1. Computer Science Day 2013, May 31 12.15-13.00Distinguished Lecture: Andy Yao, Tsinghua University 13.15-13.30 Welcome and the 'Lecturer of the Year' award 13.30-14.30 Data-Intensive Systems (Ira Assent) Computer Graphics and Image Processing (Toshiya Hachisuka) Bioinformatics (Søren Besenbacher) Use, Design and Innovation (Morten Kyng) Ubiquitous Computing and Interaction (Kaj Grønbæk) 14.30-14.45 Pause 14.45-15.45 Mathematical Computer Science (Peter Bro Miltersen) Cryptography and Security (Claudio Orlandi) Semantics and Logic (Lars Birkedal) Programming Languages (Anders Møller) Algorithms and Data Structures (Lars Arge) 15.45-Regnecentralen’s 1 year birthday party Large Auditorium, Incuba Science Park – Katrinebjerg http://cs.au.dk/csd2013

2. 2 Approximation algorithms Given minimization problem (e.g. min vertex cover, TSP,…) and an efficient algorithm that always returns some feasible solution. The algorithm is said to have approximation ratio  if for all instances, cost(sol. found)/cost(optimal sol.) ≤ 

3. General design/analysis trick Our approximation algorithms often works by constructing some relaxation providing a lower bound and turning the relaxed solution into a feasible solution without increasing the cost too much. The LP relaxation of the ILP formulation of the problem is a natural choice. We may then round the optimal LP solution. 3

4. Not obvious that it will work…. 4

5. Min weight vertex cover Given an undirected graph G=(V,E) with non- negative weights w(v), find the minimum weight subset C ⊆ V that covers E. Min vertex cover is the case of w(v)=1 for all v. 5

6. ILP formulation Find (x v ) v ∈ V minimizing  w v x v so that x v ∈ Z 0 ≤ x v ≤1 For all (u,v) ∈ E, x u + x v ≥ 1. 6

7. LP relaxation Find (x v ) v ∈ V minimizing  w v x v so that x v ∈ R 0 ≤ x v ≤ 1 For all (u,v) ∈ E, x u + x v ≥ 1. 7

8. Relaxation and Rounding Solve LP relaxation. Round the optimal solution x* to an integer solution x: x v = 1 iff x*≥½. The rounded solution is a cover: If (u,v) ∈ E, then x* u + x*≥1 and hence at least one of x u and x v is set to 1. 8

9. Quality of solution found Let z* =  w v x v * be cost of optimal LP solution.  w v x v ≤ 2  w v x v *, as we only round up if x v * is bigger than ½. Since z* ≤ cost of optimal ILP solution, our algorithm has approximation ratio 2. 9

10. Relaxation and Rounding Relaxation and rounding is a very powerful scheme for getting approximate solutions to many NP-hard optimization problems. In addition to often giving non-trivial approximation ratios, it is known to be a very good heuristic, especially the randomized rounding version. Randomized rounding of x ∈ [0,1]: Round to 1 with probability x and 0 with probability 1-x. 10

11. Approximation algorithms Given maximization problem (e.g. MAXSAT, MAXCUT) and an efficient algorithm that always returns some feasible solution. The algorithm is said to have approximation ratio  if for all instances, cost(optimal sol.)/cost(sol. found) ≤  11

12. MAX-E3-SAT Given Boolean formula in CNF form with exactly three distinct literals per clause find an assignment satisfying as many clauses as possible. 12

13. Randomized algorithm Flip a fair coin for each variable. Assign the truth value of the variable according to the coin toss. Claim: The expected number of clauses satisfied is at least 7/8 m where m is the total number of clauses. We say that the algorithm has an expected approximation ratio of 8/7. 13

14. Analysis Let Y i be a random variable which is 1 if the i’th clause gets satisfied and 0 if not. Let Y be the total number of clauses satisfied. Pr[Y i =1] = 1 if the i’th clause contains some variable and its negation. Pr[Y i = 1] = 1 – (1/2) 3 = 7/8 if the i’th clause does not include a variable and its negation. E[Y i ] = Pr[Y i = 1] ≥7/8. E[Y] = E[  Y i ] =  E[Y i ]≥(7/8) m 14

15. Remarks It is possible to derandomize the algorithm, achieving a deterministic approximation algorithm with approximation ratio 8/7. Approximation ratio 8/7 -  is not possible for any constant  > 0 unless P=NP. Very hard to show (shown in 1997). 15

16. Min set cover Given set system S 1, S 2, …, S m ⊆ X, find smallest possible subsystem covering X. 16

17. Greedy algorithm for min set cover 17

18. Approximation Ratio Greedy-Set-Cover does not give a constant approximation ratio Even true for Greedy-Vertex-Cover! Quick analysis: Approximation ratio ln(n) Refined analysis: Approximation ratio H s where s is the size of the largest set and H s = 1/1 + 1/2 + 1/3 +.. 1/s is the s’th harmonic number. s may be small on concrete instances. H 3 = 11/6 < 2. 18

19. Approximation Schemes Some optimization problems can be approximated very well, with approximation ratio 1+ε for any ε>0. An approximation scheme takes an additional input, ε>0, and outputs a solution within 1+ε of optimal. 19

20. PTAS and FPTAS An approximation scheme is a Polynomial Time Approximation Scheme, if for every fixed ε>0, the algorithm runs in polynomial time in the input length n. An approximation scheme is a Fully Polynomial Time Approximation Scheme, if the algorithm runs in time polynomial in n and in 1/ε. 20

21. Knapsack problem Given n items with weights w 1,...,w n, values v 1,...,v n and weight limit W, fit items within weight limit maximizing total value. 21

22. FPTAS for Knapsack Exercise: We have a pseudo-polynomial time algorithm for Knapsack in time O(n 2 V), where V is largest value. We use this in step 4. 22

23. More Inapproximability Unless P=NP, we can not have approximation algorithms guaranteeing the following approximation ratios: ProblemRatio Vertex Cover1.36 Set Cover c ⋅ ln(n), for some c>0 TSPAny ratio Metric TSP220/219 23