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1 Frequency Assignment for Cellulr Mobile Systems Using Constrint Stisfction Techniques Mkoto Yokooy nd Ktsutoshi Hirymz y NTT Communiction Science Lbortories -4 Hikridi, Seik-cho, Sorku-gun, Kyoto Jpn e-mil: yokoo@cslb.kecl.ntt.co.jp, z Kobe University of Mercntile Mrine 5-- Fuke-minmi-mchi, Higshind-ku, Kobe , Jpn e-mil: hirym@ti.kshosen.c.jp, Abstrct - This pper presents new lgorithm for solving frequency ssignment problems in cellulr mobile systems using constrint stisfction techniques. The chrcteristics of this lgorithm re s follows: ) insted of representing ech cll in cell ( unit re in providing communiction services) s vrible, we represent cell (which hs multiple clls) s vrible tht hs very lrge domin, nd determine vrible vlue step by step, ) powerful cell-ordering heuristic is introduced, 3) brnch-nd-bound serch tht incorportes forwrdchecking is performed, nd 4) the limited discrepncy serch isintroduced to improve the chnce of nding solution in limited mount of serch. Experimentl evlutions using stndrd benchmrk problems show tht this lgorithm cn nd optiml or semi-optiml solutions for these problems, nd most of the obtined solutions re better thn or equivlent to those of existing methods using simulted nneling, tbu serch, or neurl networks. These results show tht stte-of-the-rt constrint stisfction/optimiztion techniques re cpble of solving relistic ppliction problems when equipped with n pproprite problem representtion nd heuristics. I. Introduction With growth in the demnd of mobile telephone services, the ecient use of vilble spectrums is becoming incresingly importnt. The studies of frequency ssignment problem (lso clled chnnel ssignment problem) in cellulr mobile systems hve long history [], [6], [7], [4], [5]. Vrious AI techniques, including constrint stisfction, simulted nneling, neurl networks, tbu serch, nd GA, hve been pplied to this problem [], [4], [5], [8], []{[3], [6]. An overview of frequency ssignment problem is s follows. There exists set of geogrphiclly divided, typiclly hexgonl regions clled cells. Frequencies (chnnels) must be ssigned to ech cell ccording to the number of cll requests. There exist following three types of electro-mgnetic seprtion constrints. co-chnnel constrint: the sme frequency cnnot be ssigned to pirs of the cells tht re geogrphiclly close to ech other. djcent chnnel constrint: similr frequencies cnnot be simultneously ssigned to djcent cells. co-site constrint: ny pir of frequencies ssigned to the sme cell must hve certin seprtion. The gol is to nd frequency ssignment tht stis- es the bove constrints using minimum number of frequencies (more precisely, using the minimum spn of the frequencies). It must be noted tht there exist severl vritions of frequency ssignment problems. The benchmrk problems provided by the EUCLID-project Combintoril ALgorithms for Militry Applictions (CALMA) project re well-known in the constrint stisfction/optimiztion reserch community. This type of problem rises from militry ppliction, nd geogrphicl informtion including cells is not described in the problem speciction. Constrint stisfction/optimiztion techniques cn solve thistype of problem quite eciently. On the other hnd, ccording to [6], constrint stisfction techniques re not very eective for solving frequency ssignment problems discussed in this pper. The most strightforwrd wy for solving such problems using constrint stisfction techniques would be to represent ech cll s vrible (whose domin is vilble frequencies), then to solve the problem s generlized grph-coloring problem. However, solving rel-life, lrge-scle problems using this simple formultion seems rther dicult without voiding the symmetries between clls within one cell. In our new lgorithm, insted of representing ech cllsvrible, we represent cell s vrible tht hs very lrge domin. Furthermore, we determine the vrible vlue step by step insted of determining vrible vlue t one time. A stndrd method for solving constrintoptimiztion problems, such s prtil constrint stisfction problems [3], is depth-rst brnch-nd-bound The benchmrk problems nd technicl reports re vilble from ftp://ftp.win.tue.nl/pub/techreports/calma/

