Applie Mthemtil Sienes, Vol., 009, no. 0, 87-9 Phsing of Trffi Lights t Ro Juntion S. Mohsen Hosseini Deprtment of Mthemtil Siene Isfhn University of Tehnology, Isfhn, Irn, 856-8 m_hoseini@lumni.iut..ir H. Orooji Deprtment of Mthemtis Islmi Az University Neyshur Brnh, Neyshur, Irn orooji@mil.znu..ir Astrt A metho for solving of speil tegory of trffi prolems hs een presente in this pper. The omptiility grph orresponing to the prolem n irulr r grphs hve een introue. Comptiility grph orresponing to the prolem, spnning sugrph n irulr r grphs then utilize to reue our prolem to the solution of LP prolems. Illustrtive exmples re inlue to emonstrte the vliity n ppliility of the tehnique. Keywors: LP prolem, irulr r grph, mximl lique Introution Reent progress in mthemtis speilly in its pplitions use in oservle progress of grph theory, in wy tht it is now proper tool for reserhes in ifferent fiels like Enoing theory, Eletril networks, Opertion reserh n other fiels. In this pper we review the pplitions of irulr r grphs in solving speil tegory of trffi prolems. The pper is orgnize s follows: in setion we esrie the require efinitions n preliminry. The propose metho hs een presente in form of exmple in setion. In setion we solve n exmple of the prolem with offering metho. Preliminries Consier finite fmily of non-empty sets. The intersetion grph of this
88 S. Mohsen Hosseini n H. Orooji fmily is otine y representing eh set y vertex, two verties eing onnete y n ege if n only if the orresponing sets interset. Intersetion grphs hve reeive muh ttention in the stuy of lgorithmi grph theory n their pplitions. Well-known speil lsses of intersetion grphs inlue intervl grphs, horl grphs, irulr r grphs, n so on. Definition. A irulr r grph is the intersetion grph of fmily of rs on irle. We sy tht these rs re irulr r represention of the grph. Definition. A lique of grph G is omplete sugrph of G. Definition. A lique of grph G is mximl lique of G if it is not properly ontine in nother lique of G. Prolem sttement The prolem is to instll trffi lights t ro juntion in suh wy tht trffi flows smoothly n effiiently t the juntion. We tke speifi exmple n explin how our prolem oul e tkle. Figure isplys the vrious trffi strems, nmely,,,. Figure : Trffi strems Certin trffi strems my e terme omptile if their simultneous flow woul not result in ny ients. For instne, in Figure, strems n re omptile, wheres n re not. The phsing of lights shoul e suh tht when the green lights re on for two strems, they shoul e omptile. We suppose tht the totl time for the ompletion of green n re lights uring one yle is two minutes. We form grph G whose vertex set onsist of the trffi strems in question n we mke two verties of G jent if, n only if, the orresponing strems re omptile. This grph is the omptiility grph orresponing to the prolem in question. The omptiility grph of Figure is shown in Figure.
Phsing of trffi lights t ro juntion 89 Figure : Grph G(Comptiility grph of Figure ) We tke irle n ssume tht its perimeter orrespons to the totl yle perio, nmely 0 seons. We my think tht the urtion when given trffi strem gets green light orrespons to n r of this irle. Hene, two suh rs of the irle n overlp only if the orresponing strems re omptile. The resulting irulr r grph my not e the omptiility grph euse we o not mn tht two rs interset whenever orrespon to omptile flows. ( There my e two omptile strems ut they nee not get green light t the sme time). However, the intersetion grph H of this irulr r grph will e spnning sugrph of the omptiility grph. So we hve to tke ll spnning sugrph of G in to ount n hoose from them the spnning sugrph tht hs the most mximl lique. The proper grph H for the ove exmple is shown in Figure. The effiieny of our phsing my e mesure y minimizing the totl re light time uring trffi yle, tht is the totl wting time for ll the trffi strems uring yle. For the ske of onreteness, we my ssume tht t the time of strting, ll lights re re. The mximl lique of H re k = {,, } n k = {, }. Eh lique k i, i, orrespons to phse uring whih ll strems in tht lique reeive green lights. In phse, trffi strems, n reeive green light; in phse, n reive green light. Suppose we, ssign to eh phse k urtion i i. Our im is to etermine the s ( i 0 ) so tht the totl wting time is minimum. Further, we my ssume tht the minimum green light time for ny strem is 0 seons. Trffi strem gets re light, when the phse k reeive green light. Hene s totl re light time is. Similrly, the totl re light times of trffi strems, n, respetively, re, n. Therefore, the totl re light time of ll the strems in one yle is Z =. Our im is to minimize Z sujet to i 0, i n 0, 0, 0, = 0. The optiml solution to this prolem is = 00, = 0, min Z = 0. The phsing tht orrespons to this lest vlue woul then e the est phsing of the trffi lights.
90 S. Mohsen Hosseini n H. Orooji Figure : Grph H(Intersetion grph) Exmple The Figure isplys the vrious trffi strems, nmely,,..., g. We pply the propose metho for this exmple. The omptiility grph orresponing to the prolem is shown in Figure 5. The proper grph (the sugrph of G tht hs the most mximl lique) H for this exmple is shown in Figure 6. The mximl lique of H re k = {,, }, k = {,, }, k = {, e} n k = { e, f, g}. Our im is to minimize Z = + + + sujet to i 0, i n 0, 0, 0, 0, 0, 0, n = 0. The optiml solution to this prolem is = 80, = 0, = 0, = 0 n min Z = 60 (in seons).
Phsing of trffi lights t ro juntion 9 pe pe f g e Figure : Trffi strems e g f Figure 5: Grph G(Compility grph)
9 S. Mohsen Hosseini n H. Orooji f g e Figure 6: Grph H(Intersetion grph) 5 Consolution In this pper we present n pplition of irulr r grphs to the prolem of phsing of trffi lights. The omptiility grph orresponing to the prolem n irulr r grphs hve een introue. Illustrtive exmple is inlue to emonstrte the vliity n ppliility of the tehnique. Referenes [] Deng X., Hell P. n Hung J., Liner time represention lgorithms for proper irulr-r grphs n proper intervl grphs, SIAM Journl of Computing, to pper. [] Gvril F., Algorithms on irulr-r grphs, Networks, (97), 57-69. [] Golumi M., Algorithm Grph Theory n Perfet Grphs, (Aemi Press New York,980) [] Hell P., Bng-Jensen J. n Hung J., Lol tournments n proper irulr-r grphs, Algorithms, Leture Notes in Computer Siene, Vol. 50 (Springer-Verlg), (990), 0-09. [5] Stouffers K., Sheuling of trffi lights A new pproh, Trnsporttion Res, (968), 99-. Deng X., Hell P. n Hung J., Liner time represention lgorithms for proper irulr-r grphs n proper intervl grphs, SIAM Journl of Computing, to pper. Reeive: August, 008