Linköping University Post Print. Solving a minimum-power covering problem with overlap constraint for cellular network design

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1 Lnköpng Unversty Post Prnt Solvng a mnmum-power coverng problem wth overlap constrant for cellular network desgn Le Chen and D Yuan N.B.: When ctng ths work, cte the orgnal artcle. Orgnal Publcaton: Le Chen and D Yuan, Solvng a mnmum-power coverng problem wth overlap constrant for cellular network desgn, 2010, European Journal of Operatonal Research, (203), 3, Copyrght: Elsever Scence B.V., Amsterdam. Postprnt avalable at: Lnköpng Unversty Electronc Press

2 Solvng a mnmum-power Coverng Problem wth Overlap Constrant for Cellular Network Desgn Le Chen D Yuan Department of Scence and Technology, Lnköpng Insttute of Technology, SE , Sweden Tel: , Fax: Abstract We consder a type of coverng problem n cellular networks. Gven the locatons of base statons, the problem amounts to determnng cell coverage at mnmum cost n terms of the power usage. Overlap between adjacent cells s requred n order to support handover. The problem we consder s NP-hard. We present nteger lnear models and study the strengths of ther contnuous relaxatons. Preprocessng s used to reduce problem sze and tghten the models. Moreover, we desgn a tabu search algorthm for fndng near-optmal solutons effectvely and tme-effcently. We report computatonal results for both syntheszed nstances and networks orgnatng from real plannng scenaros. The results show that one of the nteger models leads to tght bounds, and the tabu search algorthm generates hghqualty solutons for large nstances n short computng tme. Key words: OR n telecommuncatons, Cellular networks, Coverng, Integer programmng, Tabu search 1 Introducton Moble cellular communcatons comprse an mportant applcaton area of operatonal research (OR). A well-studed problem type n the area s channel assgnment n second generaton (2G) networks [1]. A recent development of OR n 2G networks s the ntegraton of channel assgnment and base staton locaton [22]. At present, there s a Correspondng author. Emal addresses: lech@tn.lu.se (Le Chen), dyua@tn.lu.se (D Yuan).

3 rapd deployment of thrd generaton (3G) and beyond-3g cellular networks, for whch the plannng process does not nvolve channel assgnment, but other types of desgn decsons [6,7,12]. One mportant plannng task n 3G and beyond-3g networks s rado coverage. We consder a type of coverng problem arsng n ths applcaton context. Gven the locatons of base statons and the cell antenna drectons, the problem s to determne the coverage area of every cell, such that the target servce area s completely covered. The coverage area of a cell s determned by the amount of power allocated to a control sgnal for broadcastng presence of servce. A natural performance metrc s the total power allocated to these servce-coverage sgnals. Less power for coverage reduces nterference, and ncreases the power left for transmttng user data. Mnmumpower coverng tends to mnmze also the overlap between cells. Whereas excessve cell overlap should be avoded as t results n hgh nterference, some amount of overlap between a cell and ts neghborng cells s requred for supportng the handover operaton when users move from one cell to another. Moreover, coverage overlap s a necessary condton for the so called soft and softer handover states, n whch a user s smultaneously connected to two or more cells. For these reasons, t s vtal to ncorporate, n addton to coverage, overlap as a constrant for pars of cells between whch users are expected to roam. The problem we consder comes up once the locatons of base statons and antennas are known. Base staton locaton n 3G networks has been studed n a number of references. Mathematcal programmng models as well as heurstc algorthms have been proposed by, for example, Amald et al. [2,3], Matar and Schmenk [16], and Zhang et al. [23], and Zhang et al. [24]. Esenblätter et al. [5,6] present nteger models and soluton methods for optmzaton of both base staton locaton and antenna confguraton. The plannng objectves consdered n these references are network deployment cost and performance. The latter s typcally represented by the average cell load. The trade-off between nfrastructure cost and proft has been dealt wth by nteger programmng by Kalvenes [15]. In [18], Olnck and Rosenberger approached ths trade-off by a stochastc optmzaton model to take nto account demand uncertanty. Another type of plannng problem s topology optmzaton for nterconnectng nodes at varous levels of the system herarchy n the rado access network. Jüttner et al. [11] appled a meta heurstc for optmzng the nterconnecton between two node types, and a branch and bound method for solvng a specal case of the problem. A branch and cut algorthm has been developed by Fschett et al. [8] to solve the problem of optmally formng a mult-level star topology to nterconnect several types of nodes. The research n ths paper deals wth power optmzaton for coverage plannng. For 3G networks, a common practce of settng power for coverage has been the unform strategy,.e., to reserve a roughly constant amount (typcally 10-15% of the total cell power) 2

4 to the coverage sgnal n all cells. A number of prevous studes (e.g., [9,20,21]) have shown, however, that the strategy may be neffcent. Adoptng a non-unform power allocaton and optmzng ts amount can yeld consderable power savngs as well as better load balancng between cells. The problem studed n ths paper s motvated by these fndngs. Whereas power savng may not be a crucal aspect n earler 3G systems mplementng power control (that s, the power to serve each user s dynamcally adjusted to make the sgnal qualty stay at a desred level), t s of great sgnfcance n beyond-3g networks, such as hgh speed downlnk packet access (HSDPA) that targets moble Internet data. Because HSDPA does not use power control, any power savng on the control channels, n partcular the one used for coverage, wll make more resource avalable to servng user traffc and thereby mprovng data throughput. Indeed, n [10], Geerdes observed that for HSDPA, t s best to allocate as much user power as possble n order to maxmze the performance. Optmzaton of the power of the coverage sgnals has been addressed n [20,21]. The objectves are power consumpton and load balancng. Jont optmzaton of power and antenna tlt has been nvestgated n [9,19]. However, the problems studed n these works do not address suffcent overlap between adjacent cells usng explct constrants. One ssue that arses wth non-unform coverage power s handover performance, as well as soft handover consderaton n earler 3G systems. (Soft handover does not apply to HSDPA.) Under unform power, t s relatvely smple to predct handover (and soft handover) performance. More care on handover s needed for non-unform coverage power, because cell shape becomes more rregular, and because power mnmzaton has a tendency of shrnkng cell sze. As overlap s a necessary condton for handover and soft handover, ts treatment n coverage plannng, whch s a key aspect of our work, becomes essental. The work n the paper falls nto the research doman of OR applcatons. The paper targets modelng and solvng a type of coverng problem that s of relevance n moble telecommuncatons, and the approach s to develop mathematcal programmng models and soluton algorthms. Smlar to many classcal set coverng problems, the problem consdered n ths paper s NP-hard. We formulate the problem usng nteger lnear models, and study the strengths of ther contnuous relaxatons. Preprocessng technques are used for reducng problem sze and tghtenng the models. To deal wth large-scale problem nstances, we develop a tabu search algorthm, amed at fndng near-optmal solutons tme-effcently. We develop a neghborhood that effectvely deals wth the constrants of coverage and cell overlap. We apply the nteger lnear models and the tabu search algorthm to syntheszed network nstances and nstances orgnatng from real plannng cases for large ctes. The experments show that the models can be used to compute optmal or near-optmal solutons for nstances of up to moderate sze. For large-scale nstances, the tabu search algorthm s able to provde 3

