chapter1 (1).ppt

Design and Analysis of
Approximation Algorithms
Ding-Zhu Du

Text Books
• Ding-Zhu Du and Ker-I Ko, Design and
Analysis of Approximation Algorithms
(Lecture Notes).
Chapters 1-8.

Schedule
• Introduction
• Greedy Strategy
• Restriction
• Partition
• Guillotine Cut
• Relaxation
• Linear Programming
• Local Ratio
• Semi-definite Programming

Rules
• You may discuss each other on 5
homework assignments. But, do not copy
each other.
• Each homework is 10 points. 4 top scores
will be chosen.
• Midterm Exam (take-home) is 30 points.
• Final Exam (in class) is 30 points.
• Final grade is given based on the total
points (A ≥ 80; 80 > B ≥ 60 ; 60 > C ≥ 40).

Chapter 1 Introduction
• Computational Complexity (background)
• Approximation Performance Ratio
• Early results

Computability
• Deterministic Turing Machine
• Nondeterministic Turing Machine
• Church-Turing Thesis

Deterministic Turing Machine
(DTM)
Finite Control
tape
head

The tape has the left end but infinite to the
right. It is divided into cells. Each cell
contains a symbol in an alphabet Γ. There
exists a special symbol B which represents
the empty cell.
a l p h a B e

• The head scans at a cell on the tape and
can read, erase, and write a symbol on the
cell. In each move, the head can move to
the right cell or to the left cell (or stay in
the same cell).
a

• The finite control has finitely many states
which form a set Q. For each move, the
state is changed according to the
evaluation of a transition function
δ : Q x Γ → Q x Γ x {R, L}.

• δ(q, a) = (p, b, L) means that if the head
reads symbol a and the finite control is in
the state q, then the next state should be p,
the symbol a should be changed to b, and
the head moves one cell to the left.
p
a b
q

• δ(q, a) = (p, b, R) means that if the head
reads symbol a and the finite control is in
the state q, then the next state should be p,
the symbol a should be changed to b, and
the head moves one cell to the right.
q
p
a b

• There are some special states: an initial
state s and an final states h.
• Initially, the DTM is in the initial state and
the head scans the leftmost cell. The tape
holds an input string.
s

• When the DTM is in the final state, the DTM
stops. An input string x is accepted by the
DTM if the DTM reaches the final state h.
• Otherwise, the input string is rejected.
h
x

• The DTM can be represented by
M = (Q, Σ, Γ, δ, s)
where Σ is the alphabet of input symbols.
• The set of all strings accepted by a DTM
$M$ is denoted by L(M). We also say that
the language L(M) is accepted by M.

• The transition diagram} of a DTM is an
alternative way to represent the DTM.
• For M = (Q, Σ, Γ, δ, s), the transition diagram of
M is a symbol-labeled digraph G=(V, E)
satisfying the following:
V = Q (s = , h = )
E = { p q | δ(p, a) = (q, b, D)}.
a/b,D

M=(Q, Σ, Γ, δ, s) where Q = {s, p, q, h},
Σ = {0, 1}, Г = {0, 1, B}.
δ 0 1 B
s (p, 0, R) (s, 1, R) -
p (q, 0, R) (s, 1, R) -
q (q, 0, R) (q, 1, R) (h, B, R)
L(M) = (0+1)*00(0+1)*.
s p q h
1/1,R
0/0,R
1/1,R
0/0,R
0/0,R; 1/1,R
B/B,R

Nondeterministic Turing
Machine (NTM)
Finite Control
tape
head

• The finite control has finitely many states
which form a set Q. For each move, the
state is changed according to the
evaluation of a transition function
δ : Q x Γ → 2^{Q x Γ x {R, L}}.

Church-Turing Thesis
• Computability is Turing-Computability.

Multi-tape DTM
Input tape
(read only)
Storage
tapes
Output tape
(possibly, write only)

Time of TM
• TimeM (x) = # of moves that TM M takes on
input x.
• TimeM(x) < infinity iff x ε L(M).

Space
• SpaceM(x) = # of cell that M visits on the
work (storage) tapes during the
computation on input x.
• If M is a multitape DTM, then the work
tapes do not include the input tape and the
write-only output tape.

