1 1 6 6 1 1
bound = 22
Add extra cost from 1 to 2, exclude edges from 1 or into 2.

28. No edges into vertex 4 w/ 0 reduced cost.
Focus: going from 1 to 2
bound = 21
BSSF=31
Before
After 1 2 3 4 5 1 2 3 4 5 1 1
1-2 9 9 1 1 6 6 1 1
bound = 22+1
No edges into vertex 4 w/ 0 reduced cost.

29. Add cost of reducing edge into vertex 4.
Focus: going from 1 to 2
bound = 21
BSSF=31 1 2 3 4 5 1
1-2 8 1 6
bound =
Add cost of reducing edge into vertex 4.

30. Bounds for other choices
Agenda:
bound = 21
BSSF=31
1-2(23),1-4(21)
1-2
1-3
1-4
1-5
bound = 23
bound = 21

31. Two Possibilities on the Agenda1 2 3 4 5 1 2 3 4 5 1 9 1 1 6 6 1
bound = 23
bound = 21

32. Leaving Vertex 4 1 2 3 4 5 1 6 BSSF=31 1-4 bound = 21 1-4-2 1-4-3
bound = 21 1 6
1-4-2
1-4-3
1-4-5
bound = 22
bound = 21
bound = 28

33. Leaving Vertex 4 1 2 3 4 5 1 6 BSSF=31 1-4 bound = 21 1-4-2 1-4-3
Agenda:
1-4-2(22), 1-4-3(21)
1-4-5(28),1-2(23)
bound = 21 1 6
1-4-2
1-4-3
1-4-5
bound = 22
bound = 21
bound = 28

34. Leaving Vertex 3
1-4-3
BSSF=31 1 2 3 4 5
bound = 21
bound = 21

35. Leaving Vertex 3 1 2 3 4 5 BSSF=31 1-4-3 bound = 21 4-2(22), 3-2(21)
Agenda:
4-2(22), 3-2(21)
4-5(28), 1-2(23),
bound = 21
bound = 21

36. Search Tree for This Problem
1-to-2
1-to-4
b=23
b=21
4-to-2
4-to-3
4-to-5
b=22
b=21
b=28
3-to-2
b=21
2-to-5

37. Termination Criteria for a B&B Algorithm
Repeat until
Agenda is empty
Or time is up
Or BSSF cost is equal to original LB

38. Update: Branch and Bound
function BandB(v)
BSSF  quick-solution(v) // BSSF.cost holds cost
Agenda.clear()
v.b  bound(v)
Agenda.add(v, v.b)
while !Agenda.empty() and time remains and BSSF.cost != v.b do
u  Agenda.first()
Agenda.remove_first()
children = generate_children_ascending(u)
for each w in children do
if ! time remains then break
w.b  bound(w)
if (w.b < BSSF.cost) then
if criterion(w) then
BSSF  w
Agenda.prune(BSSF.cost)
else if partial_criterion(w) then
Agenda.add(w, w.b)
return BSSF

39. Assignment HW #25: Compute bound for TSP instance using today’s method
Reason about search for TSP solution