2 serch lgorithm. In brnch-nd-bound serch, we usully ssume the existence of heuristic function tht evlutes node in serch tree. This function estimtes the qulity ofthe solutions tht exist under the node in the serch tree. However, creting good heuristic function (i.e., estimting the number of required frequencies) is rther dicult in frequency ssignment problems. To solve problems eciently without good heuristic function, we use limited discrepncy serch techniques [0], [7] so tht the lgorithm cn limit the serch eorts to only the prt of the serch tree where solution is likely to exist. In this pper, we show the forml denition of the problem in Section II, then describe the lgorithm tht utilizes constrint stisfction techniques in Section III. Furthermore, we show experimentl evlutions using stndrd benchmrk problems in Section IV, nd discuss the reltion to other reserch in Section V. II. Frequency Assignment Problem A frequency ssignment problem is formlized s follows. We follow the formliztion used in [5], [6], [], [3], [5], [6]. Frequencies re represented by positive integers ; ; 3;:::. Given: Find: N: the number of cells d i, i N: the number of requested clls (demnds) in cell i c ij, i; j N: the frequency seprtion required between cll in cell i nd cll in cell j f ik, i N, k d i : the frequency ssigned to the kth cll in cell i. such tht, subject to the seprtion constrints, j f ik 0 f jl j c ij, for ll i; j; k; l except for i = j nd k = l, minimize mxf ik for ll i; k. These constrints cn be represented s n N N symmetric comptibility mtrix C. In ddition, set of requested clls cn be represented by n N-element demnd vector D. Exmple. The number of cells is N =4, C = 0 B@ CA B@ ; D = 3 CA : Positive integers (frequencies) must be ssigned to f ik,such tht their mximum is minimum, subject to the seprtion constrints C. In this exmple, there exist constrints between cell nd cell, cell nd cell 4, cell 3 nd cell 4, nmely, the required minimum seprtions re 3,, nd, respectively. In ddition, frequency ssignment problem cn be represented using grph. Figure shows grph representing problem in Exmple (co-site constrints re not described in the grph). In this grph, vertex represents cell, nd n edge represents constrint between cells. The number in the vertex represents the number of cll requests of the cell (d i ), nd the weight of n edge represents required seprtion (c ij ). This grph representtion is clled mcro-grph [4]. III. Algorithm A. Bsic Algorithm cell cell 3 3 cell 3 cell 4 Fig.. Exmple of Mcro-grph We re going to describe the bsic lgorithm developed in this pper. In this lgorithm, we rst x the number of vilble frequencies M to certin upper-bound, then nd n ssignment tht stises given seprtion constrints nd set of constrints f ik M for ll i; k using bcktrcking lgorithm. When solution is found, we setm to mxf ik 0, where f ik re frequencies used in the obtined solution, nd continue the serch process. This lgorithm cn be considered s depth-rst brnch-nd-bound lgorithm. In this lgorithm, there exists n M-element vector for ech cell. An element of this vector represents one frequency, where vlue cn be \ssigned" (the frequency is used t the cell) or \not-ssigned" (the frequency is not used). Figure shows n exmple of these vectors for the problem described in Exmple. A blnk element is \not-ssigned", nd \" mens \ssigned". This is n optiml ssignment for this problem. By using this representtion, the domin size of vrible becomes M. Since this size will be very lrge, determining the vector vlue of cell t one time is not prcticl. Therefore, we determine the vector vlue step by step. We introduce tenttive vlues used during the serch process for ech vector Such n upper-bound cn be obtined using the heuristic sequentil methods described in [5].