Time Bound
M is said to have a time bound t(n) if for
every x with |x| < n,
TimeM(x) < max {n+1, t(n)}

Complexity Class
• A language L has a (deterministic) time-
complexity t(n) if there is a multitape TM M
accepting L, with time bound t(n).
• DTIME(t(n)) = {L(M) | DTM M has a time
bound t(n)}
• NTIME(t(n)) = {L(M) | NTM M has a time
bound t(n)}

• P = U DTIME(n )
• NP = U NTIME(n )
c
c
C>0
C>0

Earlier Results on Approximations
• Vertex-Cover
• Traveling Salesman Problem
• Knapsack Problem

Performance Ratio
MAX.
for
)
(
)
(
sup
MIN
for
)
(
)
(
sup
input
Approx
inpu
Opt
r
input
Opt
input
Approx
r
input
input



Constant-Approximation
• c-approximation is a polynomial-time
approximation satisfying:
1 < approx(input)/opt(input) < c for MIN
or
1 < opt(input)/approx(input) < c for MAX

Vertex Cover
• Given a graph G=(V,E), find a minimum
subset C of vertices such that every edge
is incident to a vertex in C.

Vertex-Cover
• The vertex set of a maximal matching
gives 2-approximation, i.e.,
approx / opt < 2

Traveling Salesman
• Given n cities with a distance table, find a
minimum total-distance tour to visit each
city exactly once.

Traveling Salesman with triangular
inequality
• Traveling around a minimum spanning
tree is a 2-approximation.

Traveling Salesman with Triangular
Inequality
• Minimum spanning tree + minimum-length
perfect matching on odd vertices is 1.5-
approximation

Minimum perfect matching on odd vertices
has weight at most 0.5 opt.

Lower Bound
1+ε 1+ε
1+ε
1 1
1+ε 1+ε








n
n
n
n
as
2
3
)
1
)(
1
2
(
2
)
1
(
2



Traveling Salesman without
Triangular Inequality
Theorem For any constant c> 0, TSP has no c-
approximation unless NP=P.
Given a graph G=(V,E), define a distance table
on V as follows:






E
v
u
c
V
E
v
u
v
u
d
)
,
(
if
,
|
|
)
,
(
if
,
1
)
,
(

Contradition Argument
• Suppose c-approximation exists. Then we
have a polynomial-time algorithm to solve
Hamiltonian Cycle as follow:
C-approximation solution < cn
if and only if
G has a Hamiltonian cycle

Knapsack
}.
,
max{
Output
.
such that
Choose
.
Sort
.
any
for
Hence
.
any
for
Assume
}.
1
,
0
{
t.
s.
max
1
1
1
1
1
2
2
1
1
2
2
1
1
2
2
1
1
























k
k
i
i
G
k
i
i
k
i
i
n
n
i
i
i
n
n
n
n
c
c
c
s
S
s
k
s
c
s
c
s
c
i
opt
c
i
S
s
x
S
x
s
x
s
x
s
x
c
x
c
x
c




2-approximation
.
1
1
1
1
2 G
k
G
G
k
i
i
k
i
i
c
c
c
opt
c
c
opt
c









 


PTAS
• A problem has a PTAS (polynomial-time
approximation scheme) if for any ε > 0, it
has a (1+ε)-approximation.

Knapsack has PTAS
• Classify: for i < m, ci < a= cG,
for i > m+1, ci > a.
• Sort
• For
.
2
2
1
1
m
m
s
c
s
c
s
c


 
;
0
)
(
set
then
,
if
},
,
,
1
{






I
c
S
s
n
m
I
I
i
i




1

).
(
max
)
(
Output
.
)
(
set
and
such that
maximum
choose
then
,
If
1
1
I
c
I
c
c
c
I
c
s
S
s
m
k
S
s
I
output
k
i
i
I
i
i
I
i
i
k
i
i
I
i
i

















Proof. }.
in
1
|
}
,...,
1
{
{
*
Let opt
x
n
m
i
I i 

































1
)
(
.
1
)
(
)
(
)
(
Hence,
.
*)
(
*)
(
then
,
If
.
*)
(
then
,
If
1
*
1
*
1
output
output
output
output
k
I
i
i
m
i
i
I
i
i
m
i
i
I
c
opt
opt
I
c
a
I
c
opt
I
c
a
I
c
c
I
c
opt
s
S
s
opt
I
c
s
S
s

Time
.
)
(
time
gives
This
.
/
)
1
(
2
|
|
with
hose
consider t
only
need
we
Therefore,
.
2
)
/
)
1
(
2
(
then
,
)/
2(1
|
I
|
If
.
2
Note
)
/
1
(
/
)
1
(
2 








O
G
I
i
i
G
n
n
O
I
I
opt
c
a
c
c
opt














Fully PTAS
• A problem has a fully PTAS if for any ε>0,
it has (1+ε)-approximation running in time
poly(n,1/ε).

