1. Reuven Bar-Yehuda Gleb Polevoy Gleb Polevoy Dror Rawitz Technion Technion 1

2. Multiple interference 2 we can approximate to For small interferences

3. Interval selection with multiple interference 3 Base stations B={1,2,…,i,…,n} Interferences  i <1 Users U={1,2,…,j,…,m} Times {1,2,…,t,…,f} User j has a set of time interval requests from base station i: R ij ={I ij1,…,I ijk,….} Each request ijk has a profit P ijk >0 Optimization problem: Allocating subsets of time intervals with maximum profit s.t: At most one interval per user All intervals satisfied by a base station are independent. R ij j t

4. Main result: 7-approximation This is achieved by getting: k+1- approximation for strong interferences -approximation for weak interferences For k=2 it gives: 3+4=7 (will be shown) 4 Interval selection with multiple interference

5. Linearization & Normalization We can transform: To: Where: 5

6. Maximize s.t. Common time & One req/user One req/base station 6 R ii i t R 11 R 22 R nn t0t0

7. Maximize s.t. Bad news: NP-Hard (add width-less expensive box) Good news: FPTAS (Dynamic programming approach) Generalization to many base stations: the bipartite is a forest. Open knapsack 1  Not feasible 7

8. 8 Use open knapsack constraints at interval’s right endpoints s.t. S contains at most one interval from a user contains at most one interval per base station

9. 9 Strong interferences: w > 1/k Let Î be an interval that ends first; 1 if I in conflict with Î For all intervals I define: p 1 (I) = 0 else For every feasible x: p 1 ·x  k+1  Every Î- maximal solution is k+1 approximation. For every Î- maximal x: p 1 ·x  1 Î

10. Algorithm MaxIS( R, p ) If R = Φ then return Φ ; If  I  S p(I)  0 then return MaxIS( R - {I}, p); Let Î  R that ends first; p(Î) if I in conflict with Î  I  S define: p 1 (I) = 0else IS = MaxIS( R, p- p 1 ) ; If IS is Î- maximal then return IS else return IS  {Î}; 10 Strong interferences: w > 1/k The k+1 approximation algorithm

11. 11 Weak interferences: w ≤ 1/k Interference conflict Let Î be an interval that ends first; 0 if I not in any conflict with Î For all intervals I define: p 1 ( I ) = 1-1/k else if I same base or same user as Î w(I) else if I in interference conflict with Î For every feasible x: p 1 ·x  3-2/k  Every Î-maximal is For every Î-maximal x: p 1 ·x  1-1/k Î

12. 1/7-approximation R9 R8 w > ½ R7 w > ½ w > ½ R6 R5 w > ½ R4 R3 w > ½ w > ½ R2 R1 w > ½ w > ½ w > ½ Algorithm: GRAY = Find 1/3-approximation for gray (w>1/2) intervals; COLORED = Find 1/4-approximation for colored intervals Return the one with the larger profit Analysis: If GRAY*  3/7OPT then GRAY  1/3(3/7OPT)=1/7OPT else COLORED*  4/7OPT thus COLORED  1/4(4/7OPT)=1/7OPT

13. Interval selection with multiple interference 13 Base stations B={1,2,…,i,…,n} Interferences  i <1 Users U={1,2,…,j,…,m} Times {1,2,…,t,…,f} User j has a set of time interval requests from base station i: Rij={Iij1,…,Iijk,….} Each request ijk has a profit Pijk >0 Optimization problem: Allocating subsets of time intervals with maximum profit s.t: At most one interval per user All intervals satisfied by a base station are independent. R ij i j t

14. Frequency allocation with multiple interference 14 Base stations B={1,2,…,i,…,n} Interferences  i <1 Users U={1,2,…,j,…,m} Frequencies {1,2,…,t,…,f} User j has a set of bandwidth demands from base station i: Rij={d ij1,…,d ijk,….} Each demand d ijk has a profit p ijk >0 Optimization problem: Allocating demands with maximum profit s.t: At most one demand satisfied per user All demands satisfied by a base station are independent. |alloc(ijk)|= d ijk R ij i j t

15. Main result: 12-approximation This is achieved by getting: - approximation for strong interferences -approximation for weak interferences For k=2 it gives: 5+7=12 15 Frequency allocation with multiple interference

16. Thank you !

17. The Local-Ratio Technique: Basic definitions Given a profit [penalty] vector p. Maximize[Minimize] p·x Subject to:feasibility constraints F(x) x is r-approximation if F(x) and p·x  [  ] r · p·x* An algorithm is r-approximation if for any p, F it returns an r-approximation 17

