All-or-Nothing Demand Maximization

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

2. Satisfying costumers I: Suppliers J: Costumers x(i,j) assignment d(j): demand c(i): capacity Supplier i assigned x(i,.) s.t.x(i,J) = jx(i,j) ≤ c(i) Costumer j is satisfied ifx(I,j) = ix(i,j) ≥ d(j) Seventh 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 Main Question: How can we maximize coverage in such settings? South Harrow area, NW London (produced using Schema’s OptiPlanner) Seventh 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(i,j)= ∞ x(i,j) assignment c(s,i)=c(i) c(j,t)=d(j) Supplier i assigned x(i,.) s.t.x(i,J) = jx(i,j) ≤ c(i) Costumer j is satisfied ifx(I,j) = ix(i,j) ≥ d(j) Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms

5. Problem definition I: Suppliers J: Costumers d(j): demand x(i,j) assignment c(i): capacity pj: profit, in case of.. yj: satisfaction x(i,J) ≤ c(i) iI s.tx(i,j)≥ 0 Maxjyjpj yj {0,1} x(I,j)≥d(j)yjjJ yis r approximation ifpy≥ r py* Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms

6. -AoNDM: Our Results • AoNDM Cannot be approximated better than unless • -AoNDM Bad News: ( ) Still NP-hard… Good News: A approx. algorithm We’ll present a simpler and faster approx. algorithm Seventh 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 (1,2) 1 (2,3) 2 5 (3,4) 6 2 3 (4,5) 4 (5,6) 5 (3,6) 3 6 (5,1) 4 Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms

8. The Local-Ratio Theorem: yis an r-approximation with respect to p1 yis an r-approximation with respect to p- p1  yis an r-approximation with respect to p Proof: p1 · y  r ×p1* p2 · y  r ×p2*  p · y  r ×( p1*+ p2*)  r ×(p1 + p2 )* Seventh 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: ijEc(i)-x(i) < d(j) Seventh 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: … Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms

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

12. A (1-r)/(2-r)-Approximation (cont.) • Hence, □ Algorithm Seventh 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 maximum amount of capacity available from • A flow-based algorithm. • Slightly increased complexity Seventh Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics, and Algorithms

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

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

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

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

19. Thank You!