Bandwidth Allocation in Networks with Multiple

InterferencesReuven Bar-Yehuda

Gleb PolevoyDror Rawitz

Technion

1

Multiple interference

2

(1 )ii

1 1 2 1 2 31 (1 ) (1 )(1 ) (1 )(1 )(1 )

1

1 2 3

1 1 2 1 2 3:1 (1 ) (1 ) (1 ) 1 ii

Additive we can approximate to For small interferences

Interval selection with multiple interference

3

Base stations B={1,2,…,i,…,n}Interferences i <1Users 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.

Rijj

'' : ( ')

(1 )ijk ijk

iI S t Ii i t alloc i

t

2

1

i

n

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

1 ki i

1 ki i

1

31k

Interval selection with multiple interference

Linearization & Normalization

We can transform:

To:

Where:

5

'' : ( ')

1ijk ijk

iI S t Ii i t alloc i

w

'' : ( ')

(1 )ijk ijk

iI S t Ii i t alloc i

''

log(1 )

logi

iw

Maximize

s.t.

II SP

Common time &One req/user One req/base station

( { }) 1I Sw S I

6

Riii

t

2

1

i

n

R11

R22

Rnn

t0

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.

II SP

Open knapsack

( { }) 1I Sw S I

1

Not feasible( { }) 1w S I

7

8

1

2

1

1

2

j

i

j

m n

u

u

Base

u

u

u

Base

u

Base u

Use open knapsack constraintsat interval’s right endpoints

maxI

I SP

( { }) 1Iw S Right II S

s.t. S contains at most one interval from a user contains at most one interval per base stationIS RightI S

Same user

9

1

2

1

1

2

j

i

j

m n

u

u

Base

u

u

u

Base

u

Base u

Strong interferences: w > 1/k

Same user

Same 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 k+1

Every Î-maximal solution is k+1 approximation . For every Î-maximal x: p1 ·x 1

Î

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 ÎI S define: p1 (I) =

0 elseIS = MaxIS( R, p- p1 ) ;

If IS is Î-maximal then return IS else return IS {Î};

10

Strong interferences: w > 1/kThe k+1 approximation algorithm

11

1

2

1

1

2

j

i

j

m n

u

u

Base

u

u

u

Base

u

Base u

Weak interferences: w ≤ 1/k

Same user

Interference conflict

Let Î be an interval that ends first; 0 if I not in any conflict with Î

For all intervals I define: p1 (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: p1 ·x 3-2/k

Every Î-maximal is For every Î-maximal x: p1 ·x 1-1/k

Î Same base station

13 approximation

1k

1/7-approximationR9

R8 w½ > R7 w > ½ w½ >

R6R5 w½ >

R4R3 w > ½ w½ >

R2R1 w > ½ w > ½ w½ >

Algorithm:GRAY = Find 1/3-approximation for gray (w>1/2) intervals;COLORED = Find 1/4-approximation for colored intervalsReturn the one with the larger profitAnalysis:If GRAY* 3/7OPT then GRAY 1/3(3/7OPT)=1/7OPT elseCOLORED* 4/7OPT thus COLORED 1/4(4/7OPT)=1/7OPT

Interval selection with multiple interference

13

Base stations B={1,2,…,i,…,n}Interferences i <1Users 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.

Riji

j

11 : ( 1)

(1 )ijk ijk

iI S t I

i i t alloc i

t

Frequency allocation with multiple interference

14

Base stations B={1,2,…,i,…,n}Interferences i <1Users U={1,2,…,j,…,m}Frequencies {1,2,…,t,…,f}User j has a set of bandwidth

demands from base station i: 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

Riji

j

{1,2,... }: ( )

(1 )ijkt fijk t alloc ijk

t

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

1 ki i

1 ki i

2 1k

Frequency allocation with multiple interference

25

1k

Thank you !

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, Fit returns an r-approximation

17

The Local-Ratio Theorem:

x is an r-approximation with respect to p1

x is an r-approximation with respect to p- p1

x is 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 )*

18

Special case: Optimization is 1-approximation

x is an optimum with respect to p1

x is an optimum with respect to p- p1

x is an optimum with respect to p

19

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” p1 ;

REC = r-ApproxMax[Min]( S, p- p1 ) ;

If REC is not an r-approximation w.r.t. p1 then “fix it”;

return REC;20

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 :

….21

Single request to Single base stationI19I18I17I16I15I14I12I12I11

Maximize s.t: For each instance I:

For each freq. t:

I

IxIp )(

}1,0{Ix

)()(:

1IetIsIIx

R1j = {I1j}j

22

Single base station: How to select P1 to get optimization?

I19I18I17I16I15I14I13I12I11

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=1

P1=1

P1=1

P1=1

P1=0

P1=0

P1=0

P1=0

P1=0

23

Single base station: An Optimization Algorithm

I19I18I17I16I15I14I13I12I11 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 ÎI S define: p1 (I) =

0 elseIS = MaxIS( S, p- p1 ) ;

If IS is Î-maximal then return IS else return IS {Î};

P1=0

P1=0

P1=0

P1=0

P1=0

P1=P(Î )

P1=P(Î )

P1=P(Î )

P1=P(Î )

24

Single base station: Running Example

P(I1) = 5 -5

P(I4) = 9 -5 -4

P(I3) = 5 -5

P(I2) = 3 -5

P(I6) = 6 -4 -2

P(I5) = 3 -4

-5 -4 -2

25

Approximation for weak interferencesFA-Weak(R, p)

If R= return (, )Let be minimum in R

26

:Frequency allocation with multiple interference

1 2 1

2

ˆ1 ifˆDefine ( , , ) 2 elseif and define

2else

(1 )

ˆ ˆˆ̂ ˆ ˆDefine { : ( , , ) 0}

( , ) ( , )

ˆˆ̂return Augume

FA-W

n

e

( ,

ak

t , )

ijk

ijk ijk

i i

p i j k d j j p p p

d w

f

R ijk p i j k

S A R p

S A ijk

ˆˆ̂ijkd