5 hgh-qualty solutons usng short computng tme. In a network plannng context, the optmzaton problem we consder may have to be solved many tmes for one network. In partcular, the overlap levels are expected to be set ntally by engneerng expertse n network plannng. Smulaton can be used to assess n detal the handover performance of the optmzed coverage pattern. Based on the results of smulaton, the overlap parameters n the problem may have to be adjusted. Thus t s vtal to be able to obtan tme-effcently optmal or near-optmal solutons to coverage plannng. The remander of the paper s organzed as follows. In Secton 2 we dscuss the applcaton context and ntroduce some notaton. The optmzaton problem s formalzed n Secton 3. In Secton 4 we present two nteger lnear models, and compare the strengths of ther contnuous relaxatons. Secton 5 addresses preprocessng. The tabu search algorthm s detaled n Secton 6. We report expermental results n Secton 7. Conclusons are provded n Secton 8. 2 Defntons A cellular network conssts of a set of rado base statons. A base staton has one or several (typcally three) cells, each has ts own rado antenna. Every cell uses a broadcast channel, referred to as the common plot channel, to announce the presence of the cell and ts servce. A moble termnal s able to access the network only f t can detect at least one plot channel sgnal, of whch the rato between the receved sgnal strength and the nterference s above a threshold. If a moble termnal s covered by multple cells, the common plot channel facltates cell selecton. Typcally, the termnal s attached to the cell havng the strongest plot channel sgnal. Factors that determne the strength of a receved plot sgnal nclude transmt power, attenuaton factor between the cell antenna and the moble phone, as well as antenna gan factors. From a network plannng standpont, the parameter that can be used for adjustng cell coverage s the amount of transmt power allocated to the common plot channel. Let C = {1,...,C} denote the set of cells. The servce area s represented by a large set of test ponts, denoted by J = {1,...,J}. The (mnmum) amount of power requred to cover test pont j J by cell Cs denoted by P j. The value depends on, among other factors, the sgnal propagaton condton. We refer to [19,20] for the dervaton of the power parameter. We use J to denote the set of test ponts that can be potentally covered by cell, and let J = J. To present the mathematcal models, t s useful to denote the same nformaton from the vewpont of test ponts: We defne C j as the set 4

6 of potentally coverng cells of j, and let C j = C j. To facltate handover, coverage overlap requrement s defned for cells beng geographcally adjacent. The overlap requrement s of relevance for any two adjacent cells where some amount of handover between the two cells s expected to take place. Let J h = J J h, and J h = J h. For two adjacent cells and h, the overlap requrement specfes, wthn the area gven by J h, the mnmum amount to be covered by both cells. The overlap s measured n the number of test ponts. We denote by d h the mnmum number of ponts that must be covered by both and h. As sgnal attenuaton ncreases at least quadratcally n dstance, mnmzaton of power subject to overlap of two cells means that overlap tends to occur somewhere around the mddle of the two base statons. Overlap s not needed f two cells are adjacent but do not have users movng between (e.g., due to obstacles). Generally speakng, the overlap requrement s specfed by a set D contanng unordered pars of cells, and a postve nteger d h for each (, h) D. In Fgure 1, some of the concepts that have been ntroduced thus far are llustrated. P j P hj h J J h J h Test pont j (, h C j ) Fgure 1. An llustraton of some of the concepts n the optmzaton framework. Denote by F : J {1,...,J } a bjecton such that the sequence P,F (1), P,F (2),..., P,F (J ) s monotonously non-decreasng. Tes, f any, are broken arbtrarly. In other words, we sort P j,j J n ascendng order. For convenence, we ntroduce (j) as a short-hand notaton when and F (j) are together used as subscrpts. Hence P (j) s not the power requred to cover test pont j, but that for coverng the jth test pont n the sorted sequence. We use F to denote the nverse of F,.e., F (j) gves the poston of test pont j J n the sorted power sequence of cell. 3 The Optmzaton Problem The optmzaton problem we ntend to solve amounts to mnmzng the power needed to cover the servce area and to satsfy the overlap requrements. We refer to the problem as mnmum-power coverng wth overlap (MPCO). A formal defnton s gven below. [MPCO] Fnd a mnmum-sum vector p =(p 1,p 2,...,p C ), where p [P (1),P (J )], 5

