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Unformatted text preview: An introduction to
Approximation Algorithms Overview Introduction
Performance ratios
The vertex-cover problem
Traveling salesman problem
Set cover problem Spring 2003 Approximation Algorithmes 2 Introduction In computer science and operations research
, approximation
algorithms are algorithms used to find approximate solutions to optimization problems.
Approximation
algorithms
are
often
associated with NP-hard problems
Approximation algorithms are increasingly
being used for problems where exact
polynomial-time algorithms are known but
are too expensive due to the input size.
A typical example for an approximation
algorithm is the one vertex cover in graph. Spring 2003 Approximation Algorithmes 3 The essential techniques to design and
analyze approximation algorithms Combinatorial methods
Linear programming
Semi definite programming
Primal-dual and relaxation methods
Hardness of approximation
polynomial time. Spring 2003 Approximation Algorithmes 4 There are many important NP-Complete
problems There is no fast solution !
Suppose I need to solve an NP-hard
problem. What should I do?
A Theory says you're unlikely to find
a poly-time algorithm. Spring 2003 Approximation Algorithmes 5 Coping With NP-Hardness Brute-force algorithms. Heuristics. Develop clever enumeration strategies.
Guaranteed to find optimal solution.
No guarantees on running time. Develop intuitive algorithms.
Guaranteed to run in polynomial time.
No guarantees on quality of solution . Must sacrifice one of three desired
features. Solve problem to optimality.
Solve problem in poly-time.
Solve arbitrary instances of the problem. 6 Approximation algorithm
Approximation algorithm. Guaranteed to run in poly-time. Guaranteed to solve arbitrary instance of
the problem Guaranteed to find solution within ratio of true optimum.
Guaranteed to find "high quality" solution,
say within 1% of optimum.
Obstacle: need to prove a solution’s value is
close to optimum, without even knowing what
optimum value is! Approximation Algorithmes 7 Approximation algorithm
Challenge.: Need to prove a solution's value is close to optimum,
without even knowing what optimum value is!
Motivation: Many problems are NP-complete, so unlikely find efficient
algorithms
But we want the answer … If the input is small use backtrack. If input size is small, exponential algorithm is OK. Isolate important special case and find poly algorithms for them. Find the Near-Optimal solution in polynomial time.( either worst
case or average case)
So approximate algorithms: An algorithm returning a near-optimal solution is called
approximate algorithm
Spring 2003 Approximation Algorithmes 8 Some Characteristics of
Approximation Algorithms
Time-efficient (sometimes not as efficient as
heuristics) Don’t guarantee optimal solution Guarantee good solution within some factor of
the optimum Rigorous mathematical analysis to prove the
approximation guarantee Often use algorithms for related problems as
subroutines
- Depending on the problem , an optimal solution
may be defined as the problem of maximization
or minimization problem. Performance ratios We are going to find a Near-Optimal
solution for a given problem.(either
Worst Case or Average Case)
We assume two hypothesis : Spring 2003 Each potential solution has a positive
cost.
The problem may be either a
maximization or a minimization
problem on the cost.
Approximation Algorithmes 10 Performance ratios … If for any input of size n, the cost C
of the solution produced by the
algorithm is within a factor of ρ(n) of
the cost C* of an optimal solution:
Max ( C/C* , C*/C ) ≤ ρ(n)
We call this algorithm as an
ρ(n)-approximation
algorithm. Spring 2003 Approximation Algorithmes 11 Approximation algorithms for NPC
problems Performance ratios … Spring 2003 Approximation Algorithmes 13 Approximation Algorithm E.g.: If the total weigh of a MST(Minimum Spanning Tree) of
graph G is 20 A algorithm can produce some spanning trees, and they
are not MSTs, but their total weights are always smaller
than 25 What is the approximation ratio? 25/20 = 1.25 This algorithm is called?
