Approximation algorithm Design a case study of MRCT

Approximation algorithm Design a case study of MRCT 樹德科技大學 資訊 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, 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？

Algorithm research and NP-Complete Theorem

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

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， Knuth NPC的問題已經列不勝列了。 1974 Turing Award • 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。 Johnson

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

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

Approximation algorithms

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

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(n 1/k). – FPTAS if (1/k) not in the exponent,

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

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 2 10

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

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

Optimum communication spanning tree Problems

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 d. T(i, j)

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

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

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

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)

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

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

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

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) r – Y is a 2 -approximation >=n/2 v

• 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?

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

• 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

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

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)

• 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

• 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

Thanks Q&A