7 C, such that for all j J, set { C j : p P j }, and for all (, h) D, {j J h : p P j p h P hj } d h. Problem MPCO dffers from classcal mnmum-cost set coverng [4] n several aspects. Frst, for every cell, the coverng elements (power levels) have a structure. Specfcally, the number of ground set elements (test ponts) covered by a cell grows successvely by cell power. Second, snce every cell has to be assgned one of the possble power levels, there s a generalzed upper bound (multple choce) constrant per cell. The thrd dfference s the presence of the overlap constrants. Moreover, the par-wse cell overlap constrants n MPCO dffer from the constrants n the set multcover problem. MPCO s however smlar to many coverng problems n terms of NP-hardness. Proposton 1 MPCO s NP-hard. We provde a sketch of a proof that reduces mnmum-cost set coverng to MPCO where D =. In the reducton, every coverng element corresponds to a cell, and each ground set element of the set coverng problem s a test pont of MPCO. The MPCO nstance has C addtonal dummy test ponts. Every cell has two possble power levels. The frst level covers a dummy test pont that can be covered by the cell only. The second level covers the ground set elements n the set coverng nstance. The power of the frst level s same n all cells. The second power levels are the coverng costs n set coverng. The two nstances have equvalent sets of feasble solutons. For any two peerng solutons, the objectve functon values dffer by a constant, and the proposton follows. 4 Mathematcal Models We present two nteger lnear models for MPCO. We defne the followng sets of varables for the frst model of MPCO (denoted by F1). Note that the defnton of the x-varables follows from the observaton that we can restrct the doman of the power of cell ( C) to the dscrete set {P (1),P (2),...,P (J )}. Ths approach leads to a stronger LP relaxaton than representng power usng contnuous varables. x j = s h j = 1 f the power of cell equals P j, 0 otherwse. 1 f test pont j s covered by both cell and cell h, 0 otherwse. 6

8 [F1] mn P j x j C j J s. t. J C j k=f (j) x (k) 1, j J, (1) j J x j =1, C, (2) J k=f (j) x (k) + J J h k=f h (j) x h(k) = s h j +1, (, h) D,j J h : C j = {, h}, (3) J h x (k) + x h(k) 2s h j, (, h) D,j J h : {, h} C j, (4) k=f (j) k=f h (j) s h j d h, (, h) D, (5) j J h x j {0, 1}, C,j J, (6) s h j {0, 1}, (, h) D,j J h. (7) Constrant sets (1) and (2) ensure, respectvely, that all test ponts are covered and that exactly one of the canddate power levels s selected n every cell. Note that (2) apples also to the case where the power parameters of a cell have the same value for multple test ponts. These ponts appear consecutvely n the sorted sequence. If the optmal cell power s equal to that for coverng these ponts, then there s always an optmal soluton n whch the hghest ndex level among them s set to one. Constrants (3) and (4), state the condton that s h j can be one only f both cells and h cover j. Constrant (4) s a general formulaton of the condton. If, however, cells and h are the only two possble coverng cells of j, then (3) apples. From the LP relaxaton standpont, (3) s clearly stronger than (4) for test ponts wth two potentally coverng cells. For test ponts n (3), (1) can be removed. In addton, the s-varables n (3) can be elmnated by varable substtuton. For clarty, however, we keep these s-varables n the model. The last set of constrants (5) specfes requred overlap. We derve a less straghtforward but more effectve model F2 by further processng of data. Consder (, h) D and l {2,...,J }, for whch the correspondng test pont F (l) J h. Suppose that the power of cell s at level l 1 n the sorted power sequence,.e., the power s P (l 1) and cell covers test ponts n J h requrng power lower than P (l). Let J h ( l 1 ) denote ths set of test ponts,.e., J h ( l 1 )= {j J h : P j P (l 1) }. We defne parameter L( l 1,h) {1,...,J h } to represent the mnmum power of cell h that can meet the coverage and overlap constrants for test ponts n J h, provded that cell uses level l 1. The value of L( l 1,h)= max{l c ( l 1,h),L o ( l 1,h)}, whch are defned as follows. 7

9 Coverage. If there exst test ponts for whch and h are the only two possble coverng cells, and not all of them are covered by cell at level l 1, then L c ( l 1,h) s the smallest number n {1,...,J h } for whch the power of h, P h(l c ( l 1,h)), covers all the remanng test ponts. Otherwse L c ( l 1,h)=1. Overlap. If J h ( l 1 ) <d h, then power level P (l 1), as well as all levels below t, are nfeasble n cell. In ths case we set L o ( l 1,h)=J h +1. Suppose J h ( l 1 ) d h. Parameter L o ( l 1,h) s defned as the smallest number n {1,...,J h }, such that the correspondng power P h(l o ( l 1,h)) covers at least d h test ponts. h J h ={1,..., 12} C6 =2, C j > 2, j = 6 d h =4 Fgure 2. An llustraton of the dervaton of the second model. An llustraton of the defnton of parameter L( l 1,h) s provded n Fgure 2 usng a small example. The bars and dots show the canddate power levels and the potental set of test ponts subject to overlap, respectvely. Parameter L( l 1,h) for varous l-values s represented by the lnes wth arrows. For (, h) D, the doman of possble power levels of cell h s constraned by nequalty x (l 1) J h k=l( l 1,h) x h(k), wth the conventon that the rght-hand sde s zero f L( l 1,h)=J h +1. Next, we observe that the left-hand sde of the nequalty can be strengthened by takng the sum over all x-varables of cell for whch the power levels are less than or equal to P (l 1), because we have to select exactly one power level for every cell, and L( l1 1,h) L( l2 1,h) f l 1 l 2. We arrve at the followng model. [F2] mn P j x j C j J s. t. (1), l 1 k=1 x (k) J h k=l( l 1,h) x j {0, 1}, C,j J. x h(k), (, h) D,l {2,...,J } : F (l) J h, (8) 8

