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![Page 1: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/1.jpg)
CS 312: Algorithm Analysis
Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman
Problem: Tight Bounds
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.
Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick
![Page 2: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/2.jpg)
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
![Page 3: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/3.jpg)
Objectives
Review the Traveling Salesman Problem (TSP)
Develop a good bound function for the TSP
Reason about Tight Bounds Augment general B&B algorithm
![Page 4: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/4.jpg)
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”
cycle1
2
3 4
58
67
5
4
3
2
19
1012
Distinguish from theTSP search problem and theTSP decision problem
![Page 5: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/5.jpg)
How to solve?
If with B&B, what do we need?
![Page 6: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/6.jpg)
How to solve?
If with B&B, what do we need?
![Page 7: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/7.jpg)
Initial BSSF
1
2
3 4
5
8
6
75
4
3
2
19
10
12
How to compute?
Should be quick.
What if you have a complete graph?
What if you don’t?
![Page 8: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/8.jpg)
Simple-Minded Initial BSSF
1
2
3 4
5
8
6
75
4
3
2
19
10
12
Cost of BSSF= 9+5+4+12+1 = 31
![Page 9: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/9.jpg)
A Bound on Possible TSP Tours
We need a bound function. Lower or Upper?How to compute?
1
2
3 4
58
67
5
4
3
2
19
1012
![Page 10: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/10.jpg)
A Bound on Possible TSP Tours
We need a bound function. Lower or Upper?How to compute?
1
2
3 4
58
67
5
4
3
2
19
1012
![Page 11: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/11.jpg)
A Bound on Possible TSP Tours
1
2
3 4
5
8
6
75
4
3
2
19
10
12
What’s the cheapest way to leave each vertex?
![Page 12: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/12.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
8
6
75
4
3
2
19
10
12
Save the sum of those costs in the bound (as a rough draft).
Rough draft bound= 8+6+3+2+1 = 20
![Page 13: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/13.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
8-8=0
6
74
4
3
2
19-8=1
10
12
For a given vertex, subtract the least cost departure from each edge leaving that vertex.
Rough draft bound= 20
![Page 14: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/14.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
12
1
0
0
01
9
6
Repeat for the other vertices.What do the numbers on the edges mean now?
Rough draft bound= 20
![Page 15: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/15.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
12
1
0
0
01
9
6
Now, can we find a tighter lower bound?
Rough draft bound= 20
![Page 16: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/16.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
12
1
0
0
01
9
6
Does that set of edges now having 0 residual cost arrive at every vertex?
Rough draft bound= 20
![Page 17: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/17.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
12
1
0
0
01
9
6
In this case, those edges never arrive at vertex #3.
Rough draft bound= 20
![Page 18: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/18.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
12
1
0
0
01
9
6
We have to take an edge to vertex 3 from somewhere. Assume we take the cheapest.
Rough draft bound= 20
![Page 19: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/19.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
0
0
01
1
0
0
01
9
6
Subtract its cost from other edges entering vertex 3 and add the cost to the bound.
We have just tightened the bound.
Bound = 21
![Page 20: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/20.jpg)
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?
![Page 21: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/21.jpg)
Bound on Possible TSP Tours
1
2
3 4
5
8
6
74
4
3
2
19
10
12
Our algorithm can do this reasoning using a cost matrix.
999 9 999 8 999999 999 4 999 2999 3 999 4 999999 6 7 999 12
1 999 999 10 999
To:1 2 3 4 5From:
1
2
3
4
5
![Page 22: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/22.jpg)
Bound on Possible TSP Tours
999 1 999 0 999999 999 2 999 0999 0 999 1 999999 0 1 999 6
0 999 999 9 999
1
2
3 4
50
01
2
1
0
0
01
9
6
Reduce all rows.
To:1 2 3 4 5From:
1
2
3
4
5
![Page 23: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/23.jpg)
Bound on Possible TSP Tours
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
1
2
3 4
50
01
2
1
0
0
01
9
6
Then reduce column #3. Now we have a tighter bound.
To:1 2 3 4 5From:
1
2
3
4
5
![Page 24: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/24.jpg)
Search
Let’s start the search Arbitrarily start at vertex 1
Why is this OK? Focus on:
the bound function and the reduced cost matrix representation of
states
![Page 25: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/25.jpg)
Using this bound for TSP in B&B
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21 BSSF=31
Start at vertex 1 in graph (arbitrary)
What should our state expansion strategy be?
![Page 26: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/26.jpg)
Using this bound for TSP in B&B
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21
1-2 1-3 1-4 1-5
bound = 21+1
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
BSSF=31
Start at vertex 1 in graph (arbitrary)
bound = 21
![Page 27: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/27.jpg)
Focus: going from 1 to 2
999 999 999 999 999999 999 1 999 0999 999 999 1 999999 999 0 999 6
0 999 999 9 999
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21
1-2
bound = 22
1
2
3 4
50
00
1
1
0
0
01
96
1
2
3 4
5
01
1
0
01
96
Add extra cost from 1 to 2, exclude edges from 1 or into 2.
