Introduction. 1. Mathematical formulation. 1.1 Standard formulation

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1 Comact mathematcal formulaton for Grah Parttonng Marc BOULLE France Telecom R&D 2, Avenue Perre Marzn Lannon France Abstract. The grah arttonng roblem conssts of dvdng the vertces of a grah nto clusters, such that the weght of the edges crossng between clusters s mnmzed. We resent a new comact mathematcal formulaton of ths roblem, based on the use of bnary reresentaton for the ndex of clusters assgned to vertces. Ths new formulaton s almost mnmal n terms of the number of varables and constrants and of the densty of the constrant matrx. Its lnear relaxaton brngs a very fast comutatonal resoluton, comared wth the standard one. Exerments were conducted on classcal large benchmar grahs desgned for comarng heurstc methods. On one hand, these exerments show that the new formulaton s surrsngly less tme effcent than exected on general - arttonng roblems. On the other hand, the new formulaton aled on bsecton roblems allows to obtan the otmum soluton for about ten nstances, where only best uer bounds were revously nown. Key words. Lnear rogrammng, Grah Parttonng, Bsecton Introducton The grah arttonng s a classcal combnatoral otmzaton roblem. Ths toc has several alcatons n the telecommuncatons feld. For nstance, the frst ste n the desgn of telecommuncatons networ s to solve a arttonng roblem. The networ vertces are arttoned n order to mnmze the amount of traffc between clusters of vertces. Furthermore, the grah arttonng s an aealng otmzaton roblem, smle n ts defnton but NPcomlete (Garey, Johnson and Stocmeyer 1976). Parttonng a grah defned by ts vertces, ts edges and an edge cost functon conssts of dvdng the vertces nto clusters, such that the total weght of the edges whose endonts are n dfferent clusters s mnmzed. The number of clusters and ther sze are constraned. We consder the balanced artton of a grah, when the dfference of sze between clusters s at most one. The orgnalty of our formulaton s to use a bnary reresentaton for the ndex of clusters (assgned to vertces). The constrants n the standard arttonng formulaton must be deely reformulated to tae advantage of ths bnary reresentaton. The new formulaton results nto less number of varables, constrants and non null constrant coeffcents. It s very effcent for bsecton roblems and allowed to obtan the otmal soluton for several benchmar grahs not yet solved to otmalty. The remander of the document s organzed as follows: Secton one descrbes several new formulatons and demonstrates ther correctness. Secton two evaluates the comactness of the formulatons, and then the thrd secton resents exermental results obtaned by drectly mlementng the formulatons n a mxed nteger rogrammng solver and alyng them to classcal large benchmar grahs. 1. Mathematcal formulaton Let G=(V, E) be a grah wth vertex set V and edge set E. Let W j be the (ostve) weght of the edge (, j) between vertces and j, K the maxmum number of clusters and MaxCard the maxmum sze of each cluster ( V K. MaxCard ). The objectve s to mnmze the cut,.e. the total weght of the edges crossng the clusters. Ths s equvalent to maxmzng the total weght of the edges that are nsde the clusters. 1.1 Standard formulaton The standard straghtforward nteger rogrammng formulaton of the roblem s the followng.

