1. www.cs.technion.ac.il/~reuven 1 Throughput Maximization in 4G Cellular Networks Prof. Reuven Bar-Yehuda January 13, 2008 Technion IIT www.cs.technion.ac.il/~reuven

2. www.cs.technion.ac.il/~reuven 2 Satisfying costumers I: SuppliersJ: Costumers c(i): capacity d(j): demand 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)

3. www.cs.technion.ac.il/~reuven 3 Problem: Is there x to satisfy all costumers?: Solution: use Max Flow (and find also x) I: SuppliersJ: 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)= ∞

4. www.cs.technion.ac.il/~reuven 4 Problem: Find x to satisfying maximum costumers Difficulty: It is an NP-Hard problem, even for c=  and d=1 SetsElements c(i)=  d(j)=1 Reduction: Maximum coverage set covering

5. www.cs.technion.ac.il/~reuven 5 Problem definition I: SuppliersJ: Costumers c(i): capacity d(j): demand x(i,j) assignment x(I,j) ≥d(j)y j  j  J x(i,J) ≤ c(i)  i  I y j  {0,1} Max  j y j p j p j : profit, in case of.. y is r approximation if py ≥ r py* s.t x(i,j) ≥ 0 y j : satisfaction

6. www.cs.technion.ac.il/~reuven 6 Q: What about approximation? A: It is as difficult as Independent Set, even for c=d=1 J: Costumers c(i): capacity d(j): demand x(i,j) assignment I: Suppliers

7. www.cs.technion.ac.il/~reuven 7 Q: What about approximation for practical special cases? Our main result: Constant approximation if c(i) > d(j) I: SuppliersJ: Costumers c(i): capacity d(j): demand x(i,j) assignment y j : satisfaction

8. www.cs.technion.ac.il/~reuven 8 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 )*

9. www.cs.technion.ac.il/~reuven 9 Algorithm MAXCOVER ( I, J,E, d, p, c) x  MAXFLOW (Gf ) if {t} is a MINCUT in Gf then return x if  j  J s.t. p(j)=0 then return MAXCOVER (G0 = (I, J \ {j},E \ E(j)), d, p, c) set  = MIN j  J p(j)/d(j) for every j  J, set p1(j) =  · d(j) x  MAXCOVER (I,J,E, d, p-p1, c) for every j such that p2(j) = 0 do if mode=GREEDY then if c(N(j)) − x(N(j)) d(j) then x  GREEDY-ADDTOCOVER (G, j, x, d, c) else i.e., mode=FLOW S {j0 2 J | x(j0) = d(j0)} 21: set G0 = (I, S [ {j},E(S [ {j})) z MAXFLOW (G0f ) if {t} is a MINCUT in G0f then x FLOW-ADDTOCOVER (G, j, x, d, c) return x