10 The suffcency of constrants (8) for the overlap requrements can be realzed from the followng analyss. Consder a soluton of F2, and suppose t s not feasble to MPCO. Because of (1), the nfeasblty must be due to nsuffcent overlap. Then there exsts (, h) D, l {1,...,J }, and m {1,...,J h }, for whch x (l) = x h(m) = 1, x (k) =0for all k>land x h(k) =0for all k>m, and {j J h : P j P (l) } {j J h : P hj P h(m) } <d h. Let l be the frst number n the sequence l +1,...,J, such that F ( l) J h. Such l exsts as the overlap requrement for (, h) s not satsfed. For constrant (8) defned for l, ts left-hand sde s at least one. Moreover, parameter L( l 1,h) >mby defnton. Thus the rght-hand sde of (8) equals zero,.e., the constrant s volated. Hence the correctness of F2. F2 does not use (2). Ths constrant set s clearly necessary for the correctness of F1. For F2, they have mpact on the space of feasble solutons, but are redundant at both nteger optmum and LP optmum, as stated n the proposton below. Proposton 2 Constrant set (2) s redundant at the nteger optmum and the LP optmum of F2. Proof We show the redundancy of (2) for defnng the LP optmum only. The same result for nteger optmum can be shown smlarly. Denote by x (k), C,k {1,...,J } an optmal soluton to the LP relaxaton of F2. Suppose that J k=1 x (k) > 1 for some C. We modfy ths soluton as follows. For each cell wth J k=1 x (k) > 1, let l denote the smallest number n {1,...,J } such that x (l) > 0, x (k) =0, k <l. Thus J k=l x (k) > 1. We change the value of x (l) to zero f J k>l x (k) 1; otherwse we set ts value to 1 J k>l x (k). After the modfcaton, J k=1 x (k) 1, C. That the modfcaton does not affect the feasblty of x n (1) s obvous. Snce no varable has ts value ncreased, x also satsfes (8) for whch no value modfcaton has been made to the rght-hand sde. Consder a constrant of (8) n whch the rght-hand sde has been reduced, and suppose that the varable wth value modfed s x h(m).ifl( l 1,h) >m, then the constrant remans satsfed. Suppose L( l 1,h) m. In ths case the rghthand sde of (8) equals one, whereas ts left-hand sde s at most one. Therefore (8) s satsfed. In concluson, the modfed x s feasble n F2 and has a better objectve functon value, whch leads to a contradcton, and the proposton follows. For two cells and h wth overlap requrement, we obtan two groups of constrants of (8) by consderng the both ordered pars (, h) and (h, ). InF2, only one of them s needed. Ths suffces also for the contnuous relaxaton, n the sense that ncludng the second group of constrants does not strengthen the relaxaton. Proposton 3 In the LP relaxaton of F2, a soluton satsfyng (8), defned for (, h) D, also satsfes the correspondng constrants defned for (h, ). 9

11 Proof Consder par (h, ), m {2,...,J h } and F h (m) J h. Let l = L(h m 1,) 1. Suppose L( l 1,h) m 1. Then both coverage and overlap are satsfed for the set of test pont for whch and h are two only two potentally coverng cells, f the two power levels are l 1 and m 1, respectvely. Ths mples L(h m 1,) l 1, leadng to a contradcton. Therefore L( l 1,h) m. Utlzng ths observaton, (8) for par (, h), and Proposton 2, we obtan the nequalty J k=l(h m 1,) x (k) =1 l 1 k=1 x (k) 1 J h k=m x h(k) = m 1 k=1 x h(k) Ths nequalty s the constrant of (8) for par (h, ) and power level m of cell h, and the Proposton follows. Model F2 s not only more compact but also stronger than F1 n LP bound. The next proposton establshes ths result. Proposton 4 The bound provded by the LP relaxaton of F2 s greater than or equal to that of the LP relaxaton of F1. Proof See Appendx A. Numercally, F2 s typcally strctly better than F1 n LP bound. Our experments n Secton 7 show that the dfference between the bounds s qute sgnfcant. 5 Preprocessng We can apply preprocessng to MPCO to reduce problem sze. Frst, we use the smple observaton that there are typcally some test ponts havng only one possble coverng cell. Such test ponts mpose lower bounds on cell power. We obtan another lower bound by overlap consderaton, snce for cell par (, h) D, the power of each of the two cells must cover at least d h test ponts n J h. Let l denote the maxmum of the two derved power levels. For both F1 and F2, we can delete all power levels below l, C, as well as all test ponts j Jfor whch C j =1. Moreover, f C j > 1 and j s covered by at least one cell wth power P (l ), the coverage constrant (1) can be deleted. Removal of the coverage constrant apples also to any test pont j wth C j =2, because coverage s mposed by (3) n F1 and (8) n F2. For F1, consder cell and any test pont j that s covered by power P (l ) and not deleted n the prevous step. Consder another cell h havng j J h and an overlap requrement wth. IfP h(lh ) covers j, we can delete the s-varable and reduce d h by one. Otherwse we check f s h j s defned by constrant (4). If so, we can strengthen the constrant by replacng t wth s h j = J h k=f h (j) x h(k). Clearly, these reducton and strengthenng tests should be performed for all (, h) Dand test ponts covered by P (l ) or P h(lh ). 10

12 6 Tabu Search To deal wth large-scale nstances of MPCO, we desgn a tabu search (TS) algorthm [13], amed at fndng hgh-qualty solutons wthn short computng tme. The basc ngredents of TS are a local search procedure and a memory mechansm. Local search apples a neghborhood structure to teratvely generate and evaluate a sequence of solutons. Replacng the current soluton wth one n the neghborhood s called a move. The memory mechansm, mplemented by means of a tabu lst, ams at preventng cyclng (.e., revstng solutons that have been consdered earler n the search) and thereby avod gettng stuck n local optma. To desgn an effectve and tme-effcent TS algorthm for MPCO, the neghborhood and the content of the tabu lst should explot the structure of the coverage and overlap constrants. We say that a feasble soluton of MPCO s non-lowerable, f decreasng the power level of any sngle cell by one step makes the soluton nfeasble. The optmum of MPCO s obvously non-lowerable. We defne a neghborhood such that the TS algorthm generates a sequence of non-lowerable solutons. For convenence, we denote a soluton of MPCO by ndces of the sorted power levels of cells, n form of a vector k =(k,k 2,...,k,...,k C ),.e., the power of cell s P (k ), C. Gven a non-lowerable soluton k, we have observed that the strategy of ncreasng the power of a cell n order to allow for reducng power n some other cells does not lead to an effectve neghborhood. Consder ncreasng the power of cell. Ths may enable power reducton n cells of whch the current power levels are defned by test ponts that may be potentally covered by,.e., {h C: F k (k h ) J h }. A necessary condton for power reducton n a cell h n ths set s to set the power level of cell to at least F (F h (k h )). However, the condton s not suffcent, because there may exst overlap constrant between cell h and some other cell, for whch the constrant s currently actve for test pont F h (k h ). Indeed, cell may already cover F h (k h ) at the current level k. Hence there s no guarantee that ncreasng the power of one cell wll always lead to a power reducton of some other cell. In our TS algorthm, we defne a more effectve and convenent move by consderng power adjustment n the reverse way. Frst, we reduce the power of one cell by one step so the soluton becomes nfeasble. In step two, the power of one or several cells are ncreased to re-establsh feasblty. Note that nfeasblty after the frst step s ether because of coverage or overlap, but not both. In the former case, we ncrease the power of a cell h C F (k ) to F h (F (k )), whereas n the latter case the power of cells, for whch the overlap constrant wth s volated, are ncreased by the mnmum necessary amount to gve suffcent overlap. There s no explct restrcton on how much power may be ncreased; ths s determned by the amount necessary to restore ether coverage or overlap. In addton, several cells may have to perform a jump n ther power levels at the same tme. Ths fact contrbutes to 11