2 Comact mathematcal formulaton for Grah Parttonng 2 Formulaton A Varables:. v =1 f vertex belongs to cluster and 0 otherwse. e j =1 f edge (, j) belongs to cluster and 0 otherwse Maxmze:, j, e j W j Subject to: Each vertex belongs to only one cluster (1) v = 1 Max sze of clusters (2) v MaxCard Lnearzaton of quadratc varables (e j =v *v j ) (3) e j v ej v j ej v + v j 1, j, Decson varables v bnary varables 1.2 Slght comact formulaton The recedng formulaton contans K varables and 3K constrants for each edge. Ths can be reduced to 1 varable and 2K constrants by ntroducng for each edge a new bnary varable f j valued 1 f the edge (,j) s nsde one cluster and 0 otherwse. We show below that varables f j can be evaluated from exstng vertex varables v. f e v v j = j = j f j = 1 / v = v j We can notce that v v v v = 1. Thus, f j can be exressed n the followng way: f j = Mn 1 j ( v v ) j j As the roblem of nterest s a maxmzaton roblem n f j wth ostve weghts, the recedng equalty constrant can be turned nto an nequalty constrant, and rewrtten n the followng way. fj Mn( 1 v v j ), fj 1 v v j Ths leads to the followng formulaton. Formulaton B Varables:. v =1 f vertex belongs to cluster and 0 otherwse. f j =1 f edge (, j) belongs to one of the clusters and 0 otherwse Maxmze:, j f j W j Subject to: Each vertex belongs to only one cluster (1) v = 1 Max sze of clusters (2) v MaxCard Constrants to calculate ntra-cluster edge varables (3) fj 1 + v v j f 1 v + v, j j j, Decson varables v bnary varables

3 Comact mathematcal formulaton for Grah Parttonng Alcaton to the bsecton roblem In the case of bsecton, one sngle decson varable s enough for each vertex. Its value s 1 f the vertex belongs to the frst cluster and 0 otherwse. Formulaton B can be smlfed n the followng way. Formulaton B2 Varables:. v 1 =1 f vertex belongs to frst cluster and 0 otherwse. f j =1 f edge (, j) belongs to one of the clusters and 0 otherwse Maxmze:, j f j W j Subject to: Max sze of clusters (1) MaxCard v 1 v 1 V MaxCard Constrants to calculate ntra-cluster edge varables (2) fj 1+ v1 v j1 fj 1 v 1 + v j1, j Decson varables v 1 bnary varables 1.4 Formulaton based on bnary ndexng of clusters In the general case of -arttonng, formulatons A and B need K decson varables for each vertex. These varables theoretcally enable 2 K choces for each vertex, although only K clusters are avalable. On ths bass, we wll am at a more comact formulaton, needng less varables and constrants. For each vertex, we have to decde to whch cluster t belongs. Let be an nteger varable corresondng to the ndex of the cluster assgned to vertex (0 <K). In ts bnary reresentaton, varable can be wrtten as a lnear combnaton of the owers of 2. The coeffcents b of ths lnear combnaton wll be used as decson varables. = b 2 (0 <log2(k)) For examle, f =5, =1*2 0 +0*2 1 +1*2 2, so b 0 =1=, b 1 =0, b 2 =1. Startng from formulaton B, we wll rewrte ts constrants wth these new decson varables. Each vertex belongs to only one cluster Ths constrant s satsfed from the defnton of, ndeendently of the choce of a decmal reresentaton (varables ) or a bnary reresentaton (varables b ). Each vertex must be assgned to an exstng cluster. Ths means that (ts bnary reresentaton) must be less than K-1. Max sze of clusters The sze of each cluster must be derved from the bnary decson varables corresondng to the cluster ndexes. Let us use agan varables v reresentng the assgnment of vertex to cluster. We wll show above how to calculate these varables from the new bnary varables b. The bnary reresentaton of ndex s based on constant values B : = B 2 Varable v equals 1 f and only f varable. equals. We then have the followng equvalences. v = 1 = v = 1, b = B b \ B v 1, 1 B 1 b + B b = ( )( ) 1 =

4 Comact mathematcal formulaton for Grah Parttonng 4 v ( 1 b B + 2b B ) 1 = 1 = So v s a roduct of factors a. a = 1 b B + 2bB (f B =0, a =1-b, otherwse a =b ) Varable v can be evaluated wth a generalzaton of the lnearzaton of quadratc varables., v a ( ) v a Thus, the avablty of varables v, derved from varables b, allows to reuse the same formula for the constrants concernng the max sze of clusters. Constrants to calculate ntra-cluster edge varables Intra cluster edge varables equal 1 f the endonts of the edge are n the same cluster,.e. f the ndexes of the clusters contanng the endonts of the edge have the same bnary reresentaton. f = 1 / b = b f j j = Mn 1 j ( b b ) j As the roblem of nterest s a maxmzaton roblem n f j wth ostve weghts, the recedng equalty constrant can be turned nto an nequalty constrant, and rewrtten n the followng way. f j ( 1 b b j ), fj b b j Mn 1 Formulaton C Let B be the coeffcent of 2 n the bnary reresentaton of ndex (B s a constant). Varables:. b : coeffcent of 2 n the bnary reresentaton of cluster ndex assgned to vertex. f j =1 f edge (, j) belongs to one of the clusters and 0 otherwse. v =1 f vertex belongs to cluster and 0 otherwse Maxmze: f j W j, j Subject to: Each vertex s assgned to a cluster ndex less than K (1) b 2 K 1 Max sze of clusters (2) v MaxCard Constrants to calculate ntra-cluster edge varables (3) fj 1 + b b j f 1 b + b, j j j, Lnearzaton of varables used for vertces assgnment to clusters (4) v 1 b B + 2b B, ( ) 1 + ( 1 b B + 2b B ) 1), v, Decson varables b bnary varables 1.5 Comact formulaton based on bnary ndexng of clusters Formulaton C uses few decson varables, but ncreases the number of constrants to calculate the sze of clusters. In order to decrease the number of varables and constrants used n formulaton C, we show n ths secton how to calculate the sze of clusters drectly from the varables b (bnary reresentaton of cluster ndexes), wthout usng revous varables v.

