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Computer Science Day 2013, May 31

12.15-13.00 Distinguished 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 )    

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Computer Science Day 2013, May 31

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  1. 12.15-13.00 DistinguishedLecture: Andy Yao, TsinghuaUniversity 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 (ToshiyaHachisuka) 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 yearbirthday party Computer Science Day 2013, May 31 Large Auditorium, Incuba Science Park – Katrinebjerg http://cs.au.dk/csd2013

  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 undirectedgraphG=(V,E) with non-negative weightsw(v) , find the minimum weightsubsetC⊆Vthat covers E. • Min vertex cover is the case of w(v)=1 for all v. 5

  6. ILP formulation Find (xv)v ∈Vminimizingwvxv so that • xv∈Z • 0 ≤xv≤1 • For all (u,v) ∈ E, xu + xv≥ 1. 6

  7. LP relaxation Find (xv)v ∈Vminimizingwvxv so that • xv∈R • 0 ≤xv≤ 1 • For all (u,v) ∈E, xu + xv≥ 1. 7

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

  9. Quality of solution found • Let z* = wvxv*becost of optimal LP solution. • wvxv≤ 2 wvxv*, as weonlyround up ifxv* is biggerthan ½. • Since z* ≤cost of optimal ILP solution, ouralgorithm has approximation ratio 2. 9

  10. Relaxation and Rounding • Relaxation and rounding is a verypowerfulscheme for gettingapproximate solutions to many NP-hardoptimization problems. • In addition to oftengiving non-trivial approximation ratios, it is known to be a verygoodheuristic, especially the randomizedrounding version. • Randomizedrounding of x∈ [0,1]: Round to 1 with probabilityx and 0 with probability 1-x. 10

  11. Approximation algorithms • Given maximization problem (e.g. MAXSAT, MAXCUT) and an efficientalgorithmthatalwaysreturnssomefeasible 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 Yibe a random variable which is 1 if thei’thclausegetssatisfied and 0 if not. Let Ybe the total number of clausessatisfied. • Pr[Yi =1] = 1 if the i’thclausecontainssome variable and its negation. • Pr[Yi= 1] = 1 – (1/2)3 = 7/8 if the i’thclausedoes not include a variable and its negation. • E[Yi] = Pr[Yi = 1] ≥7/8. • E[Y] = E[ Yi] =  E[Yi]≥(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 S1, S2, …, Sm⊆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 Hswhere s is the size of the largest set and Hs = 1/1 + 1/2 + 1/3 + .. 1/s is the s’th harmonic number. s may be small on concrete instances. H3 = 11/6 < 2. 18

  19. Approximation Schemes Someoptimization problems canbeapproximatedverywell, with approximation ratio 1+εfor anyε>0. An approximationschemetakes an additional input, ε>0, and outputs a solution within1+εof optimal. 19

  20. PTAS and FPTAS An approximationscheme is a Polynomial Time ApproximationScheme, if for everyfixedε>0, the algorithm runs in polynomial time in the input length n. An approximationscheme is a FullyPolynomial Time ApproximationScheme, if the algorithm runs in time polynomial in n and in 1/ε. 20

  21. Knapsack problem Given n items with weights w1,...,wn , values v1,...,vn 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(n2V), where V is largestvalue. Weusethis in step 4. 22

  23. More Inapproximability Unless P=NP, we can not have approximation algorithms guaranteeing the following approximation ratios: 23

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