13 (1) N (k) = ; (2) for all C (3) k = k; k = k 1; j = F (k ); (4) f C j (k )=0 (5) for all h C j \{} (6) k h = F h (j); (7) for all m C (8) n = F m (k m ); (9) whle C n (k ) 2 and (J mr (k ) >d mr or F r (n) >k r,r D m) (10) (11) k m = k m 1; N (k) =N (k) {k }; (12) k h = k h; (13) else (14) for all h D : J h (k )=d h 1 (15) whle J h (k )=d h 1 (16) k h = k h +1; (17) for all m C (18) n = F m (k m ); (19) whle C n (k ) 2 and (J mr (k ) >d mr or F r (n) >k r,r D m) (20) (21) k m = k m 1; N (k) =N (k) {k }; (22) return N (k); Fgure 3. The procedure of generatng neghborng solutons n TS. the effectveness n explorng the soluton space. Power ncrease n step two may enable power reducton n some cells other than. Thus n the thrd and last step we decrease the power of these cells, arrvng at another non-lowerable soluton. Fgure 3 gves a formal descrpton of generatng the neghborhood N (k) of k. Note that the descrpton emphaszes on clarty rather than effcency of mplementaton. In the descrpton, C j (k ) and J h (k ) denote the number of cells coverng test pont j and the number of test ponts covered by both and h, respectvely, n soluton vector k. Moreover, D denotes the set of cells havng overlap requrement wth. The frst step of a move s carred out n lne 3. If ths leads to an uncovered test pont (lne 4), the power of potental cells coverng the pont s ncreased (lnes 5 6), otherwse power s ncreased n cells for whch the overlap constrant s volated (lnes 14 16). The last step of a move to ensure a non-lowerable soluton s carred out n lnes 7 10 or n lnes Note that Fgure 3 does not specfy the order of cells n whch power reducton s performed n the last step of a move. Varyng the order may result n dfferent solutons. In our mplementaton, the order s randomzed n every teraton. The second step of a move nvolves restorng ether coverage or overlap. The former may result n multple neghbors (lnes 4 12), whereas the latter (lnes 14 21) always defnes 12

14 a sngle neghbor. Let N denote the number of cells that may potentally overlap wth cell. The total number of neghbors, at maxmum, s C N. In cellular networks, N s typcally defned by cells that are geographcally adjacent to, and n real-lfe networks ths number tends to be a constant n average. Consequently the sze of the TS neghborhood grows lnearly n the number of cells. In every teraton, the TS algorthm generates all the solutons n the neghborhood. Neghborng solutons havng tabu status are excluded, except for those yeldng a total power better than the best known soluton,.e., the aspraton crteron. Whenever ths crteron does not apply, the algorthm moves to the non-tabu soluton wth mnmum total power. The content of the tabu lst conssts n cell and power ndces. Once a move s made, cells and ther power levels nvolved n the move are marked as tabu. A soluton has the tabu status f any cell and ts power level n the soluton are present n the lst. We consder two approaches for obtanng an ntal soluton n TS. The frst approach starts by settng the power levels of all cells to ther maxmum values. For a randomly chosen cell, the power s reduced to the level such that any further reducton wll volate ether a coverage or overlap constrant. Next, one of the remanng cells s randomly selected, and ts power s decreased as much as possble. Ths s repeated untl all cells have been consdered. The second approach s to utlze the LP optmum of model F2. In ths approach, the ntal soluton s constructed by settng the power level of each cell to the hghest level for whch the correspondng x-varable s strctly postve at the LP optmum. The resultng soluton, for whch both the coverage and overlap requrements are clearly satsfed, serves as the startng pont of TS. There are several possbltes to refne the basc TS algorthm, such as dversfcaton [13]. We have mplemented and tested a dversfcaton scheme. The resultng mprovement s however very margnal. The numercal results n the next secton show that our TS algorthm s able to fnd close-optmal solutons wthout havng to ncorporate any ntrcate mechansm for refnement. Ths s a strength of TS as a meta-heurstc. To obtan satsfactory results from TS, an effectve neghborhood s crucal, and the experments show that our desgn of the neghborhood structure s successful n ths regard. 7 Computatonal Experments We report expermental results for two groups of networks. The frst group contans syntheszed networks N1 N5. The second group s formed by three networks R1 R3 orgnatng from real-lfe plannng scenaros. R1 s a small downtown cellular network. Networks R2 and R3, provded by the EU MOMENTUM project [17], represent two 13

15 network plannng cases for the cty of Berln and Lsbon, respectvely. Statstcs of the networks are shown n Table 1. For each network, the table dsplays the numbers of cells, test ponts, and pars of cells havng overlap requrement. We ntroduce overlap requrement for any cell par for whch the potental overlap s above a threshold. The threshold s chosen such that the resultng number of cell pars wth overlap requrement s approxmately equal to the average number of pars of cells beng geographcally adjacent. Column Servce Area shows the sze of the servce area. Every test pont represents a small square area for whch sgnal propagaton s consdered unform. Ths area sze represented by each pont s shown n column Resoluton. For each cell, we measure ts potental sze by the number of test ponts covered under maxmum power. The last three columns present, respectvely, the mnmum, maxmum, and average values of potental coverage over cells. Table 1 Network statstcs. Network Cells Test ponts Pars Servce area [m 2 ] Resoluton [m 2 ] Potental coverage C J D Mn Max Ave. N N N N N R R R For each network, we consder two levels of overlap, defned by settng d h /J h = 10% and 20%, respectvely, for all pars (, h) D, leadng to a total of 16 nstances. In the subsequent text, subscrpts are added to network names to ndcate overlap level. We frst solve MPCO by applyng CPLEX 10.1 [14] to the two nteger lnear models F1 and F2, wth a tme lmt of one hour. In addton to examnng to what extent MPCO can be solved to optmalty usng the two models, we numercally study the strengths of ther contnuous relaxatons. These experments have been conducted on a computer wth a dual core CPU at 2.4 GHz and 7 GB RAM. In the next set of experments, we apply our TS algorthm. TS stops ether because a maxmum teraton lmt of 2000 s reached, or f the best soluton found s not mproved 14