5 Comact mathematcal formulaton for Grah Parttonng 5 Frst, let us defne the noton of masng. An ndex s masng a bt f the bnary reresentaton of has bt set to 1. For examle, the ndexes 1 (1), 3 (11) and 5 (101) are masng bt 0. Let us now evaluate the term sb0 = b 0. sb 0 stands for the number of vertces assgned to a cluster whose ndex s masng bt 0. Thus, sb = b = cardcluster = cardcluster + cardcluster / s masng bt0 More generally, sb = b = s masng bt cardcluster 1 / The noton of masng can be generalzed to bt mass. An ndex s masng a bt mas m f the bnary reresentaton of has all bts of the bt mas m set to 1. For examle, wth m=5 (101), the ndexes 5 (101), 7 (111), 13 (1101) are masng the bt mas m. Let bm m be a bnary varable, equal to 1 f cluster ndex s masng bt mas m. Let sbm sbm =. = m bm m cardcluste m r / s masng bt mas m For examle, sbm 5 = cardcluster5 + cardcluster7 + cardcluster Ths leads to the followng rooston. Prooston 1: Varables sbm m ( 0 m < K ) can be lnearly derved from the szes of clusters cardcluster ( 0 < K ). Let SBM=M.C: sbm0 a00 a01... cardcluster0 sbm1 a10 a11 cardcluster1 = sbmk 1 ak 1.0 ak 1.1 cardclusterk 1 By defnton, coeffcent a m of matrx M equals 1 f the ndex s masng the bt mas m. Prooston 2: Matrx M s reversble. Proof: Each ndex s masng the bt mas, thus a =1. If s masng m, the bnary reresentaton of s above the bnary reresentaton of m for each bnary coeffcent, and thus m. Ths can be summarzed wth the followng roertes: a = 1 a m = 0 m < As a concluson, M s an uer dagonal matrx wth all ts dagonal terms set to 1. Hence, M s reversble. Corollary 1: The szes of clusters cardcluster ( 0 < K ) can be lnearly derved from the bt mas varables sbm m ( 0 m < K ). Prooston 2 demonstrates that the sze of clusters can be calculated from the new bt mas varables bm m. We show n the followng how to derve these bt mas varables bm m from the decson varables varables b. Frst, n the secal case of m=0, sbm0 = bm 0 = cardcluster = V. Second, we can notce that when the ndexes of clusters are owers of 2, there s a corresondence between the two sets of varables: bm m =b for m=2 ( 0 < log( K )). Thrd, for m > 0 and m 2, at least 2 bts n the bnary reresentaton of m are set to 1. Thus, m=m 1 +m 2, where m 1 s a ower of 2 (only one bt set to 1 n bnary reresentaton of m 1 ) and m 2 s strctly ostve. Based on the defnton of bt mas varables bm m, we get bm m = bm 1bm 2 and bm 1 corresonds to one of the decson varables b m m m. Therefore, the new bt mas varables bm m can be calculated by recurrence from the decson varables b. Usng the lnearzaton of quadratc varables, three constrants are necessary to calculate each new bt mas varable from two other varables (two decson varables, or one decson varable and one new smaller bt mas varable). 3

6 Comact mathematcal formulaton for Grah Parttonng 6 Let us llustrate these results wth an examle for K=6: sbm cardcluster0 sbm cardcluster1 sbm cardcluster2 = sbm cardcluster3 sbm cardcluster4 sbm cardcluster5 sbm 0 = V sbm 1, sbm 2, sbm 4 are calculated wth decson varables b 0, b 1, b 2 sbm 3 and sbm 5 are calculated wth bt mas varables bm 3 and bm 5 bm 3 =b 0.b 1 (3=11=01+10 n bnary reresentaton) bm 5 =b 0.b 2 (5=101= n bnary reresentaton) Formulaton D Let B be the coeffcent of 2 n the bnary reresentaton of ndex (B s a constant). Varables:. b : coeffcent of 2 n the bnary reresentaton of cluster ndex assgned to vertex. bm m =1 f the cluster ndex ndex s matchng the bt mas m. cardcluster : sze of cluster. f j =1 f edge (, j) belongs to one of the clusters and 0 otherwse Maxmze: f j W j, j Subject to: Each vertex s assgned to a cluster ndex less than K (1) b 2 K 1 Equatons to