1. Approximation Algorithms
Pasi Fränti

2. Approximation algorithms
Optimal algorithm too slow
Heuristic algorithm too uncertain
Provide result with guarantee:

3. Bin packing problem
Consider a set of bins (capacity 1) and a set of items xi of size wi[0,1].
GOAL: pack the items into as few bins as possible
Finding the optimal solution is a NP-hard problem (exponential time)

4. 1-approximation algorithm!
Bin packing example Greedy algorithm
Items: [0.4, 0.7, 0.5, 0.2, 0.9, 0.6]
0.9
0.9
1.0
0.9
0.7
0.6
0.4
1-approximation algorithm!

5. Traveling Salesman Problem
ASSUMPTION: All pairwise distances exist and they obey triangular inequality:
w(a,c) ≤ w(a,b)+w(b,c)
GOAL: find the shortest possible route that visits each city once and returns to the origin city
Finding optimal route is slow (NP-hard)

6. Approximation algorithm
TSP-appro(V, E)
T  MST(V, E);
R  WALK-OF-TREE(T); //depth-first
 A-B-C-B-H-B-A-D-E-F-E-G-E-D-A
S  SHORTCUTS(R); // skip redundant
 A-B-C-H-D-E-F-G-A A D E B F G C H

7. Algorithm example Graph with N=8 nodesF H C B 7

8. Algorithm example Minimum spanning tree2 A D E G F H C B
11.9 2 2 2 2 2 5 8

9. Algorithm example Tree traversal2 A D E G F H C B
23.8 2 2 2 2 2 2 2 2 2 2 5 2 5
A-B-C-B-H-B-A-D-E-F-E-G-E-D-A 9

10. Algorithm example ShortcutsD E G F H C B
19.1
20 2 2 2
17 2 2 5
A-B-C-B-H-B-A-D-E-F-E-G-E-D-A 10

11. Algorithm example Final resultD E G F H C B
19.1
A-B-C-H-D-E-F-G-A 11

12. Analysis of error bound 1-approximation algorithm
Tree traversal includes all edges twice:
Cutting one link from TSP is spanning tree, but not necessarily the minimum:
Tree traversal includes all edges twice:

13. Christofides algorithm
Christofides(V, E)
T  MST(V, E);
Find Euler tour:
D  FindOddNodes(T);
M  PerfectMatch(D);
H  T+M;
R  EulerTour(H);
S  SHORTCUTS(R);

14. Christofides example Minimum spanning tree2 A D E G F H C B
11.9 2 2 2 2 2 5 14

15. Christofides example Detect the nodes with odd degreeB 15

16. Christofides example Matching the nodesB 2 2 5 16

17. Christofides example Merging MST and the matchingB 17

18. Christofides example Euler tour2 A D E G F H C B
17.0 2 2 2 2 2 2 5 2 5
A-B-C-B-H-F-E-G-E-D-A 18

19. Christofides example Shortcuts2 A D E G F H C B
15.5
2∙2 2 2 2 2 5 5
A-B-C-B-H-F-E-G-E-D-A 19

20. Christofides example ShortcutsG F H C B 20

21. Error bounds for Christofides MST + Matching
Christofides produces TSP which is at most 50% longer than the optimum:
Proof: The algorithm combines MST with the minimum-weight perfect matching:
Shortcuts can only shorten the tour. 21

22. Error bounds for Christofides Links from matching
Let v1, v2, …, v2m be the odd nodes of T. Skipping the rest of the nodes in TSP gives path that consists of two matchings:
M1 ={(v1, v2), …, (v2m-1, v2m)} M2 ={(v2, v3), …, (v2m, v1)}
Odd nodes
Due to optimality of matching:
TSP v1 v2 v3 v4
v2m
v2m-1 …
Links complementing tour
Links from matching

23. Error bounds for Christofides 0.5-approximation algorithm
Cutting one link from TSP is spanning tree, but not necessarily the minimum:
Combining this with the matching:
Thus we have:

24. Empty space for notes

25. Pseudo polynomial knapsack
Input: Weight of N items {w1, w2, ..., wn}
Cost of N items {c1, c2, ..., cn}
Knapsack limit s
Goal: Select for knapsack: {x1,x2,…xn} so that
Problem is NP hard
Pseudo polynomial solution exist
Used for approximation algorithm

26. Pseudo polynomial knapsack Dynamic programming algorithm
KNAPSACK (c[1,n], w[1,n], s} RETURNS x[1,n]
R  {(∅,0)}
FOR i 1 TO n DO
FOR (S,c)  R DO
IF ΣjS wj+wi ≤ s THEN R  R∪{(S∪{i},c+ci)}
FOR (S,c), (S’,c)  R DO
IF ΣjS wj ≤ ΣjS’ wj THEN R  R\{(S’,c)}
ELSE R  R\{(S,c)}
(S,c)  (S,c’)R for which c’ is max!
FOR i1 TO n DO IF iS THEN x[i]1 ELSE x[i]0
RETURN x;
Include if item fits in
If same cost prune out weaker solution
New comby

27. Pseudo polynomial knapsack Time complexity
Sample input:
wi={1,1,2,4,12}
ci ={1,2,2,10,4}
s=15
All combinations:
∅ {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} … 1 2 3
Time complexity:
O(nc) where c = cost of optimal solution
…but

28. Approximation algorithm Scaling the input
Original input:
Scale down input
Modified input:
Solve by DP
Solve by DP
wi ={1,1,2,4,12}
ci ={1,2,2,10,4}
S =15
wi ={1,1,2,4,12}
ci/d ={1,1,1,5,2}
S =15

29. Approximation algorithm Error bound
Optimal
for original
Optimal for scaled
∙ d/d
Truncate
xi optimal for scaled and thus bigger
z ≥ z-1
= c
Σyi≤ n

30. Approximation algorithm nd/c-approximation algorithm
Result of algorithm:
Optimal result:
Revise
Choosing d = ε∙c/n, we can get any ε-approximation
Value c is not known but upper limited: c ≤ n∙cmax
Time complexity:
Note:

31. Empty space for notes 1 2 4 A B C D 3 4 1 3 E F ?

32. Acknowledgements
Contributions by Sharon Guardado and Pekka T. Kilpeläinen to TSP example and its proof appreciated.