16 n 300 consecutve teratons, whchever becomes satsfed frst. The length of the tabu lst equals 25. We remark that ths partcular length value s not crucal to algorthm performance. The TS algorthm s run on a notebook wth a dual core CPU at 2.0 GHz. Table 2 Computatonal results of F1 and F2. (Objectve values have unt Watt, and tmes are n seconds.) Network F1 F2 LP ILP LP ILP Value Tme Value Tme Value Tme Value Tme N [33.7, 35.4] lmt N [57.2, 58.3] lmt N [116.6, 122.7] lmt N [180.9, -] lmt N [226.2, -] lmt R R [125.7, -] lmt R3 10 N [34.8, 36.9] lmt N [59.1, 60.6] lmt N [119.7, 126.2] lmt N [189.6, -] lmt N [233.3, ] lmt R R [130.3, -] lmt R3 20 In Table 2 we summarze the results of solvng F1 and F2. For each nstance, the table dsplays the objectve functon values obtaned from solvng the nteger models and ther LP relaxatons, and the soluton tmes. The objectve functon value s gven n Watt (W). We use lmt to denote that the tme lmt s reached. If a model can not be solved to nteger optmalty wthn the tme lmt, we present the nterval formed by the best lower bound and the best nteger soluton found. We use to denote that no soluton s obtaned because of excessve memory requrement. Note that the models of R3 have approxmately three tmes more numbers of varables and constrants than those 15

17 of R2. For R3, the solver process acqured almost 98% of the RAM (wthout producng any result), before the process got klled by the operatng system. F1 s clearly outperformed by F2. The former leads to nteger optmum for one network only, and for some nstances t does not enable any feasble soluton at all wthn the tme lmt. The latter delvers nteger optmum for all but one network. We note that for network R1, for whch optmum s found usng both models, the soluton tme of F2 s orders of magntude shorter. Moreover, the contnuous relaxaton of F2 yelds much sharper bounds than that of F1. The tght bounds of F2 contrbute to ts performance n obtanng nteger optmum, and n some cases the ntegralty gap of F2 s zero. In concluson, explotng the problem structure results n sgnfcant mprovement n solvng MPCO effcently. However, for network R3 (the cty of Lsbon), none of the models s able to delver solutons. Ths fact justfes the use of heurstcs for large-scale nstances of MPCO. A further observaton s that the soluton tme (of both models) tends to decrease when the overlap requrement grows from 10% to 20%. Ths s because hgher overlap requrement restrcts the set of feasble power settngs more strngently, resultng n a hgher number of power levels that are dscarded due to nfeasblty. In Table 3 we report two sets of results for the TS algorthm. The table shows the soluton value (n Watt), computng tme (excludng that needed for the ntal solutons), the number of teratons, and the relatve gap to optmum. When ntal solutons are derved by solvng LP, * denotes cases where the LP optmum s also nteger optmal. The sgn denotes that results are not avalable ether because nteger optmum or LP optmum s not known. We observe a maxmum gap of 3.4%. For the two types of ntal solutons, the average gaps are 1.4% and 0.6%, respectvely. The computatonal effort requred by the algorthm s moderate. Thus the proposed TS algorthm s a promsng approach to deal wth large-scale nstances. Startng from the LP soluton does enable TS to fnd better results. However, as ths scheme requres the LP soluton pror to runnng TS, the overall computng tme s longer. In Fgure 4 we use the optmal solutons of MPCO to llustrate and compare the two cell overlap levels for network R2 (cty of Berln). The small dots represent the locatons of the base staton stes. The cells are represented by short lnes showng the drectons of the cell antennas. For each test pont, ts color denotes the number of cells coverng t. From the fgure, we observe that the overlap does have a form that facltates handover between cells. If two cell antennas are co-located at the same ste, the overlap has a beam shape, and ts drecton s somewhere between the drectons of the two antennas. For two geographcally adjacent cells of dfferent stes, the overlap appears n some regon between the two stes. When the overlap level requrement goes from 10% to 20%, the overlap regons become more contnuous, and there s a clear ncrement n the number of test ponts beng covered by more than two cells. 16

18 Table 3 Computatonal results of TS. (Objectve values have unt Watt, and tmes are n seconds.) Network Intal soluton: heurstc Intal soluton: LP 8 Conclusons Value Tme Iter. Gap (%) Value Tme Iter. Gap (%) N * * * * N N N N R * * * * R R N * * * * N * * * * N N N R * * * * R R We have studed a type of coverng problem wth applcaton n cellular network desgn, where cell overlap s necessary for enablng handover of users between cells. We have developed two nteger lnear programmng models that dffer n the level of explotng problem structure. Comparsons of the two models have been conducted both theoretcally and numercally. In addton, we have presented and evaluated a tabu search algorthm. The computatonal experments show that our second nteger lnear model performs very well n delverng sharp lower bounds as well as optmal solutons to nstances up to medum sze, and the tabu search algorthm s a promsng approach for obtanng close-to-optmal solutons usng lttle computng tme. One nterestng lne of further research s to extend the current work to a verson of the 17

