1. Using Homogeneous Weights for Approximating the Partial Cover Problem
Reuven Bar-Yehuda

2. The Partial Set Cover problem
Given:
Find: min
s.t.
The partial set cover problem

3. The partial set cover problem
Minimal t-cover
C is a t-cover (a feasible solution)
if and
C is a minimal t-cover
if , is not a t-cover
The partial set cover problem

4. Example (a simple graph)2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3
The partial set cover problem

5. The partial set cover problem
Example: 13-cover 2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3
The partial set cover problem

6. Example: minimal 13-cover2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3
The partial set cover problem

7. Example: optimal 13-cover2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3
The partial set cover problem

8. Example (a hyper-graph)V 1 6 2 4 a 7 5 10 2 b c
e={a,b,c,d} d
The partial set cover problem

9. Example: minimal 11-cover6 2 4 a 7 b 5 c 10 d 2
The partial set cover problem

10. Example: optimal 11-cover6 2 4 a 7 b 5 c 10 d 2
The partial set cover problem

11. Homogeneous Weight function
Definition:
Property: any minimal cover is
a “good” approximation
The partial set cover problem

12. Homogeneous Weight function
Given:
Claim: C is a minimal t-cover
(define: )
The partial set cover problem

13. The partial set cover problem
Proof:
Lemma 1: If C is a t-cover then
Lemma 2: If C is a minimal t-cover then
The partial set cover problem

14. The partial set cover problem
Proof:
Lemma 1: If C is a t-cover then
If √
Else
The partial set cover problem

15. The partial set cover problem
Proof:
Lemma 2 (a simple graph): If C is a minimal t-cover then
Case 1:
Case 2:
(case 1)
The partial set cover problem

16. The partial set cover problem
Proof:
Lemma 2 (a simple graph): If C is a minimal t-cover then
Case 3:
The partial set cover problem

17. L2: If C is a minimal t-cover then
Case 3:
Define:
The partial set cover problem

18. The partial set cover problem
To complete the proof:
Let , and
C is a minimal t-cover 
But 
The partial set cover problem

19. Example: hyper-graph, C is a minimal t-cover, δ(C) > 2t4 1 a
1,9 b
1,9 c
2,5 d
3,5 e
5,4
The partial set cover problem

20. The partial set cover problem
Proof:
Lemma 2 (hyper-graph): If C is a minimal t-cover then
Case 1:
Case 2:
(case 1)
The partial set cover problem

21. L2: If C is a minimal t-cover then
Case 3:
Define:
The partial set cover problem

22. The partial set cover problem
To complete the proof:
Let , and
C is a minimal t-cover 
But 
The partial set cover problem

23. Homogeneous Weight function
Claim: C is a minimal t-cover, C* optimal
Proof:
By Lemma 2:
By Lemma 1:
The partial set cover problem

24. The partial set cover problem
Weight Decomposition
Local-Ratio Theorem:
C is an approximation w.r.t.
C is an approximation w.r.t.
( )
The partial set cover problem

25. The partial set cover problem
Algorithm Cover ( )
If return
If return “no solution”
If return Cover( )
Cover( )
The partial set cover problem

26. The partial set cover problem
For each
C = Cover( )
If is a t-cover
Return C
The partial set cover problem

27. The partial set cover problem
Example 2 4 19 1 1 1 1 10 1 3 2 23 1
The partial set cover problem

28. The partial set cover problem
Example 2
4,4
19,9 1 1 1
1,6 10
ω,δ 1 3
2,3
23,9 1
The partial set cover problem

29. The partial set cover problem
Example 2
10/3
35/2 1 1 1 10 ω2 1 3
3/2
43/2 1
The partial set cover problem

30. The partial set cover problem
Example
10/3,3
35/2,3 2 1 10
ω,δ 1
3/2,2
43/2,3
The partial set cover problem

31. The partial set cover problem
Example 2
13/12
61/4 1 10 ω2
77/4 1
The partial set cover problem

32. The partial set cover problem
Example
13/12,1
61/4,1 2 10
ω,δ
77/4,1
The partial set cover problem

33. The partial set cover problem
Example 2 4 19 1 1 1 1 10 1 3 2 23 1
The partial set cover problem

34. The partial set cover problem
Example 2 4 19 1 1 1 1 10 1 3 2 23 1
The partial set cover problem

35. The partial set cover problem
Algorithm Cover
Claim: Algorithm Cover is a approximation
for the t-cover problem
Proof:
(L3: Algorithm Cover has at most iterations)
By induction on the number of iterations.
Base: for the empty set is optimal
Step: by the Local-Ratio Theorem
The partial set cover problem

36. The partial set cover problem
Time complexity
Lemma 4: Algorithm Cover can be implemented
in time
Proof:
vertex deletion  adjacent edges deletion
For each such edge,
l(e) is subtracted from its |e| vertices 
At most iterations, each iteration
The partial set cover problem

37. The partial set cover problem
Related work
t-VC: (simple graph)
Bshouty and Burroughs, 1998, 2-approximation
Hochbaum, Here:
t-VC with edge lengths: (simple graph)
Hochbaum, 1998, 3-approximation
Here: approximation
The partial set cover problem

38. The partial set cover problem
Related work
t-SC:
Burroughs, 1998, ( )-approximation
t-SC with edge lengths:
Here: approximation time
The partial set cover problem