1.25 approximation algorithm. Spring 2003 Approximation Algorithmes 14 Approximation scheme
Approximation scheme is an
approximation algorithm that takes as
input an instance of the problem and Є>0
such that for any fixed Є>0, the scheme is
(1+Є)-approximation algorithm. There are 2 schemes
1. polynomial time approximation scheme
2. Fully Polynomial-time approximation
scheme Spring 2003 Approximation Algorithmes 15 Approximation schemePolynomial-time approximation
scheme scheme is a
We say that an approximation polynomial time approximation scheme if for any
fixed Є>0 , the scheme runs in time polynomial in
the size n of its input instance
Polynomial-time approximation scheme is such
algorithm that runs in time polynomial in the size
of input. As the Є decreases the running time of the
algorithm can increase rapidly: For example it
might be O(n2/Є) If Є decreases by constant factor the running
time should not increase by more than a
constant factor. ie., the running time to be
polynomial in 1/Є as well as in n. Spring 2003 Approximation Algorithmes 16 Approximation scheme- Fully
Polynomial-time approximation
scheme We have Fully Polynomial-time
approximation scheme when its
running time is polynomial not only
in n (input size instance) but also in
1/
Є
For example it could be O((1/Є)3n2) Therefore any constant –factor decrease in
Є can be achieved with a corresponding
constant factor increase in the Running
Time. Spring 2003 Approximation Algorithmes 17 Some examples: Vertex cover problem.
Traveling salesman problem.
Set cover problem. Spring 2003 Approximation Algorithmes 18 The vertex-cover problem A vertex-cover of an undirected
graph G is a subset of its vertices
such that it includes at least one
end of each edge.
The problem is to find minimum size
of vertex-cover of the given graph.
This problem is an NP-Complete
problem. Spring 2003 Approximation Algorithmes 19 The vertex-cover problem
… Finding the optimal solution is hard
(its NP!) but finding a near-optimal
solution is easy.
There is an 2-approximation
algorithm: Spring 2003 It returns a vertex-cover not more
than twice of the size optimal
solution.
Approximation Algorithmes 20 The vertex-cover problem
…
APPROX-VERTEX-COVER(G) 1
2
3
4
5
6 C←Ø
E′ ← E[G]
while E′ ≠ Ø
do let (u, v) be an arbitrary edge of E′
C ← C U {u, v}
remove every edge in E′ incident on u
or v
7 return C Spring 2003 Approximation Algorithmes 21 The vertex-cover problem
… Near Optimal
size=6
Spring 2003 Approximation Algorithmes Optimal
Size=3
22 Demo Compare this cover to23the
one from the example Polynomial Time
•C
• E’ E O(n2)
• while E’ do
– let (u,v) be an arbitrary edge of E’
– C C {u,v}
O(n2)
– remove from E’ every edge
incident to either u or v
• return C O(1)
O(n) 24 Correctness
The set of vertices our algorithm returns
is clearly a vertex-cover, since we
iterate until every edge is covered. 25 Vertex-cover problem
and a 2-approximation
algorithm Vertex-cover problem
and a 2-approximation
algorithm
1
b
1
b c
2 3
d 4
a 5
e 6
f g
7 Are the red vertices a vertex-cover?
No. why?
Edges (5, 6), (3, 6) and (3, 7) are not covered by it Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 Are the red vertices a vertex-cover?
No. why?
Edge (3, 7) is not covered by it 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 Are the red vertices a vertex-cover?
Yes
What is the size?
4 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 Are the red vertices a vertex-cover?
Yes
What is the size?
7 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 Are the red vertices a vertex-cover?
Yes
What is the size?
5 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 Are the red vertices a vertex-cover?
Yes
What is the size?
3 7 Vertex-cover problem
and a 2-approximation
algorithm Vertex-cover problem Given a undirected graph, find a vertex
cover with minimum size. Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 A minimum vertex-cover 7 Vertex-cover problem
and a 2-approximation
algorithm Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 It is then a vertex cover
Size? 6 How far from optimal one? Max(6/3, 3/6) = 2 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 7 Vertex-cover problem
and a 2-approximation
algorithm
1 2 3 4 5 6 It is then a vertex cover
Size? 4 How far from optimal one? Max(4/3, 3/4) = 1.33 7 Vertex-cover problem
and a 2-approximation
algorithm
THEOREM:35.1 APPROX-VERTEX-COVER(G) is a 2approximation algorithm
AIM: When the size of minimum vertex-cover
is s The vertex-cover produced by APPROXVERTEX-COVER is at most 2s Proof :
APPROX-VERTEX-COVER(G) runs in Polynomial time.
Let ‘A’ denote the set of edges that were picked in
line -4 of Algorithm.
In order to cover the edges in A, an optimal cover
C*(U* denoted in proof) must include at least one
end point of each edge in A.(By example C*
includes b,e,d)
No two edges in A share an END POINTS,since once
an edge is picked , all the edges incident on it are
deleted from E‘.(By example –A ={ (b,c),(e,f), (d,g)}
)
Thus no two edges in A are covered by the same
vertex from C* and we have the lower bound.