3 cell cell cell 3 cell Fig.. Exmple of Cell Representtion element. Avector element cn be one of the following three vlues, \ssigned", \free", or \forbidden". The vlue \ssigned" mens the frequency is used t the cell, \forbidden" mens the frequency cnnot be used t the cell due to seprtion constrints, nd \free" mens the frequency is vilble t the cell. We show the outline of the bsic lgorithm in Figure 3. In the initil stte, ll vector elements of cell re free, nd stck is empty. This lgorithm is bsiclly depth-rst brnch-nd-bound lgorithm tht incorportes forwrd-checking [9]. Therefore, s long s the cell-ordering heuristic nd the frequencyordering heuristic re exhustive, this lgorithm cn eventully nd n optiml solution. The worst-cse time complexity iso(m S ), where M init init is P the initil upper-bound of frequencies, nd S = i= d N i. B. Cell-ordering Heuristic Now, we re going to describe the cell-ordering heuristic used in Step 3 of the min procedure. A commonly used rule of thumb for selecting vrible in constrint stisfction problems is to select the vrible tht is most strongly constrined. For exmple, we cn select the vrible with the smllest domin fter constrint propgtion. A simple extension of this heuristic clcultes the verge number of free frequencies per one cll for ech cell. More speciclly, let us represent the number of vector elements tht re \ssigned" for cell i s ssign i, nd the number of vector elements tht re \free" s free i. We clculte the vlue free i =(d i 0 ssign i ) for ech celli (we cll this vlue verge vilble frequencies, AAF), nd select the cell with the smllest AAF. For exmple, when selecting cell in Exmple, AAF is / for cell nd cell 3, for cell, /3 for cell 4. As result, cell 4 is selected rst. However, we found tht using only the AAF heuristic is not very eective. This is becuse it tends to select cell tht simply hs mny demnds in the shllow serch nodes in the serch tree, nd does not ppropritely consider the strength of the constrints mong cells. For exmple, let us consider the sitution fter ssigning frequencies of three clls of cell 4 (Figure 4). In the gure, \x" mens vector element tht becomes \forbidden". In this cse, the AAF for min procedure. If the demnds re fully stised for ll cells, then solution is found, cll procedure reduce-frequency. Go to Step of this procedure.. If demnd cnnot be stised for cell, cll procedure bcktrck, go to Step of this procedure. 3. Select cell whose demnds re not fully stised using cell-ordering heuristic. 4. Select frequency for ssigning one cll of cell using frequency-ordering heuristic, nd set the vector element of the cell corresponding to frequency to \ssigned". Push (\choice", cell, frequency) tostck. Propgte constrints, i.e., set the vectorelements tht interfere with frequency of cell to \forbidden". Go to Step of this procedure. procedure bcktrck. If stck is empty, then there is no solution. Finish the lgorithm nd return bestsolution.. Pop n element (g, cell, frequency) from stck. 3. If g=\choice", then set the vector element of the cell corresponding to frequency to \forbidden", unpropgte constrints. Push (\forbidden", cell, frequency) to stck, nd nish this procedure. 4. If g=\forbidden", set the vector element of the cell corresponding to frequency to \free". Go to Step of this procedure. procedure reduce-frequency. Record the current ssignment s bestsolution.. Set mx-frequency to the mximl frequency used in the current ssignment. 3. If no cell uses mx-frequency, for ech cell, set ech vector element lrger thn or equl to mx-frequency to \forbidden", set M to mx-frequency0, nish this procedure. 4. Pop n element (g, cell, frequency) from stck. 5. If g=\choice", then set the vector element of the cell corresponding to frequency to \free", unpropgte constrints. Go to Step 3 of this procedure. 6. If g=\forbidden", set the vector element of the cell corresponding to frequency to \free". Go to Step 3 of this procedure. Fig. 3. Bsic Algorithm

