Bandwidth Allocation in Networks with Multiple Interferences Reuven Bar-Yehuda GlebPolevoy DrorRawitz Technion
Multiple interference we can approximate to For small interferences
Interval selection with multiple interference 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. Rij j t
Interval selection with multiple interference 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)
Linearization & Normalization We can transform: To: Where:
Common time • & One req/userOne req/base station R11 Maximize s.t. R22 Rii i Rnn t t0
Open knapsack Not feasible 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. 1
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
Strong interferences: w >1/k Same user Î Same user Same time LetÎbe an interval that ends first; 1 if I in conflict withÎ For all intervals I define: p1(I) = 0 else For every feasible x: p1 ·x k+1 Every Î-maximalsolution is k+1 approximation . For everyÎ-maximal x: p1·x 1
Strong interferences: w >1/k • The k+1 approximation algorithm Algorithm MaxIS( R, p ) If R = Φ then returnΦ ; If ISp(I) 0 then returnMaxIS( R- {I}, p); Let ÎRthat ends first; p(Î) if I in conflict with Î IS define: p1(I) = 0 else IS= MaxIS( R, p- p1) ; If IS isÎ-maximal then returnIS else return IS {Î};
Weak interferences: w ≤1/k Same base station Î Same user • Interference conflict LetÎbe an interval that ends first; 0 ifInot in any conflict with Î For all intervals I define: p1(I) = 1-1/kelse if I same base or same user as Î w(I)else if I in interference conflict with Î For every feasible x: p1 ·x 3-2/k Every Î-maximalis For everyÎ-maximalx: p1·x 1-1/k
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 > ½w > 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* 3/7OPTthen GRAY 1/3(3/7OPT)=1/7OPTelse COLORED* 4/7OPTthus COLORED 1/4(4/7OPT)=1/7OPT
Interval selection with multiple interference 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. i Rij j t
Frequency allocation with multiple interference 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 stationi: Rij={dij1,…,dijk,….} • Each demand dijk has a profit pijk >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)|= dijk i Rij j t
Frequency allocation with multiple interference 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
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The Local-Ratio Technique: Basic definitions Given a profit [penalty] vector p. Maximize[Minimize]p·x Subject to: feasibility constraints F(x) x isr-approximationif F(x) andp·x[]r·p·x* An algorithm is r-approximationif for any p, F it returns an r-approximation
The Local-Ratio Theorem: xis an r-approximation with respect to p1 xis an r-approximation with respect to p- p1 xis an r-approximation with respect to p Proof: (For maximization) p1 · x r ×p1* p2 · x r ×p2* p · x r ×( p1*+ p2*) r ×(p1 + p2 )*
Special case: Optimization is 1-approximation xis an optimum with respect to p1 xis an optimum with respect to p- p1 xis an optimum with respect to p
A Local-Ratio Schema for Maximization[Minimization] problems: Algorithm r-ApproxMax[Min]( Set, p ) If Set = Φ then returnΦ ; If I Setp(I) 0 then returnr-ApproxMax( Set-{I}, p ) ; [If I Setp(I)=0 then return {I} r-ApproxMin( Set-{I}, p ) ;] Define “good” p1 ; REC = r-ApproxMax[Min]( S, p- p1) ; If REC is not an r-approximation w.r.t. p1 then “fix it”; return REC;
The Local-Ratio Theorem: Applications Applications to some optimization algorithms (r = 1): ( MST) Minimum Spanning Tree (Kruskal) ( SHORTEST-PATH) s-t Shortest Path (Dijkstra) (LONGEST-PATH) s-t DAG Longest Path (Can be done with dynamic programming) (INTERVAL-IS) Independents-Set in Interval Graphs Usually done with dynamic programming) (LONG-SEQ) Longest (weighted) monotone subsequence (Can be done with dynamic programming) ( MIN_CUT) Minimum Capacity s,t Cut (e.g. Ford, Dinitz) Applications to some 2-Approximation algorithms: (r = 2) ( VC) Minimum Vertex Cover (Bar-Yehuda and Even) ( FVS) Vertex Feedback Set (Becker and Geiger) ( GSF) Generalized Steiner Forest (Williamson, Goemans, Mihail, and Vazirani) ( Min 2SAT) Minimum Two-Satisfibility (Gusfield and Pitt) ( 2VIP) Two Variable Integer Programming (Bar-Yehuda and Rawitz) ( PVC) Partial Vertex Cover (Bar-Yehuda) ( GVC) Generalized Vertex Cover (Bar-Yehuda and Rawitz) Applications to some other Approximations: ( SC) Minimum Set Cover (Bar-Yehuda and Even) ( PSC) Partial Set Cover (Bar-Yehuda) ( MSP) Maximum Set Packing (Arkin and Hasin) Applications Resource Allocation and Scheduling: ….
Single request to Single base station I19 I18 I17 I16 I15 I14 I12 I12 I11 Maximize s.t: For each instance I: For each freq. t: R1j = {I1j} j
Single base station: How to select P1 to get optimization? P1=0 P1=1 P1=0 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: p1(I) = 0 else For every feasible x: p1 ·x 1 Every Î-maximal is optimal. For every Î-maximal x: p1 ·x 1 P1=0 P1=0 P1=1 P1=0 P1=1 P1=1
Single base station: An Optimization Algorithm P1=P(Î ) P1=0 P1=0 I19 I18 I17 I16 I15 I14 I13 I12 I11 Î time Algorithm MaxIS( S, p ) If S = Φ then returnΦ ; If ISp(I) 0 then returnMaxIS( S - {I}, p); Let ÎS that ends first; p(Î) if I in conflict with Î IS define: p1(I) = 0 else IS= MaxIS( S, p- p1) ; If IS isÎ-maximal then returnIS else return IS {Î}; P1=0 P1=0 P1=P(Î ) P1=0 P1=P(Î ) P1=P(Î )
Single base station: Running Example P(I5) = 3 -4 P(I6) = 6 -4 -2 P(I3) = 5 -5 P(I2) = 3 -5 P(I1) = 5 -5 P(I4) = 9 -5 -4 -4 -5 -2
:Frequency allocation with multiple interference Approximation for weak interferences FA-Weak(R, p) If R= return (, ) Let be minimum in R