Spring 2003 Approximation Algorithmes 41 NOTE:
To decide the size of the optimal cover we
utilize the lower bound on the optimal cover
.
By relating the size of the solution returned
to the lower bound , we obtain our
approximation ratio. Spring 2003 Approximation Algorithmes 42 Vertex-cover problem
and a 2-approximation
algorithm The set-Cover Generalization of vertex-cover
problem.
We have given (X,F) : Spring 2003 X : a finite set of elements.
F : family of subsets of X such that
every element of X belongs to at least
one subset in F.
Solution C : subset of F that Includes
all the members of X.
Approximation Algorithmes 44 The set-covering problem
X: a b d e F:
f1: a b f2: b
f3: c f4: d h
e f5: a c h {f1, f3, f4} is a subset of F
covering X
{f1, f2, f3, f4} is a subset of F
covering X
{f2, f3, f4, f5} is a subset of F
covering X
Here, {f1, f3, f4} is a minimum
cover set The set-covering problem The set-Cover … Minimal
Covering
set
size=3 Spring 2003 Approximation Algorithmes 48 The set-Cover …
GREEDY-SET-COVER(X,F)
1M←X
2C←Ø
3 while M ≠ Ø do
4 select an SЄF that maximizes |S חM|
5 M←M–S
6 C ← C U {S}
7 return C
Spring 2003 Approximation Algorithmes 49 The set-Cover …
1st chose 3rd chose 2nd chose
Greedy
Covering
set
size=4
4th chose Spring 2003 Approximation Algorithmes 50 The set-Cover … This greedy algorithm is polynomialtime ρ(n)-approximation algorithm ρ(n)=H(max{|S| : S Є F})
Hd =∑ d
i=1 The proof is beyond of scope of this
presentation. Spring 2003 Approximation Algorithmes 51 Traveling salesman problem Given an undirected weighted
Graph G we are to find a minimum
cost Hamiltonian cycle.
Satisfying triangle inequality or not
this problem is NP-Complete. Spring 2003 We can solve Hamiltonian path. Approximation Algorithmes 52 Traveling salesman problem Exact exponential solution: Branch and bound Spring 2003 Lower bound:
(sum of two lower degree of vertices)/2 Approximation Algorithmes 53 Traveling salesman problem
3
3 a
7 b
4 e c
6 2 Spring 2003 6 4 5 A: 2+3
B: 3+3
C: 4+4
D: 2+5
E: 3+6
= 35
Bound: 17,5 d Approximation Algorithmes 54 Traveling salesman problem
17.5 All Terms 17.5 18.5
ab ~ab 20.5 17.5 ab,ac => ~ac,ad ab 21 18.5 ~ab,~ac ab,ac 23
Ab,~ac,~ad 18
23 Ab,~ac,ad Ab,~ac,ad~ac,bc Spring 2003 21
Ab,~ac,ad,~ac,~bc Approximation Algorithmes 55 Traveling salesman problem Near Optimal solution Spring 2003 Faster
More easy to impliment. Approximation Algorithmes 56 Traveling salesman problem
with triangle inequality.
APPROX-TSP-TOUR(G, c)
1 select a vertex r Є V[G] to be root.
2
compute a MST for G from root r using
Prim Alg.
3 L=list of vertices in preorder walk of that MST.
4 return the Hamiltonian cycle H in the order L. Spring 2003 Approximation Algorithmes 57 Traveling salesman problem
with triangle inequality. MS
T roo
t Pre-Order
walk Spring 2003 Approximation Algorithmes Hamiltonia
n Cycle 58 Traveling salesman problem This is polynomial-time 2approximation algorithm. (Why?) Preorder Solutio
n
Spring 2003 Because: APPROX-TSP-TOUR is O(V2) C(MST) ≤ C(H*)
C(W)=2C(MST)
C(W)≤2C(H*)
C(H)≤C(W)
C(H)≤2C(H*) Optimal Approximation Algorithmes 59 Traveling salesman problem In
General Theorem:
If P ≠ NP, then for any constant ρ≥1,
there is no polynomial time ρapproximation algorithm. c(u,w) ={uEw ? 1 : ρ|V|+1}
ρ|V|+1+|V|-1>ρ|V| Selected
edge not in
E
Spring 2003 Rest of
edges Approximation Algorithmes 60 ...
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- Spring '19
- Computational complexity theory, NP-complete problems, Approximation algorithm, NP-complete, Algorithmes
