Planning maximum capacity Wireless Local Area Networks

Transcription

1 Edoardo Amaldi Sandro Bosio Antonio Capone Matteo Cesana Federico Malucelli Di Yuan Planning maximum capacity Wireless Local Area Networks

2 Outline Application 3 combinatorial optimization problems Complexity issues Hyperbolic formulations and solution approaches Quadratic formulations and solution approaches Linearization and model strenghthening Preliminary computational results Concluding remarks

3 Wireless Local Area Networks (WLANs) (Cabled) Local Area Networks Dramatic size increase Difficult cable management Cannot cope with users' mobility Introduction of wireless connections

4 Wireless Local Area Networks (WLANs) Users connected to the network via antennas (access points, hot spots) WLANs allow: to substitute cables in offices and departments (easier and more flexible management) at very low cost to provide network services in public areas (airports, business districts, hospitals, etc.)

5 WLAN planning J = {1,,n} candidate sites where antennas can be installed I = {1,,m} "test points" (TPs) or possible users positions For each j J: Ij I subset of test points covered by antenna j Goal: select a subset of candidate sites S J with covering constraints: each test point must be covered by at least one antenna without covering constraints: a test point is not necessarily covered

6 Solution quality measures Transmission protocol: a user can "talk" if all interfering users are "silent" "Talking probability" = 1/(# of the interfering users) 1/4 1/4 1/3 1/3 1/5 1/6 1/3 Network capacity = sum of the "talking probabilities" of all users

7 Objective functions For any S J, let I(S) denote the subset of users covered by S Network capacity c(s) = Network fairness i I(S) 1 j S:i I j Ij f(s) = min i I 1 j S:i I j Ij Intuitively solutions with small non-overlapping subsets should be privileged

8 Combinatorial Optimization problems Maximum capacity unconstrained WLAN P: {max c(s): S J } Maximum capacity covering WLAN PC: {max c(s): S J, j S Ij = I } Maximum fairness WLAN PF: {max f(s): S J } PF implies full coverage, since any solution covering all users dominates those not covering some users (which have fairness =0)

9 Example: third floor of our department Candidate sites Test points uniformly distributed

10 Minimum cardinality set covering Practitioner solution (dense)

11 Practitioner solution (sparse) Maximum capacity solution (PC)

12 Numerical results # Access Points Capacity Efficiency Min. card. Set Covering Practitioner dense Practitioner sparse Maximum capacity Efficiency = Capacity/(# Access Points)

13 Computational complexity Proposition: P, PC, and PF are NP-hard Reduction Exact Cover by 3-sets: Given a set X ( X =3q) and a collection C of n 3-element subsets Cj, j=1,,n, of X, does C contain an exact cover of X, i.e., C' C s.t. every element of X occurs in exactly one element of C? I = X, J = {1,,n}, {I 1,,I n } = C, S { j: Cj C' } P (PC) has a solution S with c(s)=q iff C' is an exact cover PF has a solution S with f(s)=1 iff C' is an exact cover

14 Mathematical Programming Formulations Data: users/subsets incidence matrix a ij = 1 if i Ij, j J 0 otherwise Variables: x j = 1 if j S, 0 otherwise selection of subset Ij y ih = 1 if i and h appear together in a selected subset, 0 otherwise union definition z i = 1 if i is covered, 0 otherwise coverage of user i

15 Max capacity covering WLAN (PC) PCH: max 1 y i I ih h I a ij x j 1 i I full coverage j J a ij a hj x j y ih i,h I, j J definition of y ih y ih 0 i,h I x j {0,1} j J Hyperbolic sum 0-1 constrained problem

16 Max capacity unconstrained WLAN (P) PH: max i I h I z i y ih a ij x j z i i I j J definition of z i a ij a hj x j y ih i,h I, j J definition of y ih 0 z i 1 i I y ih 0 i,h I x j {0,1} j J Hyperbolic sum 0-1 constrained problem

17 Max fairness WLAN (PF) 1 PFH: max min i I y ih h I a ij x j 1 i I full coverage j J a ij a hj x j y ih i,h I, j J definition of y ih y ih 0 i,h I x j {0,1} j J Hyperbolic bottleneck 0-1 constrained problem

18 Solving Hyperbolic formulations Problems PH and PCH cannot be solved by standard techniques nor the algorithms studied for Hyperbolic unconstrained 0-1 problems [Hansen, Poggi de Aragão, Ribeiro 90; 91] can be extended to the constrained case max { i a io + a ij x j j, x b io + b ij x j {0,1}} j j

19 Problem PFH can be solved by a sequence of mixed integer linear systems PFH: max { β : β SFH(β)} Fairness β [0,1] SFH(β): 1 β ( y ih ) i I h I a ij x j 1 i I j J a ij a hj x j y ih i,h I y ih 0 i,h I, x j {0,1} j J Optimal β can be found by binary search (solving a sequence of SFH(β)) Otherwise let α = 1/β and minimize α

20 Quadratic formulation (1) c j = i I j 1 I j = 1 q jk = I j I k I j I k - I j I k I j - I j I k I k (-1 q ij 0) I j I j I k I k