19 5.825 x 106 Overlap requrement of cell pars = 10% x 106 Overlap requrement of cell pars = 20% x x 10 6 Fgure 4. An llustraton of cell overlap n the solutons for network R2 (Berln). problem where the power s restrcted to a relatvely small number of possble levels. Ths problem verson becomes of relevance as an approxmaton of the general problem consdered n ths paper, or n case a quantzaton has been performed n calculatng the power. If power s confned to a small dscrete set, many test ponts are expected to have the same power value; such test ponts may be clustered together. Thus one research topc s the development of stronger preprocessng schemes and faster soluton approaches that are talored to ths specfc problem verson. A second topc s to nvestgate whether or not the power restrcton wll open up a polynomal tme approxmaton. Acknowledgments The authors wsh to thank Dr. Fredrk Gunnarsson and Dr. Iana Somna, Ercsson Research, Sweden, and the MOMENTUM project group, for provdng some of the test data. The work of the frst author s funded by the Swedsh Research Councl and CENIIT, Lnköpng Insttute of Technology, Sweden. The work of the second author s part of the European FP7 project IAPP@Ranplan. We thank the two anonymous referees for ther valuable comments and suggestons. A Proof of Proposton 4 To avod a proof of excessve length, we assume that the bjecton F s unque for all C. The proof can be extended to the general case, although t would requre 18

20 addtonal techncal detals. Before movng nto the proof tself, we sketch the underlyng dea. We ntroduce Jh 2 to denote the set of ponts n J h havng and h as the only two possble coverng cells, that s, Jh 2 = {j J h : C j = {, h}}. Let the LP optmum of F2 be x, we defne, followng (3) (4), s h j = J k=f (j) x (k) + J h k=f h (j) x h(k) 1, (, h) D,j Jh 2, and s h j =1/2( J k=f (j) x (k) + J h k=f h (j) x h(k)), (, h) D,j J h \Jh 2. We then show that s satsfes the overlap requrement (5). Note that, for each j J h, there s a constrant of (8) defned for j = F (l), cf. Fgure 2. Due to Proposton 2, the left-hand sde equals 1 J k=l x (k), resultng n the followng nequalty. J k=l x (k) + J h k=l( l 1,h) x h(k) 1 (A.1) Suppose we defne all the s-varables by (4). Ths gves n fact a model that s vald but weaker than F1. In ths case, we can smply take, among the ponts n J h, the frst d h ponts n the sorted sequence of cell. For each pont, the frst term of (A.1) equals one. Take the d h ponts defned correspondngly for cell h. Next, observe that each sum of (A.1) for these ponts appears exactly once n (5), and the proposton follows. The proof s more complcated f (3) s present. In the proof, we frst consder the pars of power levels defned by (A.1) for the remanng J h d h ponts of cell. Applyng (A.1) drectly does not gve a proof, because there may exst one or several l l, wth L( l 1,h)=L( l 1,h), requrng the use of a sum n (5) more than once (cf. Fgure 2). However, note that (A.1) remans vald for a power level lower than l for, or a power level lower than L( l 1,h) for h. We therefore modfy the pars of power levels n ths drecton, and show the exstence of a sequence of modfcatons, such that each sum of (A.1) for the modfed levels s used no more than once n the left-hand sde of (5). We start the proof by some preparaton. Consder any (, h) D. Let j 1,j2,...,jJ h be the sequence of test ponts n J h, such that ther postons n the sorted power sequence of cell, F (j 1 ), F (j 2 ),...,F (j J h ), s monotonously ncreasng. For cell h, order the same test ponts n a smlar manner, and denote the resultng sequence by jh,j 1 h,...,j 2 J h h. For any k {d h +1,...,J h }, parameter L( F (j k) 1,h) defned for constrant (8) clearly corresponds to a power level of cell h for coverng one of the ponts n J h,.e., L( F (j k ) 1,h)=F h (jm h ), where m s an nteger between d h and J h. To smplfy the notaton, we use L(j k,h) to replace L( F (j k ) 1,h) n the remander of the proof. 19

21 We ntroduce a set A = {(j k,f h(l(j k,h))),k = d h +1,...,J h } J h J h. Each element (j, a j ) As referred to as an assocaton because t assocates one test pont j J h to another test pont a j J h. Test ponts j and a j may concde. There are J h d h elements n A. The lemma below states some propertes of A. Lemma 5 (1) For any j k, k {d h +1...,J h }, there s exactly one element (j, a j ) Awhere j = j k (2) For any. j k, k {d h +2...,J h }, and d h +1 t<k, F h(a j t ) F h(a j k ). (3) For any (j, a j ) Awhere j Jh 2 2 and there are m test ponts n Jh requrng power greater than or equal to P j n cell, F h(a j ) F h(j d h+m h ). (4) For any k {d h +1,...,J h } for whch jh k J h 2, f there are m test ponts n Jh 2 requrng power greater than or equal to P h,j k h n cell h, then the number of elements (j, a j ) Awhere F h (a j) F h (jk h ) s at least m. Proof The frst property follows mmedately from the constructon of A. The second property s smply a reformulaton of L(j t,h) L(jk,h) for any two ponts jk and j t where t<k. Consder the thrd clamed property, and assume that ts condton s satsfed. Suppose cell uses a power level strctly lower than F (j). Then there are m test ponts n J h that have to be covered by cell h. Moreover, these ponts cannot contrbute to overlap between cells and h. Consequently cell h must addtonally cover at least d h other ponts n J h to satsfy the overlap requrement. Hence L(j, h) used to defne constrant (8) for pont j must correspond to a power level of cell h allowng for coverng d h + m ponts n J h, and the result follows. To prove the last property, we assume, wthout any loss of generalty, that the m test ponts n Jh 2 requrng power greater than or equal to P h,jh k n cell h are jk h, jk+1 h,..., jh k+t 1, jh k+t,..., jh k+m 1, among whch the frst t ponts (0 t mn{m, d h }) n ths sequence are n the set {j 1,...,jd h }. Parameters L(j k+t,h),...,l(j k+m 1,h) are all greater than or equal to F h(j h), k because these parameters must ensure that cell h can cover jh k+t,...,jh k+m 1. Hence the m t elements n set A defned for j = jh k+t,...,j = j k+m 1 h all satsfy the property. Next, suppose cell uses level F (jd h ), and hence covers exactly d h test ponts n J h. If the power level used by cell h s strctly below F h(j h), k than the overlap s at most d h t, because jh k,jk+1 h,...,jh k+t 1 are covered by cell but not cell h. We follow the sequence of ponts j d h+1,j d h+2,...,j J h one by one, untl we encounter a test pont, say j r, such that jr / {jh k+t,...,jh k+m 1 }. At ths stage, the maxmum possble overlap s d h t, and thus L(j r,h) F h (jk h ). Pont jr must exst because both sequences j 1,j 2,...,j J h and jh,j 1 h,...,j 2 J h h are of the same length. We can contnue followng the sequence, and repeat the same argument for exactly t test ponts, for whch the assocatons satsfy the property. These t elements, together wth the m t elements dscussed earler, complete the proof. 20