18. The Local-Ratio Theorem: x is an r-approximation with respect to p 1 x is an r-approximation with respect to p- p 1  x is an r-approximation with respect to p Proof: ( For maximization) p 1 · x  r × p 1 * p 2 · x  r × p 2 *  p · x  r × ( p 1 *+ p 2 *)  r × ( p 1 + p 2 )* 18

19. Special case: Optimization is 1-approximation x is an optimum with respect to p 1 x is an optimum with respect to p- p 1 x is an optimum with respect to p 19

20. A Local-Ratio Schema for Maximization[Minimization] problems: Algorithm r-ApproxMax[Min]( Set, p ) If Set = Φ then return Φ ; If  I  Set p(I)  0 then return r-ApproxMax( Set-{I}, p ) ; [ If  I  Set p(I)=0 then return {I}  r-ApproxMin( Set-{I}, p ) ; ] Define “good” p 1 ; REC = r-ApproxMax[Min]( S, p- p 1 ) ; If REC is not an r-approximation w.r.t. p 1 then “fix it”; return REC; 20

21. The Local-Ratio Theorem: Applications Applications to some optimization algorithms (r = 1): ( MST) Minimum Spanning Tree (Kruskal) MST ( SHORTEST-PATH) s-t Shortest Path (Dijkstra) SHORTEST-PATH (LONGEST-PATH) s-t DAG Longest Path (Can be done with dynamic programming)(LONGEST-PATH) (INTERVAL-IS) Independents-Set in Interval Graphs Usually done with dynamic programming)(INTERVAL-IS) (LONG-SEQ) Longest (weighted) monotone subsequence (Can be done with dynamic programming)(LONG-SEQ) ( MIN_CUT) Minimum Capacity s,t Cut (e.g. Ford, Dinitz) MIN_CUT Applications to some 2-Approximation algorithms: (r = 2) ( VC) Minimum Vertex Cover (Bar-Yehuda and Even) VC ( FVS) Vertex Feedback Set (Becker and Geiger) FVS ( GSF) Generalized Steiner Forest (Williamson, Goemans, Mihail, and Vazirani) GSF ( Min 2SAT) Minimum Two-Satisfibility (Gusfield and Pitt) Min 2SAT ( 2VIP) Two Variable Integer Programming (Bar-Yehuda and Rawitz) 2VIP ( PVC) Partial Vertex Cover (Bar-Yehuda) PVC ( GVC) Generalized Vertex Cover (Bar-Yehuda and Rawitz) GVC Applications to some other Approximations: ( SC) Minimum Set Cover (Bar-Yehuda and Even) SC ( PSC) Partial Set Cover (Bar-Yehuda) PSC ( MSP) Maximum Set Packing (Arkin and Hasin) MSP Applications Resource Allocation and Scheduling : …. 21

22. Single request to Single base station I19 I18 I17 I16 I15 I14 I12 I11 Maximize s.t: For each instance I: For each freq. t: R 1j = {I 1j } j 22

23. Single base station: How to select P 1 to get optimization? I19 I18 I17 I16 I15 I14 I13 I12 I11 Î time Let Î be an interval that ends first; 1 if I in conflict with Î For all intervals I define: p 1 (I) = 0 else For every feasible x: p 1 ·x  1 Every Î- maximal is optimal. For every Î- maximal x: p 1 ·x  1 P1=1P1=1 P1=1P1=1 P1=1P1=1 P1=1P1=1 P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 23

24. Single base station: An Optimization Algorithm I19 I18 I17 I16 I15 I14 I13 I12 I11 Î time Algorithm MaxIS( S, p ) If S = Φ then return Φ ; If  I  S p(I)  0 then return MaxIS( S - {I}, p); Let Î  S that ends first; p(Î) if I in conflict with Î  I  S define: p 1 (I) = 0else IS = MaxIS( S, p- p 1 ) ; If IS is Î- maximal then return IS else return IS  {Î}; P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 P1=0P1=0 P 1 =P (Î ) 24

25. Single base station: Running Example P(I 1 ) = 5 -5 P(I 4 ) = 9 -5 -4 P(I 3 ) = 5 -5 P(I 2 ) = 3 -5 P(I 6 ) = 6 -4 -2 P(I 5 ) = 3 -4 -5 -4 -2 25

26. Approximation for weak interferences FA-Weak(R, p) If R=  return ( ,  ) Let be minimum in R 26 :Frequency allocation with multiple interference