21 Quadratic formulation (1) QPC: max 1 xqx + cx 2 Ax 1 x {0,1} n QP: max 1 xqx + cx 2 x {0,1} n Linear contribution: capacity of a non overlapping subset Quadratic contribution: penalty due to the overlapping of two subsets

22 Quadratic formulation (1) QPC and QP are equivalent to PC and P if each element belongs to at most 2 subsets In the other cases QPC and QP underestimate network capacity QP can be approached by pseudoboolean techniques QPC is a Quadratic Set Covering problem Semidefinite Programming Combinatorial optimization approaches Bounding techniques derived from QAP (e.g. Gilmore and Lawler)

23 Quadratic formulation (1) P j : subproblem obtained by fixing x j =1 w j = max 1 2 q jk x k k J a ik 1 k J x k {0,1} i I \ I j k J due to nonpositiveness of coefficients qjk Pj is a Set Covering W = max (w j + c j ) x j j J a ij 1 j J x j {0,1} i I j J After some fixing, W can be computed by a Set Covering

24 Quadratic formulation (1) Claim: W is an upper bound for QPC It is an upper bound also when we use relaxations instead of computing the exact solution of the set covering problems

25 Quadratic formulation (2) Tradeoff between network capacity and cost p jk = I j I k (0 p I j I k jk 1) approximate measure of the capacity: tends to favor non overlapping subsets g j = installation cost QPC': max { 1 2 xpx - α gx: Ax 1, x {0,1}n } QP': max { 1 2 xpx - α gx, x {0,1}n } tradeoff parameter α>0

26 Quadratic formulation (2) QP' can be solved in polynomial time (min cut computation) Auxiliary graph G = (N, A) with capacities γ + j j γ j s p jk p kj t γ + k k γ k γ + j = max {0, 1 2 k J p jk - α g j } γ - j = max {0,- 1 2 k J p jk + α g j }

27 The minimum capacity s-t cut corresponds to the solution x maximizing the objective function of QP' [Hammer 65] 1 γ + j j γ j s p jk t p kj γ + k k γ k 0

28 Quadratic formulation (2) The Lagrangian relaxation of QPC' can be solved efficiently Minimization of a piecewise convex function At each iteration the computation of a min cut gives the value of the Lagrangian function

29 Computational results Quadratic vs. Hyperbolic Small instances ( J = 10, I = 100, 300) Subsets = circles in the plane (radii 50m, 100m, 200m) Comparison of the objective functions: Hyperbolic, Quadratic, Fairness, # installed access points Exact solutions computed by enumeration Simple heuristic algorithms Average on 10 instances

30 Full coverage: exact solutions cs=10 Fairness exact Hyperbolic exact Quadratic exact #AP Hyperbolic Quadratic fairness #AP Hyperbolic Quadratic fairness # AP Hyperbolic Quadratic fairness R=50m R=100m R=200m R=50m R=100m R=200m Without covering constraints cs=10 Hyperbolic exact Quadratic exact # AP Hyperbolic Quadratic # AP Hyperbolic Quadratic R=50m R=100m R=200m R=50m R=100m R=200m

31 Full coverage: comparison exact and heuristic solutions cs=10 Hyperbolic Quadratic exact heuristic exact heuristic R=50m R=100m R=200m R=50m R=100m R=200m Full coverage: comparison exact and heuristic solutions cs=10 Hyperbolic Quadratic exact heuristic exact heuristic R=50m R=100m R=200m R=50m R=100m R=200m

32 Linearization: idea A test point i may be covered by different activated antennas Introduce a 0-1 variable ξir for each test point i and each subset r of possible activated antennas covering i Exponentially many variables, depending on the cardinality of overlaps

33 Linearization: notation Ji subset of candidate sites covering i S(i) = 2 J i \ {Ø} set of antennas configurations covering i we include the emptyset if the total coverage is not required J(r) subset of candidate sites of configuration r For each configuration r in S(i) we can compute the contribution of test point i to the total network capacity Kir = 1 j J(r) Ij (Kir can be computed also according to the quadratic formulation)

34 Linearization: model max i I Kir ξir r S(i) ξir = 1 i I (one configuration per test point) r S(i) ξir = xj r:j J(r) xj {0,1} i I, j Ji j J (consistency in configuration selection) ξir 0 i I, r S(i) note that only x variables are binary

35 Instance generator Generated on a geometric base (2D) Avoided test points covered by a single candidate site

36 Computational results (1) instance QUADRATIC HYPERBOLIC J / I /id Integer LP Integer LP time value time value time value time value 50/25/ /50/ /100/ /37/ /75/ /150/ /50/ /100/ times in seconds on a 2.8 GHx Xeon

37 Strenghthening equalities Consider a pair of test points i and h and a subset S of candidates sites among those covering both i and h i S h The selected configurations in S for i and h must coincide ξir = r:j J(r) S ξhr r:h J(r) S

38 Computational results (2) instance QUADRATIC HYPERBOLIC J / I /id Integer LP Integer LP time value time value time value time value 100/50/ S = S = S =2/var /100/ S = S = S =2/var var3: generated variables for subsets of at most 3 candidate sites

39 Concluding remarks New interesting combinatorial optimization problems Hyperbolic 0-1 formulations Quadratic formulations (good approximation) Linearization and strenghthening Column generation? Frequency assignment