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Approximation algorithm Design a case study of MRCT

Approximation algorithm Design a case study of MRCT. 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu). 1988 – before studying algorithms. 2000 – after studying algorithms. Ron Rivest. Leonard Adleman. Adi Shamir. RSA. Last year, after Prof. Chang went to NSYSU for a speech,

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Approximation algorithm Design a case study of MRCT

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  1. Approximation algorithm Design a case study of MRCT 樹德科技大學 資訊工程系 吳邦一 (B. Y. Wu)

  2. 1988 – before studying algorithms 2000 – after studying algorithms

  3. Ron Rivest Leonard Adleman Adi Shamir RSA Last year, after Prof. Chang went to NSYSU for a speech, A student asked me for a question: 為何做演算法的人皆白髮而做security的易禿頭？

  4. Algorithm research and NP-Complete Theorem

  5. NP-hard: the barrier • Since the results of Cook (1971) , Levin(1973) & Karp (1972), many important problems have been shown to be NP-hard. Karp 1985 Turing Award Cook 1982 Turing Award Levin

  6. The NPC Theorem • The name “NP-Complete” is due to Knuth（高德納） • Garey and Johnson在1979年所著的Computers and Intractability: A Guide to the Theory of NP-Completeness書中蒐列了數以百計的重要NPC問題，到今天，NPC的問題已經列不勝列了。 • According to Wikipedia（維基百科）, 在2002 年的一項調查中，一百位研究者裡面有61位相信NP不等於P，9位相信NP=P，22位不確定，而有8位研究者認為此問題在目前的假設基礎下是無法證明的。 Knuth 1974 Turing Award Johnson

  7. For an NP-Complete or NP-hard problem, it is not expected to find an efficient algorithm. Or maybe you need the 1,000,000 USD award • In 70s, the life-cycle of a problem • Defined • NP-hard • Heuristic or for special data

  8. 艱困而逐漸褪色 • Life finds the ways • Approximation • Online • Distributed • Mobile • New models • Quantum computing • Bio-computing

  9. Approximation algorithms

  10. Approximation algorithms • For optimization (min/max) problem • Heuristic vs. approximation algorithms • Ensuring the worst-case quality • The error • Relative and Absolute • A function k of input size n. A k-approximation: • minimization: sol/opt<=k;maximization: opt/sol<=k • The ratio is always >1

  11. 最高境界: Polynomial time approximation scheme • Some algorithms are of fixed ratio • Approximation scheme: allow us to make trade-off between time and quality • The more time, the better quality • PTAS: for any fixed k>0, it finds a (1+k)-approximation in polynomial time. • Usually (1/k) appears in the time complexity, e.q. O(n/k), O(n1/k). • FPTAS if (1/k) not in the exponent,

  12. The first PTAS (Not sure) • In Ronald L. Graham’s 1969 paper for scheduling problem (Contribution also due to Knuth and another)

  13. An example -- TSP • Starting at a node, find a tour of min distance traveling all nodes and back to the starting node. 6 8 2 15 10 5 3 10 2

  14. The doubling tree algorithm • Find a minimum spanning tree • Output the Euler tour in the doubling tree of MST 6 6 8 8 2 2 15 15 10 10 5 5 3 3 10 10 2 2

  15. The error ratio • MST<=TSP • MST is the minimum cost of any spanning tree. • A tour must contain a spanning tree since it is connected. • It is a 2-approximation

  16. Optimum communication spanning tree Problems

  17. OCT: definition • Input: • an undirected graph with nonnegative edge lengths • a nonnegative requirement for each pair of vertices • Output: • a spanning tree minimizing the total communication cost summed over all pairs of vertices, in which the cost of a vertex pair is the distance multiplied by their requirement, that is, we want to minimize Σ λi,j dT(i,j)

  18. First studied by T.C. Hu 1974 SICOMP First approximation appeared in Wong 1980

  19. A way to a PTAS A case study of the MRCT problem Optimum Communication Spanning Trees

  20. Minimum routing cost spanning trees • A spanning tree with minimum all-to-all distance • NP-hard in the strong sense • Tree with short edges may have large routing cost

  21. Approximation– comparing with a trivial lower bound • A lower bound • d(T,u,v)>=d(G,u,v) (樹上距離<=原圖最短路徑) • Opt>=Σd(G,u,v) • The median of G: a node m min Σvd(G,m,v) • Since min<=mean, Σvd(G,m,v)<=(1/n) Σd(G,u,v)

