AllorNothing Demand Maximization Reuven BarYehuda Technion Joint work

All-or-Nothing Demand Maximization Reuven Bar-Yehuda Technion Joint work with David Amzallag Danny Raz and Gabriel Scalosub

Satisfying costumers I: Suppliers c(i): capacity J: Costumers x(i, j) assignment d(j): demand Supplier i assigned x(i, . ) Costumer j is satisfied s. t. x(i, J) = j x(i, j) ≤ c(i) if x(I, j) = i x(i, j) ≥ d(j) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 2

Motivating Example l l l Future 4 G: Technology enables having several stations cover a client “Cover-by-many” Larger demands Main Question: How can we maximize coverage in such settings? South Harrow area, NW London (produced using Schema’s Opti. Planner) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 3

Problem: Is there x to satisfy all costumers? : Solution: use Max Flow (and find also x) I: Suppliers J: Costumers c(i, j)= ∞ c(s, i)=c(i) x(i, j) assignment c(j, t)=d(j) Supplier i assigned x(i, . ) Costumer j is satisfied s. t. x(i, J) = j x(i, j) ≤ c(i) if x(I, j) = i x(i, j) ≥ d(j) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 4

Problem definition I: Suppliers x(i, j) assignment c(i): capacity Max j yjpj J: Costumers d(j): demand pj: profit, in case of. . yj: satisfaction s. t x(i, j) ≥ 0 x(i, J) ≤ c(i) yj {0, 1} x(I, j) ≥d(j)yj i I j J y is r approximation if py ≥ r py* May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 5

Our Results l Ao. NDM Cannot be approximated better than unless -Ao. NDM: -Ao. NDM l ( ) Bad News: Still NP-hard… Good News: A approx. algorithm We’ll present a simpler and faster approx. algorithm May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 6

Hardness of Approximation l Reduction from Maximum Weight Independent Set 1 (1, 2) 5 6 2 (2, 3) (3, 4) (4, 5) (5, 6) (3, 6) 3 (5, 1) 1 2 3 4 5 6 4 Theorem: Ao. NDM Cannot be approximated better than unless May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 7

The Local-Ratio Theorem: y is an r-approximation with respect to p 1 y is an r-approximation with respect to p- p 1 y is an r-approximation with respect to p Proof: p 1 · y r × p 1* p 2 · y r × p 2* p · y r × ( p 1*+ p 2*) r × ( p 1 + p 2 )* May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 8

A (1 -r)/(2 -r)-Approximation Our Goal: Find a good decomposition of p l x, y is greedy-maximal if it cannot be extended: l i. e. i’s free space: c(i)-x(i) is not enough to satisfy a new costumer j i. e: ij E c(i)-x(i) < d(j) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 9

A (1 -r)/(2 -r)-Approximation (cont. ) Lemma: Assume CP x for S is a . Then any greedy-maximal approx. Proof: … May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 10

A (1 -r)/(2 -r)-Approximation (cont. ) x(i)/c(i) < 1 -r i is utilized Utilized } OPT ≤ p(S) S Satisfied } OPT Ŝ ≤ c(Utilized) ≤ x(Utilized)/(1 -r) ≤ May 2007 p(S)/(1 -r) Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 11

A (1 -r)/(2 -r)-Approximation (cont. ) l Hence, □ Algorithm If return Set For every j “try” adding j to the cover Return x May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 12

A (1 -r)-Approximation l is wasteful: Does not exhaust the capacity of l Solution: Add clients to the cover, while using the maximum amount of capacity available from l A flow-based algorithm. May 2007 • Slightly increased complexity Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 13

A (1 -r)-Approximation (cont. ) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 14

A (1 -r)-Approximation (cont. ) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 15

A (1 -r)-Approximation (cont. ) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 16

A (1 -r)-Approximation (cont. ) May 2007 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 17

Future Work l l May 2007 Is there a constant factor approximation independent of r? Is there a good approximation algorithm for 1 -Ao. NDM? Hardness reduction: demand > capacity • Hardness phase transition: ? ? Online? Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms 18

Thank You!