Algorithms and Networks 20142015 Hans L Bodlaender Johan M M van Rooij Capproximation Optimization problem output has a value that we want to maximize or minimize An algorithm A is an ID: 395563
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Slide1
Approximation Through Scaling
Algorithms
and Networks 2014/2015
Hans L.
Bodlaender
Johan M. M. van RooijSlide2
C-approximation
Optimization problem: output has a value that we want to
maximize
or
minimize
An algorithm A is an
C-approximation algorithm
, if for all inputs X, we have that (with OPT the optimal value)
value(A(X)) / OPT ≤ C (
minimisation
), or
OPT / value(A(X)) ≤ C (
maximisation
)Slide3
PTAS
A
Polynomial Time Approximation Scheme
is an
algorithm, that gets two inputs:
the
“usual” input X, and a real value
e
>0.
For each fixed
e
>0, the algorithm
Uses
polynomial time
Is an (1+
e
)-approximation algorithmSlide4
FPTAS
A
Fully
Polynomial Time Approximation Scheme
is an algorithm, that gets two inputs: the “usual” input X, and a real value
e
>0, and
For each fixed
e
>0, the algorithm
Is an (1+
e
)-approximation algorithm
The algorithm uses time that is polynomial in the size of X and 1/
e
.Slide5
Example:
Knapsack
Given:
n
items each with a positive integer weight w(
i
) and integer value v(
i
) (1 ≤
i
≤
n
), and an integer B.
Question: select items with total weight at most B such that the total value is as large as possible.Slide6
Dynamic Programming for Knapsack
Let P be the maximum value of all items.
We can solve the problem in O(n
2
P) time with dynamic programming:
Tabulate M(
i,Q
): minimum total weight of a subset of items 1, …,
i
with total value Q for Q at most
nP
M(0,0) =
0
M(0,
Q
) =
∞
, if Q>0
M(i+1,Q) = min{ M(
i,Q
), M(
i,Q
-v(i+1)) + w(i+1)
}Slide7
Scaling for Knapsack
Take input for knapsack
Throw away all items that have weight larger than B (they are never used)
Let c be some constant
Build new input: do not change weights, but set new values v’(
i
)
=
ë
v(
i
) / c
û
Solve scaled instance with DP optimally
Output this solution: approximates solution for original instanceSlide8
The Question is….
How do we set c, such that
Approximation ratio good enough
Polynomial timeSlide9
Approximation ratio
Consider optimal solution Y for original problem, value OPT
Value in scaled instance: at least OPT/c – n
At most
n
items, for each v(
i
)/c -
ë
v(
i
)/c
û
<1
So, DP finds a solution of value at least OPT/c –n for scaled problem
So, value of approximate solution for original instance is at least c*(OPT/c –n) = OPT -
ncSlide10
Setting c
Set c =
e
P
/(2n)
This is a
FPTAS
Running time:
Largest value of an item is at most P/c = P / (
e
P
/(2n)) = 2n/
e
.
Running time is O(n
2
* 2n/
e
) =
O(n
3
/
e
)
Approximation: … next Slide11
e
-approximation
Note that each item is a solution (we removed items with weight more than B).
So OPT ≥ P.
Algorithm gives solution of value at least:
OPT –
nc
= OPT – n(
e
P
/ (2n) ) = OPT –
e
/2 P
OPT / (OPT –
e
/2 P) ≤ OPT / (OPT –
e
/2 OPT)
= 1/(1-
e
/2) ≤ 1+
eSlide12
Comments
Technique works for many problems
.. But
not always …