calculate the sze of clusters (2) cardcluster V b = = cardcluster / s masng bt bmm = / s masng bt mas m cardcluster m ( m 2 Max sze of clusters (3) cardcluster MaxCard Constrants to calculate ntra-cluster edge varables (4) fj 1 + b b j f 1 b + b, j j j, Lnearzaton of quadratc varables bm m (bm m =b bm m ) (5) bm m b bmm bm m' bm m b + bm m' 1 Decson varables b bnary varables, m ( m 2 2. Evaluaton of the sze of formulatons The szes of formulatons are evaluated n table 1 on the bass of the roblem dmensons: V : sze of V E : sze of E K: number of requred clusters. ) )

7 Comact mathematcal formulaton for Grah Parttonng 7 Formulaton Constrants Varables Bnary Non null coeffcents varables A (standard) V + 3 E K + V K + E K V K 2 V K + 7 E K B (A wth fewer constrants) K V + 2 E K + K B2 (B for K=2) 2 E + K C (bnary reresentaton) D (comact bnary reresentaton) Table 1. Sze of formulatons V (1+K+Klog(K)) + 2 E log(k) + K V (1+3(K-1-log(K))) + 2 E log(k) + 2K V K + E V + E V (K+log(K)) + E V (K-1) + E + K V K 2 V K + 6 E K V V K + 6 E V log(k) V (log(k)+2k+3klog(k)) + 6 E log(k) V log(k) V (2log(K) + 8(K-1-log(K))) + 6 E log(k) + K(K+1) Formulaton B s slghtly more comact than formulaton A. In the case of bsecton, formulaton B2 and D are almost dentcal. In ths case, usng only one bnary varable for cluster assgnment s actually the same as usng the bnary reresentaton of cluster ndexes. Thus, formulaton D can be seen as a generalzaton of formulaton B2 for greater K. Formulaton C s not nterestng. Ths s only an ntermedate ste to ntroduce formulaton D. Comarson of the sze of formulatons B and D The comactness of formulatons B and D vares n ooste way for V and E. E For the number of constrants: constr( D) constr( B) 1 + V 2 K log( K) log( K) 2V 1 K If E 3/2 V, the number of constrants n formulaton D s smaller than n formulaton B, and when E / V gets larger, the reducton factor n the number of constrants becomes close from log(k)/k. The number of varables s smlar n the two formulatons. However, the number of bnary varables s reduced wth a factor log(k)/k n formulaton D. E 4 1 K + 1 For the non null coeffcents: coef ( D) coef ( B) 1 + V 3 K log( K) log( K) 6V 1 K If E V, the number of non null coeffcents n formulaton D s smaller than the number of non null coeffcents n formulaton B, and when E / V gets larger, the reducton factor n the number of non null coeffcents becomes close from log(k)/k. As a concluson, formulaton D s more comact than formulaton B f E 3/2 V, and when E / V gets larger, the comactness reducton factor becomes close from log(k)/k for the number of constrants, of non null coeffcents and of bnary varables. Table 2 dslays a numercal examle for V =100, E =1000 and K=10. It shows a sgnfcant decrease n sze n the new formulaton for every dmenson of the roblem. Formulaton Constrants Varables Bnary varables Non null coeffcents A B D Table 2: Sze of formulatons for V =100, E =1000 and K=10 Remar: The value of the lnear relaxaton of formulatons A, B, B2, C and D s all edges of the grah. W j, j,.e. the sum of the weghts of

8 Comact mathematcal formulaton for Grah Parttonng 8 3. Resoluton wth a solver We comare the dfferent formulatons on the bass of numercal exerments by alyng a solver on a benchmar. 