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!