4 both cell nd cell 3 is 4. However, selecting cell 3 does not help to reduce the serch spce, since it does not hve ny constrints between other cells except cell 4, which is lredy fully ssigned. To tke into ccount the strength of constrints mong cells, we inventednevlution vlue for cell i clled generlized weighted degree (GWD) dened s follows. X c ij (ssign j +) j In this formul, ssign j is the number of ssigned frequencies of nother cell j, ndc ij is the weight of the edge 3. We select the cell tht hs the mximl GWD vlue. For exmple, in Figure 4, the GWD for cell is, while the GWD for cell 3 is 4. Therefore, we prefer cell to cell 3. However, we found tht using only the GWD heuristic is lso not very eective since it does not tke into ccount the demnds, nd it tends to mke poor decisions in the deep serch nodes in the serch tree. Thus, we decided to combine these two heuristics. Since we prefer cell with smller AAF nd lrger GWD, we use the evlution vlue obtined by AAF/GWD, nd select the cell with miniml AAF/GWD. cell cell cell 3 cell x x x x x x x x x x x xxx xxxx Fig. 4. Cell Sttus during Serch Process C. Frequency-ordering Heuristic We re going to describe the frequency-ordering heuristic used in Step 4 in the min procedure. The simplest wy would be to select the rst (smllest) free frequency (rst-free). A more sophisticted method would be to consider the impct of selecting frequency for other cells. Nmely, for ech frequency, we clculte the number of vector elements of other cells tht turn from \free" to \forbidden" by selecting the frequency, nd select the one tht minimizes this vlue (lest-impct). If the number of frequencies is lrge, this clcultion cn be very costly. However, we cn reduce the number of possibilities using co-site constrints. For exmple, let us ssume we re selecting the rst frequency for cell 4 in Exmple, nd the number of possible frequencies is. Since there re three demnds for cell 4, we cnnot ssign frequencies lrger 3 We dd one to ssignj, otherwise this vlue becomes 0 for ll cells in the initil stte. thn 6 for the second cll, otherwise there is no free frequency for the third cll. Then, if we ssign 6 to the second cll, we hve only one possibility, i.e., for the rst cll. We found tht the lest-impct heuristic is more powerful thn the rst-free heuristic in nding solution. On the other hnd, the rst-free heuristic cn nd better solution ( solution with smller mximl frequency) if it cn nd solution. Therefore, in our lgorithm, we rst use the rst-free heuristic in the 0-discrepncy serch described in the next subsection. When the 0-discrepncy serch fils to nd solution, we switch to the lest-impct heuristic. D. Limiting Serch Eorts Although this method uses constrint propgtion (forwrd-checking) to reduce the serch spce, the serch tree of rel-life, lrge-scle problem is still too lrge to perform n exhustive serch. We need to limit the serch eorts to only the prt of the serch tree where solution is likely to exist. One method for limiting the serch eorts is the limited discrepncy serch [0]. In the limited discrepncy serch, the serch process initilly chooses only the best nodes ccording to given heuristic t ech decision point in serch tree (this is clled 0- discrepncy serch). If solution cnnot be obtined by the 0-discrepncy serch, the serch process is llowed to select sub-optiml node only once t decision point (-discrepncy serch). If solution cnnot be obtined, then the serch process is llowed to select sub-optiml nodes twice (-discrepncy serch). The number of llowed discrepncies is incresed one by one. In our lgorithm, we limit the number of discrepncies for the frequency selection (in Step 4 of the min procedure). More speciclly, if solution cnnot be found by selecting the best frequencies ccording to the frequency-ordering heuristic, the serch process is llowed to choose the second-best or the thirdbest frequencies. The originl limited discrepncy serch lgorithm is developed for serching binry tree, i.e., the brnching fctor is two. As shown in [0], there re severl lterntives for modifying the limited discrepncy serch lgorithm to non-binry serch tree. In our lgorithm, we weight discrepncies depending on the order in the heuristic, i.e., the second-best vlue is counted s -discrepncy, nd the third-best vlue is counted s -discrepncy. If the current limit of discrepncies is, the lgorithm is llowed to choose third-best vlue once, or choose the second-best vlue twice 4. Furthermore, since the lest-impct heuristic is less informtive in shllow nodes in the serch tree, we use 4 Another wy istocountllvlues except the best s - discrepncy. This is not prcticl in our frequency ssignment problems since the brnching fctor of tree node cn be very lrge.