22 We defne a second set of assocaton G J h J h. The next lemma concerns the exstence of G wth the specfed propertes. Lemma 6 There exsts a set of assocatons G J h J h satsfyng the followng propertes. I. For any j k, k {d h +1...,J h } and j k Jh 2, there s exactly one element (j, g j ) Gwhere j = j k and F h(g j ) F h(j d h h II. For any j )+1. k, k {d h +1...,J h }, F h (a j k) F h (g j k). III. For any jh k, k {d h +1...,J h 1} and jh k J2 h, there s at least one element (j, g j ) n G where g j = jh. k IV. For any jh k, k {d h +1...,J h 1}, there s at most one element (j, g j ) n G where j Jh 2 and g j = jh. k Proof We start by settng G = A. That G satsfes Property I follows drectly from Propertes 1 and 3 n Lemma 5. Set G also satsfes Property II by equalty. For convenence, we use a set M of ponts referred to as marked (see below). Intally M =.If G does not satsfy Property III, we make some modfcatons to ts elements, resultng n the fulfllment of the property. We start from j d h+1 h, and consder the sequence of test ponts j d h+1 h,j d h+2 h,...,j J h h. Let jk h be the frst pont not satsfyng Property III,.e., jh k Jh 2 and G does not contan any element (j, g j ) where g j = jh. k Consder sequence j J h,j J h 1,..., and let j r be the frst pont for whch F h (g j r) >F h (jk h ). (The nequalty s strct because no pont s currently assocated wth jh k.) We set g j r = jh k,.e., replace the element (j r,g j r) Gwth (jr,jk h ), and mark jr by settng M = M {j r}.we then repeat the procedure for the next pont n jh k+1,...,j J h h not satsfyng Property III, untl ths property holds for G. We show that the above update procedure s always feasble. Assume that the update s g j r = jh k. Later, we need to fnd another pont j r because Property III s not satsfed for j k h. As the procedure treats j d h+1 h,j d h+2 h,...,j J h h n the gven order, F h (j k h ) >F h (jk h ). Moreover, F h(g j r )=F h(j h). k Therefore j r j r. As a result, a pont n j J h,...,j d h+1 can be selected and marked at most once as j r n the update procedure,.e., jr wll not be used for a later update (whch wll break agan Property III at jh k ). So once Property III becomes satsfed at jh, k t wll reman satsfed for jh k n all subsequent updates. Moreover, as long as there s a pont jh k for whch Property III does not hold, we can always fnd the pont j r. The reason s that G ntally satsfes Property 4 n Lemma 5, and the updates wll not change ths property of G. The above procedure wll obvously not cause G to volate Propertes I-II. Next, we show that Property 2 n Lemma 5 holds after the updatng procedure. Ths property s satsfed by G before the procedure starts. Consder the frst update made for pont jh k, and the update s to set g j r = jh. k We only need to examne the property for j r and another 21

23 arbtrary pont j q n j d h+1,...,j J h, because the update wll not affect the property for other pars of ponts. Assume q>r. In ths case F h (g j q) <F h (jk h ), as otherwse jr would not have been chosen. After the update F h (g j r)=f h (jk h ), so the property remans satsfed for j r and jq. Consder the case q<r, and thus F h (g j q) F h (g j r) >F h (jk h ) before the update. The update changes the second nequalty to equalty, and hence the frst nequalty remans vald. Therefore Property 2 s satsfed after the update. The same argument apples to the subsequent updates, and the result follows. Consder Property 3 n Lemma 5. We can mmedately conclude that the property s satsfed by any pont j Jh 2 f j / M, because g j s not changed. If j Jh 2 M, the property may not hold. In ths case we make the observaton that Property IV s satsfed for g j. Ths s because any pont n j d h+1,...,j J h s marked at most once, and the ponts n j d h+1 h,...,j J h h causng updates and markng are all dfferent. To summarze, the new G has Propertes I-III, and Propertes 1, 2, and 4. Moreover, f a pont j n Jh 2 does not satsfy Property 3, then g j wll satsfy Property IV. If Property IV does not hold for G, we make some further modfcatons, wthout changng the fulfllment of Propertes I-III. Note that volaton of Property IV means that there are at least two ponts j k,j r Jh, 2 such that g j k = g j r. We process ponts j J h,j J h 1,...,j d h+1 n the gven order, and stop when we encounter a pont j k Jh, 2 for whch there exsts another pont j r Jh 2, r<k, and g j k = g j r = jh t. We fnd a pont j q h n jd h+1 h,j d h+2 h,...,jh t 1, wth the condton that there does not exst any element (j, g j ) Gwhere j Jh 2 and g j = j q h. We update G by settng g j k = j q h. Ths s repeated untl Property IV becomes vald for G. We show the exstence of pont j q h as follows. Because Property IV s not satsfed at g j k = g j r = jh t, none of jk and j r s n set M, and consequently Property 3 s satsfed at both j k and jr 2. Let m 1 be the number of test ponts n Jh requrng power strctly greater than P,j k n cell, then sequence j d h+1 h,j d h+2 h,...,j t 1 h contans at least m ponts. Among them, at most m 1 are assocated wth the m 1 ponts n Jh 2 requrng power strctly greater than P,j k n cell. Moreover, because of Property 2, none of the ponts n j d h+1 h,j d h+2 h,...,jh t 1 s assocated wth any j k where d h +1 k <k. Thus there s at least one pont that can be chosen as j q h. After the update for jk, Property 3 s clearly stll vald for all ponts n ({j k 1,j k 2,...,j d h+1 } Jh) 2 \M. Moreover, Property 2 remans satsfed for any par of these ponts. Therefore we can repeat the aforementoned update as long as Property IV does not hold at some pont. Once an update s made for j k, subsequent updates wll not alter ths property at g j k because of the crteron used for selectng j q h. Furthermore, the updates wll obvously not affect Propertes I-III. Hence eventually G wll satsfy Propertes I-IV. 22