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All-or-Nothing Demand Maximization. Reuven Bar-Yehuda Technion Joint work with David Amzallag Danny Raz and Gabriel Scalosub. Satisfying costumers. I: Suppliers. J: Costumers. x( i , j ) assignment. d(j): demand. c(i): capacity. Supplier i assigned x( i ,.) - PowerPoint PPT Presentation
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All-or-Nothing Demand Maximization
Reuven Bar-YehudaTechnion
Joint work withDavid Amzallag Danny Raz and Gabriel Scalosub
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 2
Satisfying costumers
I: Suppliers J: Costumers
c(i): capacity d(j): demandx(i,j) assignment
Costumer j is satisfied
if x(I,j) = i x(i,j) ≥ d(j)
Supplier i assigned x(i,.)
s.t. x(i,J) = j x(i,j) ≤ c(i)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 3
Motivating Example
Future 4G:
Technology enables having several stations cover a client
“Cover-by-many” Larger demands
South Harrow area, NW London (produced using Schema’s OptiPlanner)
Main Question:
How can we maximize coverage in such settings?
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 4
Problem: Is there x to satisfy all costumers?: Solution: use Max Flow (and find also x)
I: Suppliers J: Costumers
c(s,i)=c(i) c(j,t)=d(j)x(i,j) assignment
Costumer j is satisfied
if x(I,j) = i x(i,j) ≥ d(j)
Supplier i assigned x(i,.)
s.t. x(i,J) = j x(i,j) ≤ c(i)
c(i,j)= ∞
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 5
Problem definition
I: Suppliers J: Costumers
c(i): capacityd(j): demandx(i,j)
assignment
x(I,j) ≥d(j)yj jJ
x(i,J) ≤ c(i) iI
yj {0,1} Max j yjpj
pj: profit, in case of..
y is r approximation if py ≥ r py*
s.t x(i,j) ≥ 0
yj: satisfaction
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 6
Our Results
AoNDM Cannot be approximated betterthan unless
-AoNDM Bad News:
( ) Still NP-hard…
Good News:A approx. algorithmWe’ll present a simpler and faster approx. algorithm
-AoNDM:
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 7
Hardness of Approximation
Reduction from Maximum Weight Independent Set
Theorem: AoNDM Cannot be approximated better than
unless
1
2
3
4
56
1
2
3
4
5
6
(1,2)
(2,3)
(3,4)
(4,5)
(5,6)
(3,6)
(5,1)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 8
The Local-Ratio Theorem:
y is an r-approximation with respect to p1
y is an r-approximation with respect to p- p1
y is an r-approximation with respect to p
Proof: p1 · y r × p1*
p2 · y r × p2*
p · y r × ( p1*+ p2*)
r × ( p1 + p2 )*
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 9
A (1-r)/(2-r)-Approximation
Our Goal:
Find a good decomposition of p x,y is greedy-maximal if it cannot be
extended: i.e. i’s free space: c(i)-x(i)
is not enough to satisfy a new costumer j
i.e: ijE c(i)-x(i) < d(j)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 10
A (1-r)/(2-r)-Approximation (cont.)
Lemma:
Assume . Then any greedy-maximal
CP x for S is a approx.
Proof: …
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 11
A (1-r)/(2-r)-Approximation (cont.)
Utilized Satisfied
x(i)/c(i) < 1-r i is utilized } OPTS ≥ p )S)
} OPTŜ ≥ c )Utilized)
≥ x)Utilized)/(1-r)
≥ p)S)/(1-r)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 12
A (1-r)/(2-r)-Approximation (cont.)
Hence,
□
Algorithm
If
return
If
return
Set
Set
For every j “try” adding j to the cover
Return x
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 13
A (1-r)-Approximation
is wasteful:Does not exhaust the capacity of
Solution:Add clients to the cover,while using the maximumamount of capacityavailable from
A flow-based algorithm.• Slightly increased complexity
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 14
A (1-r)-Approximation (cont.)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 15
A (1-r)-Approximation (cont.)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 16
A (1-r)-Approximation (cont.)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 17
A (1-r)-Approximation (cont.)
May 2007Seventh Haifa Workshop on Interdisciplinary Applications of
Graph Theory, Combinatorics, and Algorithms 18
Future Work
Is there a constant factor approximation independent of r?
Is there a good approximation algorithm for
1-AoNDM? Hardness reduction: demand > capacity
• Hardness phase transition: ? ?
Online?
Thank You!
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