5 modied version of the depth-bounded limited discrepncy serch described in [7]. In our lgorithm, discrepncy is llowed only in the nodes whose depth is shllower thn or equl to given depthlimit. In ddition, we introduce modied version of the bounded bcktrcking described in [7] to llow quick recovery from mistkes deep in the tree. In the originl version of the bounded bcktrcking, the lgorithm is llowed to perform bcktrcking up to xed level. However, in the frequency ssignment problems, the brnching fctor of node my vry signicntly. Therefore, llowing xed level of bcktrcking is inpproprite, since the serched subtree cn be too lrge or too smll. Therefore, we set the limit to the totl number of bcktrcking in the subtree. The lgorithm is llowed to perform certin number of bcktrcking in subtree. If the number of bcktrcking exceeds the limit, the subtree is discrded. In Figure 5, we show n exmple of the nodes visited in serch tree. We ssume tht the heuristic prefers left brnches, nd there exists no solution in this serch tree. Furthermore, we ssume the depthlimit is two nd the limit of the totl number ofbck- trcking is one. In the 0-discrepncy serch, the lgorithm follows the left brnches nd reches node, which is ded-end. Since the lgorithm is llowed to perform bcktrcking once, it goes to node, which is nother ded-end. Since the lgorithm reches the limit of bcktrcking, it discrds the subtree nd increses the number of llowed discrepncies. In the -discrepncy serch, the lgorithm rst follows the left, then the right brnch, nd visits node 3 nd node 4. Note tht discrepncies re llowed only t the nodes within the depth-limit. The lgorithm discrds the subtree, nd visits node 5 nd node 6. In the -discrepncy serch, the lgorithm visits node 7 nd node 8. serches including [4]{[6], [], [5], [6]. These problems re formulted bsed on n re in Phildelphi, Pennsylvni. The network consists of cells s shown in Figure 6. There re mny vritions for setting constrints nd demnds. The prmeter settings used in our evlutions re described in Tble I. In the tble, \Nc" mens the squre of required distnce for cochnnel constrints, ssuming tht the distnce between djcent cells is. For exmple, if Nc=, while cell nd cell 5 cn use the sme frequency (the distnce is 4), cell nd cell 4 cnnot (the distnce is 3). \cc" represents the seprtion required for djcent chnnel constrints, nd \c ii " represents co-site constrints. The demnd vectors used in the tble re s follows (cse 3 nd cse 4 re obtined bymultiplying nd 4 to cse, respectively): cse : ( ) cse : ( ) cse 3: ( ) cse 4: ( ) Fig. 6. Cellulr Geometry of Phildelphi Problems depth-limit Fig. 5. Limited Discrepncy Serch with Bounded Bcktrcking IV. Evlutions We use benchmrk problems clled Phildelphi problems, which hve been used widely in previous re- TABLE I Specictions for Phildelphi Problems Instnce Nc cc cii Demnd Vector P 5 cse P 7 5 cse P3 7 cse P4 7 7 cse P5 5 cse P6 7 5 cse P7 7 cse P8 7 7 cse P9 5 cse 3 P0 5 cse 4 Tble II shows the theoreticl lower-bounds reported in [], [5] nd the results obtined with our constrint stisfction method (CS). In this method,

6 to nish the lgorithm execution within resonble mount of time, we set the limit of the visited nodes to 0,000. We terminte the execution when the lgorithm exceeds this limit, nd use the best solution obtined so fr. Also, we set the depth-limit where the discrepncy is llowed to 0, nd the number of llowed bcktrcking to 00. To the extent oftheuthors' knowledge, the best published results for these problems hve been obtined by FASoft [], [6] nd [4]. FASoft is n integrted pckge of vrious methods for solving frequency ssignment problems, such s heuristic sequentil methods, methods using constrint stisfction techniques, Simulted Anneling, GA, tbu serch, etc. We show the results obtined with Simulted Anneling (SA) nd tbu serch (TS) reported in []. These two methods re the most ecient mong the vrious components of FASoft. Furthermore, we show the best results obtined with set of heuristic sequentil methods (SE) reported in [5], nd the results obtined with neurl networks (NN) reported in [4] (\..." in the tble mens tht the result is not reported). As shown in the tble, our lgorithm obtins optiml solutions for the instnces of P, P, P3, P4, P7, P8, P0, nd obtins semi-optiml solutions tht re very close to the optiml for other problem instnces. Moreover, this method cn obtin better or equivlent solutions compred with existing methods for ll problem instnces except P5 (where NN is better), TABLE II Comprison of Solution Qulity (Phildelphi Problems) Instnce Lower CS TS SA SE NN Bound P P P P P P P P P P Furthermore, to exmine the eciency of the proposed lgorithm in lrger-scle problems, we show the evlution results for the benchmrk problems presented in [4], []. There re 7 7 symmetriclly plced cells (49 cells in ll) in these problems. Problem prmeters re described in Tble III, where \c ij " is the miniml frequency seprtion between ny pir of cells whose distnce is less thn p N c, except for djcent cells. The demnd vector is: ( ). This vector is rndomly generted from uniform distribution between 0 nd 30. There re 976 clls in totl. TABLE III Specictions for Kim's Benchmrk Problems Instnce Nc cij cc cii K 7 3 K K TABLE IV Comprison of Solution Qulity (Kim's Benchmrk Problems) Instnce CS NN SE K K K Tble IV shows the results obtined with our new method (CS). The prmeter settings of the lgorithm re identicl to those of the Phildelphi Problems. For comprison, we show the results described in [4], i.e., the results obtined using neurl networks (NN), nd the best results obtined with set of heuristic sequentil methods (SE). Our method obtins much better solutions thn those of NN for K nd K3. Tble V shows the totl execution time of this lgorithm. Although the execution time is obtined by nive LISP implementtion on Sun Ultr 30 Model 300 (Ultr SPARC-II 96MHz), we cn see tht very high-qulity solutions re obtined within resonbly short running time. Since the optimlity of the obtined solution is gurnteed before reching the limit of the visited nodes in P3 nd P4, the execution time for these instnces is very short. It must be noted the primry evlution criterion in these benchmrk problems is the solution qulity. The execution time of the lgorithms seems less importnt nd not often reported in the literture. One reson for this is tht most lgorithms (including our lgorithm) show rpid improvements in the erly stge of the serch process, then the improvements sturte very quickly, nd the solution qulity cnnot be signicntly improved even fter very long execution time (e.g., dy). V. Discussion In [], [6], it is reported tht optiml solutions for the Phildelphi problems cn be obtined using cliques. In this method, mximl or some lrge clique in micro-grph (where ech cll is represented s vertex) is identied rst, then frequencies re ssigned to the clls in the clique. Subsequently, this prtil solution is itertively extended by dding clls to the prtil solution until it becomes complete solution. One drwbck to this method is tht nding the mximl clique is nother NP-complete problem nd time-consuming (note tht micro-grph is much lrger thn mcro-grph). Furthermore, this clique

7 TABLE V Algorithm Execution Time Instnce Execution Time (sec) P 49.3 P 59.4 P3 0. P4 0. P P P P P9.9 P K 79.4 K 33.0 K method is not fully utomted, i.e., it requires humn tril-nd-error selections of cliques nd extension methods. Our method nd this clique method re not mutully exclusive, nmely, our method cn be used for nding prtil solution for the clique nd extending the prtil solution. A mcro-grph is introduced in [4] to solve frequency ssignment problems. However, this method uses mcro-grph not to directly solve problem, but to obtin upper-bounds of the problem. A similr problem representtion used in this pper is introduced in [4], [5] for solving the problem using neurl networks. VI. Conclusion nd Future Issues We hve developed new lgorithm for solving frequency ssignment problems in cellulr mobile systems. This lgorithm is bsiclly depth- rst brnch-nd-bound procedure