Computer Science Day 2013, May 31
12.15-13.00
Distinguished Lecture: Andy Yao, Tsinghua University
13.15-13.30
13.30-14.30
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)
14.30-14.45
14.45-15.45
15.45-
Regnecentralen’s 1 year birthday party
Large Auditorium, Incuba Science Park – Katrinebjerg
http://cs.au.dk/csd2013
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.) ≤ 
2
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 nonnegative 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 (xv)v ∈ V minimizing  wv xv so that
• xv ∈ Z
• 0 ≤ xv ≤1
• For all (u,v) ∈ E,
xu + xv ≥ 1.
6
LP relaxation
Find (xv)v ∈ V minimizing  wv xv so that
• xv ∈ R
• 0 ≤ xv ≤ 1
• For all (u,v) ∈ E,
xu + xv ≥ 1.
7
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.
8
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.
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 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).
15
Min set cover
•Given set system S1, S2, …, Sm ⊆ 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 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.
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 w1,...,wn ,
values v1,...,vn 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(n2V), where
V is largest value.
22
We use this in step 4.
More Inapproximability
Unless P=NP, we can not have
approximation algorithms guaranteeing
the following approximation ratios:
Problem
Vertex Cover
Ratio
1.36
Set Cover
TSP
Metric TSP
c⋅ln(n), for some c>0
Any ratio
220/219
23