  22. Y : a shortest path tree rooted at m • d(Y,i,j)<=d(Y,i,m)+d(Y,m,j) • Σd(G,u,v)<=2nΣvd(G,m,v)<=2*OPT • A shortest path tree rooted at the median is a 2-approximation of the MRCT. m j j i

  23. To find an approx. • A lower bound of the optimum • An algorithm • Analyze the worst-case ratio

  24. Solution decomposition • 假設T是一個OPT, 我們將T做一些處理, 得到另一個解Y, 使得 • Y的cost不至於與T相差太多 • Y屬於某一種特殊類別的解, 而這類別中的最佳解是可以在polynomial time 求得的 • 注意: 我們無法得知Y, Y並不會出現在algorithm中, 只在分析中扮演一個中繼的角色

  25. Metric MRCT • For easy to understand, we consider only the metric case • The input is a metric graph: a complete graph with edge length satisfying the triangle inequality

  26. Metric MRCT • 假設T是OPT, r是T的centroid • 一個tree的centroid是去掉它的話, 剩下的subtree均不會超過一半的node • 在計算cost時, d(T,r,v)至少被計算n次 • opt>=nΣvd(T,r,v) • Let Y: the star centered at r • C(Y)= 2(n-1)Σvd(Y,r,v) • Y is a 2-approximation r >=n/2 v

  27. 利用solution decomposition証得 • 存在一個star是2-approximation • 以窮舉法嘗試所有的star (n個)並取出最好的, 必然是一個2-approximatin • Can we do better?

  28. δ-separator • Separator of a tree: • Centroid is a ½ separator • How the 2-approx. algorithm works? • Guess (try all possible) the separator • Connect the others greedily • Distance increases only for nodes in the same branch -- we don’t pay too much

  29. To get better result, we try to generalize the centroid to general δ-separator • Indeed, when δ↘, the error↘ • But it costs too much to obtain the exact δ-separator for δ<1/2. • For example, a 1/3-separator may have n/3 nodes 1/3-separator n/3 n/3

  30. 屬下犧牲了上司也該犧牲 • We don’t need a perfect separator • Only some critical nodes are necessary • Leaves of the separator (確保下屬有個好的依歸) • Branch nodes of the separator(確保結構) δ-separator

  31. To a k-Star • k-star: a tree with at most k internal nodes • Need some other work to show the ratio(通常這樣的話代表了背後有慘不忍睹的內容)

  32. Solution decomposition • 從一個OPT開始，我們設法將他改造成一個k-star，並證明此k-star是一個不錯的approximation • 設計一個演算法可以求得最好的k-star，既然他是最好，當然不比那個改造的差 • 精緻的分析是重要的，「好，要說的出口」

  33. 3-star =>1.5-approximation • k-star => (k+3)/(k+1)-approxiamtion • The best k-star for fixed k can be found in polynomial time • We have a PTAS

  34. 一些經驗之談 • Evolutionary tree reconstruction • 給一個n個物種的距離矩陣，找一個tree以此n個物種為leaves, 使得兩兩物種之間在樹上的距離>=給定的距離且最小化距離總合 • 這個問題比較難，因為樹的中間節點是可以任意給的 • Steiner tree vs. Spanning tree

  35. 花了不少時間study Steiner tree • 先做做Spanning 的case • MRCT • 找到separator的方法 • (15/8)-approx => 1.577 =>1.5 =>4/3+ • 兩種extension • 這個方法在general graph上不可能做到比4/3+更好了

  36. 困難點在於受限於shortest path tree • 如果是metric graph就有可能做到更好 • 但是metric graph的case還不知是不是NP-hard • 對於證明NPC實在是很厭煩了 • 把Garey & Johnson的書翻了又翻 • 遠在天邊 近在眼前 • 把general case transform 到metric case • 不只解決NP-hard的疑問, 證明了metric上的approx. 可以用在general case 上

  37. 找到k-star的方法 • 意外的插曲 • 研究是很競爭的 • 提心吊膽 難以入眠 • 謎底揭曉的那一刻 • 1997年，我做到了兩年來作夢都夢不到的事 • 更多的extension

  38. 做研究是在千百次失敗中期待一次成功 • 做行政是在千百次成功中等待一次失敗 • 研究之路很迷人，如果有人結伴而行則更加美好（當學生很幸福啊！） • 李老師告訴我說：沒有計畫，只有方向 • 研究如此，人生何嘗不是

  39. Thanks Q&A

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