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台大資工系 呂學一 http://www.csie.ntu.edu.tw/~hil/algo/ 1

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2 計算 難題 人生

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Input: A graph G Output: A smallest vertex subset of G that covers all edges of G. 3

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Clever ways to solve computational problems 5

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Being clever in solving a computation problem could make a big difference. 6

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It makes a big difference 8

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9 http://www.grand-illusions.com/simulator/montysim.htm

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Coffee and Milk 11

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The Factorization Problem Input: a number N Output: “yes” if N is a prime number; A factorization of N if N is not a prime number. For example, N = 323264989793317. Output = 18672511 * 17312347. 17

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Is there an efficient recipe for the Factorization Problem? 18

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The security of many encryption schemes is based upon the assumption that the factorization problem is difficult. 19

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20 Rivest ShamirAdleman

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21 Challenge Number Prize ($US) Challenge Number Prize ($US) RSA-576$10,000RSA-896$75,000 RSA-640$20,000RSA-1024$100,000 RSA-704$30,000RSA-1536$150,000 RSA-768$50,000RSA-2048$200,000

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18819881292060796383869723946165043980716 35633794138270076335642298885971523466548 531906060650474304531738801130339671619969 23212057340318795506569962130516875930765 0257059 22

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3980750864240649373971255005503864911990 6436234252670840638518957594638895726176 8583317 472772146107435302536223071973048224632914 6953020971164598521711305207112563635903975 27 At the same time, Adi Shamir gave two talks at NTU (Dec. 4, 2003 ) 23

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F. Bahr, M. Boehm, J. Franke, T. Kleinjung 1634733645809253848443133883865090859841783670033 092312181110852389333100104508151212118167511579 190087128166482211312685157393541397547189678996851 5493666638539088027103802104498957191261465571 Efforts: 30 2.2GHz-Opteron-CPU years over five months of calendar time. 25

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251959084756578934940271832400483985714292821262040 32027777137836043662020707595556264018525880784406 9182906412495150821892985591491761845028084891200728 449926873928072877767359714183472702618963750149718 246911650776133798590957000973304597488084284017974 29100642458691817195118746121515172654632282216869987 549182422433637259085141865462043576798423387184774 4479207399342365848238242811981638150106748104516603 7730605620161967625613384414360383390441495263443219 0114657544454178424020924616515723350778707749817125 772467962926386356373289912154831438167899885040445 364023527381951378636564391212010397122822120720357 26

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www.rsasecurity.com/rsalabs/challenges/factoring/ 27

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Input: A graph with edge lengths Output: A shortest tour visiting each node of the input graph exactly once. 31

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Clay Mathematics Institute (Cambridge, MA, USA) offered US$1,000,000 for each of seven open problems on May 24, 2000 at Paris. | Birch and Swinnerton-Dyer Conjecture | Hodge Conjecture | Navier-Stokes Equations | P vs NP | Poincare Conjecture | Riemann Hypothesis | Yang- Mills Theory |Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincare Conjecture Riemann Hypothesis Yang- Mills Theory 33

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Input: A graph G Output: A smallest vertex subset of G that covers all edges of G. 39

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Clay Mathematics Institute (Cambridge, MA, USA) offered US$1,000,000 for each of seven open problems on May 24, 2000 at Paris. | Birch and Swinnerton-Dyer Conjecture | Hodge Conjecture | Navier-Stokes Equations | P vs NP | Poincare Conjecture | Riemann Hypothesis | Yang- Mills Theory |Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincare Conjecture Riemann Hypothesis Yang- Mills Theory 41

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Being clever in solving computational problems could make a big difference. 42

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Approximation Algorithms ( 近似演算法 ) 44

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放下對完美的堅持, 往往就可以找到新的出路。 知所進退, 則近道矣 ! 45

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康熙大學士兼禮部尚書張英 一紙書來只為牆， 讓他三尺又何妨。 長城萬里今猶在， 不見當年秦始皇。 47

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路燈問題 (Vertex Cover) 48

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Input: A graph G Output: A smallest vertex subset of G that covers all edges of G. 49

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Input: A graph G Output: A near smallest vertex subset of G that covers all edges of G. 52

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Initially, let S be an empty set. Repeat until G has no edges: Arbitrarily choose an edge (u, v) of G. Insert u and v into S. Delete all edges of G incident to u or v. Output S. 53

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Q1: Is the output vertex set indeed a vertex cover of the input graph? Q2: Does the algorithm run in polynomial time? Q3: Is the quality of the output solution close to optimal? 55

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Yes. 56

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It is not difficult to see that this algorithm runs in linear time. O(n + m) time, where n is the number of vertices and m is the number of edges in G. 57

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The output vertex cover has size at most 2 times that of any optimal vertex cover. 58

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Criterion 1: feasibility Always output a feasible solution. Criterion 2: tractability Always runs in polynomial time. Criterion 3: quality The solution’s quality is always provably not too far from that of an optimal solution. 60

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That 2-approximation was known for 30 years. It remains the best known approximation algorithm for the vertex cover problem! Finding a 1.166-approximation is known to be NP- hard. Even a 1.9-approximation would be a significant breakthrough. 向公園路燈管理局致敬 62

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掃黑的藝術 (maximum cut) 63

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Input: A graph G Output: A partition of G’s nodes into A and B that maximizes the number of edges between A and B. 64

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Repeat the following randomized subroutine for, say, 100 times, and then output the best cut among them. For each node v of G, put v into A with probability ½. 66

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Any partition is a feasible solution. 67

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Throwing a fair coin is “easy” to simulate by computers. As a matter of fact, the existence of pseudo-random generator implies NP≠P. 68

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One can prove that this simple algorithm is a 2- approximation with very high probability (something like 1-1/2 100 ). 69

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For about 20 years, the above 2-approximation was the best known result for maximum cut. 70

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Goemans and Williamson, ACM STOC 1994 71

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1/0.878-approximation for MAXCUT Initiate a series of research in Operations Research, Scientific Computing, and Approximation Algorithms. 72

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73 Integer Linear Program Integral solution Linear Program Fractional solution relaxation Rounding Approximation Solver

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74 Quadratic Program Scalar solution Semi-definite Program Vector solution relaxation Rounding Approximation Solver

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Being clever in solving problem can make a big difference Most of the real life problems cannot be solved perfectly in reasonable amount of time. Approximate solutions might be a nice alternative. 75

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God, grant us the serenity to accept the things we cannot change, the courage to change the things we can, and the wisdom to know the difference. 76

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