3.1 Benchmar nstances The nstances used for tests were ntroduced by (Johnson, Aragon, McGeoch and Schevon 1989). These nstances fall nto two categores: random grahs (resented n table 3) and random geometrc grahs (resented n table 4). In random grahs, edges are randomly generated n order to reach an average vertex degree. In random geometrc grahs, vertces are randomly located n square [0, 1]x[0, 1] and edges are created f ther length s less than a dstance d (d comuted n order to reach a gven average vertex degree). All weghts of edges are set to 1. V \ degree G G G G G G G G G G G G G G G G Table 3. Random grahs V \ degree U U U U U U U U Table 4. Random geometrc grahs 3.2 Resoluton envronment We used Clex 6.5 lbrary routnes to solve the mxed nteger formulaton drectly. For lnear relaxaton, we used the smlex dual method that roved much faster than the smlex rmal method for all formulatons. For exact resoluton (mxed nteger rogrammng), we actvated the roundng heurstc to set the ntal value of bnary varables, and the otons of memory swang of the branch and bound tree, not to be lmted by the hyscal memory of the machne. The target machne s a PC Dell Worstaton 400 wth Pentum II 300 MHz and 256 MB RAM, oeratng under Wndows/NT 4.0 system. Ths machne s rated 12.2 n SPECnt95 benchmar. 3.3 Slghtly stronger formulaton We can restrct the cluster ndexes of the vertces n the followng way: the frst vertex must belong to frst cluster the second vertex must belong ether to frst cluster, ether to second cluster the th vertex must belong to one of the frst clusters These constrants can be added to each formulaton: Each vertex wth ndex <K s assgned to a cluster wth ndex less than b 2 0 K 1 Wth these new constrants, the formulatons are less degenerated because ermutatons of vertex assgnment to clusters are now constraned. Exerments showed that the lnear relaxatons of formulatons were mroved wth only a few ercent comared wth the revously establshed relaxaton value. Exact resolutons were besdes twce faster thans to these new constrants. 3.4 Comarson of formulatons n lnear relaxaton resoluton The value of the bound obtaned wth lnear relaxaton s not reorted here. The only crteron used for comarson s CPU tme n seconds. We used random grah G that have 124 vertces and an average vertex degree 10 (620 edges) for ths comarson, whose results are dslayed n table 5.

9 Comact mathematcal formulaton for Grah Parttonng 9 Formulaton 3 clusters 4 clusters 5 clusters 6 clusters 8 clusters 16 clusters A B C D Table 5. CPU tme for -arttonng of nstance G wth lnear relaxaton Formulaton B s u to twce slower than formulaton A, although t s slghtly more comact. Formulaton D, based on bnary reresentaton of cluster ndexes, s much faster than standard formulaton A, wth a CPU tme reducton factor larger than ts comactness reducton factor log(k)/k 3.5 Comarson of formulatons n exact resoluton Exact resoluton s tractable wth all formulatons only wth small sze nstances. We frst evaluated all formulatons on two 2-arttonng and one 3-arttonng roblem. Results dslayed n table 6 show that standard formulaton A s clearly the slowest one. Formulaton D, whch s almost dentcal to formulaton B2, s the fastest on bsecton roblems. Surrsngly, n ste of ts comactness, formulaton D based on the bnary reresentaton of cluster ndexes s less effcent than formulaton B on the 3-arttonng roblem Formulaton Grah G clusters Grah G clusters Grah U clusters A B B C D Table 6. Exact resoluton CPU tme for 2 and 3-artonng of nstances G and U In order to confrm that trend, we roceeded wth other exerments of -arttonng and comared formulatons B and D on small random grahs wth 30 to 50 vertces, tractable wthn reasonable CPU tme. Table 7 reorts the number of bnary varables and the CPU tme for formulatons B and D, for each -arttonng roblem. In ste of ts comactness, formulaton D s always less tme effcent than formulaton B on general -arttonng roblems. Tests wth other solvers, usng dfferent resoluton strateges, would be necessary to confrm these results. Grah( V x E ) Clusters Formulaton B Formulaton D Bnary varables