tht incorportes forwrd-checking. In this lgorithm, we represent cell s vrible with very lrge domin, nd determine the vrible vlue step by step. Furthermore, we hve developed powerful cell-ordering heuristic nd introduced the limited discrepncy serch to cope with lrge-scle problems. Experimentl evlutions using stndrd benchmrk problems showed tht for most of the problem instnces, this lgorithm cn nd better or equivlent solutions compred with existing optimiztion methods. These results imply tht stte-of-the-rt constrint stisfction/optimiztion techniques re cpble of solving relistic ppliction problems, if we choose the pproprite problem representtion nd heuristics. There is plenty of room for improvement in this lgorithm. Currently, vrious lgorithm prmeters (e.g., the depth-limit, the limit of the number of bcktrcking) re djusted by hnd. It would be desirble if these prmeters cn be dynmiclly tuned ccording to the chrcteristics of the solved problem instnces. Our future works lso include introducing locl-consistency lgorithms tht re stronger thn forwrd-checking, introducing the limited discrepncy serch to the cell-ordering heuristic, nd using hybrid type lgorithms of bcktrcking nd itertive improvement (e.g., [8]). References [] Box, F.: A Heuristic Technique for Assignment Frequencies to Mobile Rdio Nets, IEEE Trnsctions on Vehiculr Technology, Vol. 7, No., (978) 57{64 [] Crlsson, M. nd Grindl, M.: Automtic Frequency Assignment for Cellulr Telephones Using Constrint Stisfction Techniques, Proceedings of the Tenth Interntionl Conference on Logic Progrmming (993) 647{663 [3] Freuder, E. C. nd Wllce, R. J.: Prtil Constrint Stisfction, Articil Intelligence, Vol. 58, No. {3, (99) {70 [4] Funbiki, N., Okutni, N., nd Nishikw, S.: A Threestge Heuristic Combined Neurl Network Algorithm for Chnnel Assignment in Cellulr Mobile Systems, IEEE Trnsctions on Vehiculr Technology (000): (to pper) [5] Funbiki, N. nd Tkefuji, Y.: A Neurl Network Prllel Algorithm for Chnnel Assignment Problems in Cellulr Rdio Networks, IEEE Trnsctions on Vehiculr Technology, Vol. 4, No. 4, (99) 430{437 [6] Gmst, A.: Some Lower Bounds for Clss of Frequency Assignment Problems, IEEE Trnsctions on Vehiculr Technology, Vol. 35, No., (986) 8{4 [7] Hle, W. K.: Frequency Assignment: Theory nd Appliction, Proceedings of the IEEE, Vol. 68, No., (980) 497{53 [8] Ho, J. K., Dorne, R., nd Glinier, P.: Tbu Serch for Frequency Assignment in Mobile Rdio Networks, Journl of Heuristics, Vol. 4, (998) 47{6 [9] Hrlick, R. nd Elliot, G. L.: Incresing Tree Serch Eciency for Constrint Stisfction Problems, Articil Intelligence, Vol. 4, (980) 63{33 [0] Hrvey, W. D. nd Ginsberg, M. L.: Limited discrepncy serch, Proceedings of the Fourteenth Interntionl Joint Conference on Articil Intelligence (995) 607{63 [] Hurley, S., Smith, D. H., nd Thiel, S. U.: FASoft: A system for discrete chnnel frequency ssignment, Rdio Science, Vol. 3, No. 5, (997) 9{939 [] Kim, S. nd Kim, S. L.: A Two-Phse Algorithm for Frequency Assignment in Cellulr Mobile Systems, IEEE Trnsctions on Vehiculr Technology, Vol. 43, No. 3, (994) 54{548 [3] Kunz, D.: Chnnel Assignment for Cellulr Rdio Using Neurl Networks, IEEE Trnsctions on Vehiculr Technology, Vol. 40, No., (99) 88{93 [4] Pennotti, R. J. nd Boorstyn, R. R.: Chnnel Assignments for Cellulr Mobile Telecommunictions Systems, Proceedings of Ntionl Telecommunictions Conference (976) 6:5{{6:5{5 [5] Sivrjn, K. N., McEliece, R. J., nd Ketchum, J. W.: Chnnel Assignment in Cellulr Rdio, Proceedings of 39th IEEE Vehiculr Technology Society Conference (989) 846{850 [6] Smith, D. H., Hurley, S., nd Thiel, S. U.: Improving Heuristics for the Frequency Assignment Problem, Europen Journl of Opertionl Reserch, Vol. 07, (998) 76{86 [7] Wlsh, T.: Depth-bounded Discrepncy Serch, Proceedings of the Fifteenth Interntionl Joint Conference on Articil Intelligence (997) 388{393 [8] Yokoo, M.: Wek-commitment Serch for Solving Constrint Stisfction Problems, Proceedings of the Twelfth Ntionl Conference on Articil Intelligence (994) 33{38