Computer Science Day 2013, May Distinguished Lecture: Andy Yao, Tsinghua University Welcome and the 'Lecturer of the Year' award 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) Pause 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) Regnecentralen’s 1 year birthday party Large Auditorium, Incuba Science Park – Katrinebjerg

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.) ≤ 

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

Not obvious that it will work…. 4

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. 5

ILP formulation Find (x v ) v ∈ V minimizing  w v x v so that x v ∈ Z 0 ≤ x v ≤1 For all (u,v) ∈ E, x u + x v ≥ 1. 6

LP relaxation Find (x v ) v ∈ V minimizing  w v x v so that x v ∈ R 0 ≤ x v ≤ 1 For all (u,v) ∈ E, x u + x v ≥ 1. 7

Relaxation and Rounding Solve LP relaxation. Round the optimal solution x* to an integer solution x: x v = 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 x u and x v is set to 1. 8

Quality of solution found Let z* =  w v x v * be cost of optimal LP solution.  w v x v ≤ 2  w v x v *, as we only round up if x v * is bigger than ½. Since z* ≤ cost of optimal ILP solution, our algorithm has approximation ratio 2. 9

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. 10

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) ≤  11

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

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

Analysis Let Y i 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[Y i =1] = 1 if the i’th clause contains some variable and its negation. Pr[Y i = 1] = 1 – (1/2) 3 = 7/8 if the i’th clause does not include a variable and its negation. E[Y i ] = Pr[Y i = 1] ≥7/8. E[Y] = E[  Y i ] =  E[Y i ]≥(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). 15

Min set cover Given set system S 1, S 2, …, S m ⊆ X, find smallest possible subsystem covering X. 16

Greedy algorithm for min set cover 17

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 H s where s is the size of the largest set and H s = 1/1 + 1/2 + 1/ /s is the s’th harmonic number. s may be small on concrete instances. H 3 = 11/6 < 2. 18

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. 19

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. 20

Knapsack problem Given n items with weights w 1,...,w n, values v 1,...,v n and weight limit W, fit items within weight limit maximizing total value. 21

FPTAS for Knapsack Exercise: We have a pseudo-polynomial time algorithm for Knapsack in time O(n 2 V), 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: ProblemRatio Vertex Cover1.36 Set Cover c ⋅ ln(n), for some c>0 TSPAny ratio Metric TSP220/219 23