Fully FTAS for Knapsack
exists.
subset
such
no
if
)
,
(
}}.
,...,
1
{
,
|
{
min
and
,
such that
i}
...,
{1,
in
indeces
of
subset
a
is
)
,
(
)
,
(
)
,
(
)
,
(
nil
j
i
c
i
I
j
c
s
s
S
s
j
c
j
i
c
I
i
i
I
i
i
j
i
c
i
i
j
i
c
i
i
j
i
c
i
i

















Pseudo Polynomial-time Algorithm
for Knapsak
• Initially,
otherwise
,
if
{1}
,
0
if
)
,
1
(
,
1
for
1
1













nil
c
j
j
j
c
c
c
, ... c
j n
sum 

).
,
1
(
)
,
(
Set
.
but
,
)
,
1
(
2.
Case
);
,
1
(
)
,
(
Set
,
)
,
1
(
1.
Case
do
to
0
for
do
to
1
for
)
,
1
(
j
i
c
j
i
c
S
s
s
nil
c
j
i
c
j
i
c
j
i
c
nil
c
j
i
c
c
j
n
i
i
c
j
i
c
k
k
i
i
i
sum














 



}.
)
,
(
|
max{
Output
}.
{
)
,
1
(
)
,
(
set
else
)
,
1
(
)
,
(
set
then
If
.
)
,
1
(
and
,
,
)
,
1
(
4.
Case
}.
{
)
,
1
(
)
,
(
Set
.
)
,
1
(
and
,
,
)
,
1
(
3.
Case
)
,
1
(
)
,
1
(
)
,
1
(
)
,
1
(
nil
j
i
c
j
opt
i
c
j
i
c
j
i
c
j
i
c
j
i
c
s
s
s
nil
j
i
c
S
s
s
nil
c
j
i
c
i
c
j
i
c
j
i
c
nil
j
i
c
S
s
s
nil
c
j
i
c
i
j
i
c
k
k
c
j
i
c
k
k
i
c
j
i
c
k
k
i
i
i
c
j
i
c
k
k
i
i
i
i
i












































Time
• outside loop: O(n)
• Inside loop: O(nM) where M=max ci
• Core: O(n log (MS))
• Total O(n M log (MS))
• Since input size is O(n log (MS)), this is a
pseudo-polynomial-time due to M=2
3
log M

h
n
h
n
h
h
h
h
i
n
n
n
n
i
i
c
h
opt
x
c
x
c
x
c
c
S
h
n
h
n
O
x
x
S
x
s
x
s
x
s
x
c
x
c
x
c
M
h
n
c
c
)
1
1
(
show
We
.
Let
)).
)
1
(
log(
)
1
(
(
in time
computed
be
can
solution
optimal
Its
}.
1
,
0
{
s.t.
'
'
'
max
Consider
)
1
(
'
Set
2
2
1
1
4
2
2
1
1
2
2
1
1





















 





)
1
1
1
(
1
)
1
(
)
*
)
1
)
1
(
(
*
)
1
)
1
(
((
)
1
(
)
*
'
*
'
(
)
1
(
)
'
'
(
*
*
*
Suppose
1
1
1
1
1
1
1
1
.
2
2
1
1





























h
opt
h
M
opt
h
n
M
x
M
h
n
c
x
M
h
n
c
h
n
M
x
c
x
c
h
n
M
x
c
x
c
x
c
x
c
c
x
c
x
c
x
c
opt
n
n
n
n
n
h
n
h
n
h
n
h
h
n
n






Lecture 3 Complexity of
Approximation
• L-reduction
• Two subclass of PTAS
• Set-cover ((ln n)-approximation)
• Independent set (n -approximation)
c

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