1. Partial Sublinear Time Approximation and Inapproximation for Maximum Coverage
Bin Fu
Department of Computer Science
University of Texas Rio Grande Valley Texas, USA

2. Hard to find, Easy to check
blind monkey

3. Hamiltonian Path Hamiltonian path goes through each node exactly once
HAMPATH={G| G is a directed graph with a Hamiltonian path}

4. P versus NP Polynomial time: P: polynomial time decidable problems.
NP: polynomial time verifiable problems.

5. Polynomial Time Verifier
verifier V(w,c)
w: input
c: certificate of length poly(|w|).
where |w| is length of w. For example, |C++|=3

6. Polynomial Time Verifier
A verifier V(w,c) for a language L is an algorithm V,
V(w,c) runs in polynomial time poly(|w|) for input w, and c with a polynomial length
w is in L if and only if V(w,c) accepts for some c with a polynomial length.

7. Algorithms toward NP-hardness
1.Approximation algorithm
2. Fixed Parameter Algorithm
3.Heurisitc Algorithm

8. NP-Complete A problem H is NP-complete if for every B in NP,
Every problem in NP can be reduced to an NP-complete problem in a polynomial time. NP
NP-complete P

9. Polynomial Time Reduction
Assume that A and B are two sets.
A is polynomial time mapping reducible to A if a polynomial time computable function f exists such that

10. Introduction
Approximation algorithms are used to get a solution close to the (optimal) solution of an optimization problem in polynomial time

11. Definition
An algorithm is an α-approximation algorithm for an optimization problem Π if
The algorithm runs in polynomial time.
The algorithm always produces a solution that is within a factor of α of the optimal solution.

12. Maximum Cover (MC)
A collection T of finite m sets S1, S2, …, Sm, and integer k.
Find k of them with largest union.

13. Decision Version of Maximum Cover
A collection T of finite m sets S1, S2, …, Sm, integers k and t.
Decide if there are k sets from them with union size at least t.

14. c-Approximation for Max Opt.
An approximation produces a solution T if

15. Example of Maximum Coverage
Input: k=2 with sets:
S1 = { 1, 2, 3 }
S2 = { 2, 7, 8 }
S3 = { 1, 4, 5, 6, 7, 8}
S4 = { 4, 5, 6, 8 }
Output: Optimal Solution is S1, S3

16. MC Hardness MC is NP-Hard The decision version of MC is NP-complete:
Is it possible to find k sets with union size at least t?

17. Greedy Algorithm Repeat k times
Pick one set to cover the largest number
of uncovered elements

18. Greedy Algorithm Performance
Approximation Ratio for MC via greedy
For any fixed there is no poly. time ratio approximation to MC unless P=NP (Feige, J.ACM 1998).

19. Example of Maximum Coverage
Step 1: Select the largest set
S3 = { 1, 4, 5, 6, 7, 8}
Step2: Select the set S1 such that S1-S3 is the largest.

20. Our Input Model Each Set : 1) Membership query ?
2) Generating a random element in
3) The size

21. Greedy Algorithm k times: Pick one set with largest number
of uncovered elements
Largest |B-A| B A

22. Approximate Union
Union size
Union size in MC B A

23. Input Size of Maximum Coverage
S3 = { 1, 4, 5, 6, 7, 8} n=5, m=4
S4 = { 4, 5, 6, 8 }

24. Randomized Greedy Algorithm
Approximate |B-A| using random samples from B
Estimate the percentage to be in B-A B A

25. Randomized Greedy Algorithm
Approximate |B-A| using random samples from B
Estimate the percentage to be in B-A B A

26. Approximate |B-A| Let w be the random samples in B.
Let t be the items in B-A among w samples. B A

27. Randomized Greedy Algorithm
Accuracy B A

28. Approximate
Assume A B

29. Approximate
Proof

30. Randomized Greedy Algorithm
Repeat k times
Pick one set to cover approximate
largest number of uncovered elements

31. Classical Ratio Analysis
Let OPT be the optimal solution size.
Let be selected via greedy.
The first size
Assume the union of first t sets

32. Classical Ratio Analysis
The t+1-th set
Assume the union of first t+1 sets

33. Classical Ratio Analysis
Function is increasing.
Limit
Bound
Ratio

34. Ratio Analysis Let OPT be the optimal solution size.
Let be selected with
Then

35. Ratio Analysis Let OPT be the optimal solution size. Let be selected.
There is a

36. Ratio Analysis Let OPT be the optimal solution size.
Let be selected with

37. Monte Carlo Algorithm Put the circle into a square A
Generate n random points in A
Compute the number of points m in the circle
(m/n)*|A| is the approximate area size of the circle

38. Randomized algorithm
blind monkey

39. Randomized algorithm
blind monkey

40. Input length for Sorting Problem
Input: a list of numbers: 5, 3, 1, 7, 6
Input length n=5
Output: the sorted list 1, 3, 5, 6, 7

41. Time: number of steps Super linear: n(log n) (sorting) Sublinear:
log n (binary search at sorted list)

42. Input length for MC m: number of sets
n: the number of elements in the biggest set
The total input size can be mn

43. Partial Sublinear Time for MC
for some functions f(.) or g(.)

44. Classical Algorithm
Old algorithm time
Approximation ratio

45. New Partial Sublinear Time
Our algorithm time
Approximation ratio

46. Hoeffiding Bound
Therorem: Let be independent 0,1-random variables such that Then for
, and

47. Hoeffiding Bound
Therorem: Let be independent 0,1-random variables such that Then for
, and

48. Chernoff Bound
Therorem: Let be independent 0,1-random variables such that Then for
, and

49. Chernoff Bound
Therorem: Let be independent 0,1-random variables such that Then for
, and

50. Union Bound
Probability inequality:

51. Paper address
arXiv

52. Future work
More partial sublinear time algorithms.

53. Thanks
Question?