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' award13.30-14.30 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 Pause14.45-15.45 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

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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.) ≤

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.

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Not obvious that it will work….

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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.

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ILP formulation

Find (xv)v ∈ V minimizing wv xv so that

• xv ∈ Z

• 0 ≤ xv ≤1

• For all (u,v) ∈ E, xu + xv ≥ 1.

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LP relaxation

Find (xv)v ∈ V minimizing wv xv so that

• xv ∈ R

• 0 ≤ xv ≤ 1

• For all (u,v) ∈ E, xu + xv ≥ 1.

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Relaxation and Rounding

• Solve LP relaxation.

• Round the optimal solution x* to an integer solution x: xv = 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 xu and xv is set to 1.

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Quality of solution found

• Let z* = wv xv* be cost of optimal LP solution.

• wv xv ≤ 2 wv xv*, as we only round up if xv* is bigger than ½.

• Since z* ≤ cost of optimal ILP solution, our algorithm has approximation ratio 2.

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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.

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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) ≤

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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.

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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.

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Analysis

• Let Yi 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[Yi =1] = 1 if the i’th clause contains some variable and its negation.

• Pr[Yi = 1] = 1 – (1/2)3 = 7/8 if the i’th clause does not include a variable and its negation.

• E[Yi] = Pr[Yi = 1] ≥7/8.

• E[Y] = E[ Yi] = E[Yi]≥(7/8) m 14

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).

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Min set cover

•Given set system S1, S2, …, Sm ⊆ X, find smallest possible subsystem covering X.

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Greedy algorithm for min set cover

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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 Hs where 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.

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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.

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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.

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Knapsack problem

• Given n items with weights w1,...,wn , values v1,...,vn and weight limit W,

• fit items within weight limit maximizing total value.

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FPTAS for Knapsack

Exercise: We have a pseudo-polynomial time algorithm for Knapsack in time O(n2V), where V is largest value.We use this in step 4. 22

More Inapproximability

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

Problem RatioVertex Cover 1.36Set Cover c⋅ln(n), for some c>0TSP Any ratioMetric TSP 220/219

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