BSSF=31
Before After
![Page 28: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/28.jpg)
999 999 999 999 999999 999 1 999 0999 999 999 1 999999 999 0 999 6
0 999 999 9 999
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21
bound = 22+1
1
2
3 4
50
00
1
1
0
0
01
96
1
2
3 4
5
01
1
0
01
96
No edges into vertex 4 w/ 0 reduced cost.
Focus: going from 1 to 2
BSSF=31
Before After
1-2
![Page 29: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/29.jpg)
999 999 999 999 999999 999 1 999 0999 999 999 0 999999 999 0 999 6
0 999 999 8 999
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21
bound = 21+1+1
1
2
3 4
5
01
0
0
01
86
Add cost of reducing edge into vertex 4.
Focus: going from 1 to 2
BSSF=31
1-2
![Page 30: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/30.jpg)
Bounds for other choices
999 999 999 999 999999 999 1 999 0999 999 999 0 999999 999 0 999 6
0 999 999 8 999
999 1 999 0 999999 999 1 999 0999 0 999 1 999999 0 0 999 6
0 999 999 9 999
bound = 21
bound = 23
999 999 999 999 999999 999 1 999 0999 0 999 999 999999 0 0 999 6
0 999 999 999 999
bound = 21
1-2(23),1-4(21)BSSF=31
1-2 1-3 1-4 1-5
Agenda:
![Page 31: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/31.jpg)
Leaving Vertex 4
999 999 999 999 999999 999 1 999 0999 0 999 999 999999 0 0 999 6
0 999 999 999 999
bound = 21
1
2
3 4
50
00
1 0
0
0
6
999 999 999 999 999999 999 0 999 0999 999 999 999 999999 999 999 999 999
0 999 999 999 999
999 999 999 999 999999 999 999 999 0999 0 999 999 999999 999 999 999 999
0 999 999 999 999
999 999 999 999 999999 999 0 999 999999 0 999 999 999999 999 999 999 999
0 999 999 999 999
1-4-2 1-4-3 1-4-5
bound = 22 bound = 21 bound = 28
BSSF=311-4
![Page 32: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/32.jpg)
Leaving Vertex 4
999 999 999 999 999999 999 1 999 0999 0 999 999 999999 0 0 999 6
0 999 999 999 999
bound = 21
1
2
3 4
50
00
1 0
0
0
6
999 999 999 999 999999 999 0 999 0999 999 999 999 999999 999 999 999 999
0 999 999 999 999
999 999 999 999 999999 999 999 999 0999 0 999 999 999999 999 999 999 999
0 999 999 999 999
999 999 999 999 999999 999 0 999 999999 0 999 999 999999 999 999 999 999
0 999 999 999 999
bound = 22 bound = 21 bound = 28
1-4-2(22), 1-4-3(21)1-4-5(28),1-2(23)
BSSF=311-4
1-4-2 1-4-3 1-4-5
Agenda:
![Page 33: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/33.jpg)
Leaving Vertex 3
999 999 999 999 999999 999 999 999 0999 0 999 999 999999 999 999 999 999
0 999 999 999 999
bound = 21
1
2
3 4
50
00
0
0
999 999 999 999 999999 999 999 999 0999 999 999 999 999999 999 999 999 999
0 999 999 999 999
1-4-3-2
bound = 21
BSSF=311-4-3
![Page 34: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/34.jpg)
Leaving Vertex 3
999 999 999 999 999999 999 999 999 0999 0 999 999 999999 999 999 999 999
0 999 999 999 999
bound = 21
1
2
3 4
50
00
0
0
999 999 999 999 999999 999 999 999 0999 999 999 999 999999 999 999 999 999
0 999 999 999 999
bound = 21
4-2(22), 3-2(21)4-5(28), 1-2(23),
BSSF=31
1-4-3-2
Agenda:
1-4-3
![Page 35: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/35.jpg)
Search Tree for This Problem
b=21
b=23 b=21
b=22 b=21 b=28
b=21
1-to-2 1-to-4
4-to-2 4-to-3 4-to-5
3-to-2
2-to-5
![Page 36: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/36.jpg)
Termination Criteria for a B&B Algorithm
Repeat until Agenda is empty Or time is up Or BSSF cost is equal to original LB
![Page 37: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/37.jpg)
Update: Branch and Boundfunction BandB(v)
BSSF quick-solution(v) // BSSF.cost holds costAgenda.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 doif ! time remains then breakw.b bound(w)
if (w.b < BSSF.cost) thenif criterion(w) then
BSSF wAgenda.prune(BSSF.cost)
else if partial_criterion(w) thenAgenda.add(w, w.b)
return BSSF
![Page 38: CS 312: Algorithm Analysis Lecture #34: Branch and Bound Design Options for Solving the Traveling Salesman Problem: Tight Bounds This work is licensed.](https://reader035.fdocuments.us/reader035/viewer/2022062717/56649e4c5503460f94b410f9/html5/thumbnails/38.jpg)
Assignment
HW #25: Compute bound for TSP instance using
today’s method Reason about search for TSP solution