CPU tme Bnary varables CPU tme G1(24x26) G1(24x26) G1(24x26) G2(28x66) G2(28x66) G3(47x60) G3(47x60) Table7. Exact resoluton CPU tme for several -arttonng roblems on three small random grahs 3.6 Otmum soluton for large grahs Deste the dsaontng results for general -arttonng roblem, the resoluton tmes obtaned wth bsecton roblems were romsng. Ths was the reason for conductng new exerments to obtan as many as ossble otmum solutons on benchmar grahs. These grahs are large sze grahs whch are used to evaluate and to comare heurstc solvng methods. The urose of these last exerments s not to evaluate the dfferent formulatons, but to fnd otmal solutons on benchmar grahs. That s why we chose what seemed to be the fastest formulaton when used wth our straghtforward mlementaton n Clex solver. For bsecton roblems, we used formulaton B2, whch s almost dentcal to formulaton D n ths case. For -arttonng roblems, we chose formulaton B.

10 Comact mathematcal formulaton for Grah Parttonng 10 Bsecton The best nown values come from (Johnson, Aragon, McGeoch, Schevon 1989) for small nstances ( V 250) and from (Batttt and Bertoss 1999) for large nstances ( V 500). These uer bounds corresond to the best values obtaned by any heurstc method. Resolutons wth formulaton B2 were conducted on art of the nstances. Some of these resolutons were erformed untl otmalty was reached. Some other ones were stoed f otmalty was untractable. The last ones were not conducted at all when smaller smlar nstances were not solvable. The results are dslayed n the tables 8 and 9. The otmal cuts found wth exact resoluton are n bold face, wth the CPU tme (seconds). For the other nstances, the best nown uer bounds found by heurstc methods are reorted. V \ degree Cut CPU Cut CPU Cut CPU Cut CPU Table 8. Results for bsecton of random grahs wth formulaton B2 V \ degree Cut CPU Cut CPU Cut CPU Cut CPU Table 9. Results for bsecton of random geometrc grahs wth formulaton B2 Otmal cuts were obtaned on 9 out of the 24 nstances. These results obtaned wth formulaton B2 rove the otmalty of the corresondng uer bounds revously reached by heurstc methods. These new results (roven otmalty on 9 nstances) rovde an absolute crteron to evaluate heurstcs. It s surrsng that such large nstances could be solved to otmalty. The reason s that the solved nstances have a very low cut, and that ther lnear relaxaton s n ths case, very close to the otmum. Comutaton tme needed to rove otmalty ncreases qucly wth the sze of the grahs and the value of the cut. Random geometrc grahs are easer to solve than random grahs wth the nteger rogrammng formulaton. These grahs are more structured and thus the soluton sace s less degenerated than n the case of random grahs. Ths leads to a qucer exloraton of the branch and bound tree durng the resoluton. K-arttonng Otmal values were obtaned only wth two small random grahs, usng formulaton B. Results are reorted n tables 10 and 11 Grah G Otmal cut CPU tme 2 clusters clusters clusters clusters Table10. Results for -arttonng of nstance G wth formulaton B Grah G Otmal cut CPU tme 2 clusters clusters Table 11. Results for -arttonng of nstance G wth formulaton B These results rovde a benchmar for the -arttonng roblem. Comutaton tme ncreases extremely qucly wth the number of clusters. We used formulaton B because t was more effcent than formulaton D wth Clex solver. Formulaton D stll remans very romsng because of ts reduced sze (artcularly ts reduced number of bnary varables). We exect that usng formulaton D wth other solvers or wth a secally desgned resoluton method could mrove sgnfcantly the effcency of the resoluton, and therefore brng new otmum values.

11 Comact mathematcal formulaton for Grah Parttonng Qualty of the state of the art heurstcs The grah bsecton roblem has been extensvely studed n the ast, and many heurstcs have been exermented. For examle, algorthms such as (Kernghan-Ln 1970) or (Fducca-Mattheyes 1982) are frequently used to locally mrove bsectons. Many meta-heurstcs have also been used such as smulated annealng (Kratrc, Gellat and Vecch 1983) evaluated by (Johnson, Aragon, McGeoch and Schevon 1989), genetc algorthms used by (Bu and Moon 1996) or tabu search (Glover 1989) enhanced and adated to bsecton roblem by (Battt and Bertoss 1999). The multlevel aroach s secally ftted to very large grahs and constraned comutaton tme. It has been resented and studed by (Hendrcson and Leland 1995), (Monen and Demann 1997), (Pellegrn and Roman 1996), (Karys and Kumar 1998). These famles of heurstcs reresent a range of otons for the trade-off between comutaton tme and qualty of the soluton. In the lght of the 9 otmal solutons roven n ths aer, we shortly revew the results of some of the revously resented heurstcs, from the ont of vew of the qualty of solutons. The multlevel aroach that favors extremely short comutaton tme at the exense of the qualty of solutons s not evaluated. The frst evaluaton deals wth the results of (Johnson, Aragon, McGeoch and Schevon 1989), that ntroduced the benchmar nstances studed n ths aer and extensvely studed heurstcs based on smulated annealng. Table 12 comares otmal values wth best values found by the authors n 1989 wth ther smulated annealng based method. These early wors allowed to reach 6 of the 9 otmum cut szes. The 3 non otmal values are ndcated wth stars n table 12. Grah Otmal cut Best Smulated Annealng cut sze G G G G *52 U *4 U U U *3 U Table 12. Otmal versus best values for annealng method The second evaluaton s based on results resented by Bu and Moon 1996 for ther method based on genetc algorthms (BFS-GBA), and Battt and Bertoss 1999 for ther method based on tabu search (RRTS). The nstances wth less than 500 nodes are now consdered as too easy and are not taen nto account by the communty. Comutaton tmes have been scales wth resect to SecINT95 to be comarable. We comare the mnmal and average cuts of 1000 heurstc runs wth the otmal cut. Ths evaluaton shows that state of the art heurstcs have reached the otmal cut szes for these 6 nstances. The average cut sze of RRTS method s otmal for all nstances, excet for G Comutaton tme needed to rove the otmal soluton wth the exact resoluton method s about 1000 tmes that of heurstc methods. Grah Otmal resoluton BFS-GBA RRTS, 1000 n ter Otmal CPU Mn Average CPU Mn Average CPU G ,97 0,22 *51 52,06 0,98 U ,65 0, ,83 U ,68 0, ,32 U ,58 0, ,59 U ,78 0, ,05 U ,78 1, ,03 3,08 Table 13: Otmal versus best nown values for two state of the art methods based on genetc algorthms and tabu search Concluson The new formulaton of -arttonng roblem s more comact wth an average reducton factor of log(k)/k comared wth the standard formulaton. In artcular, the number of bnary varables used n the new formulaton s exactly reduced by a factor log(k)/k. Ths comact formulaton s romsng and could be the bass for effcent exact

12 Comact mathematcal formulaton for Grah Parttonng 12 resoluton methods or boundng methods. A straghtforward use of a commercal solver was conclusve for bsecton roblems, but resulted nto an ncreased comutaton tme for the resoluton of general -arttonng roblems. Ths new formulaton s nterestng, but further wor wll be necessary to conduct exerments wth other solvers or to desgn a dedcated exact resoluton method that could tae advantage of the comactness of the formulaton. Exerments were conducted on some large sze benchmar grahs, classcally used to comare heurstc methods. For all these nstances, revous wor reorted best nown values,.e. uer bounds. These exerments, based on the new formulaton for bsecton and on an almost standard formulaton for general -arttonng roblems, allowed to obtan more than 10 otmum values on nstances u to 1000 vertces and 5000 edges. These results rovde a strong evaluaton crtera to comare heurstc methods. References R. Battt and A.A. Bertoss, Greedy, Prohbton, and Reactve Heurstcs for Grah Parttonng, IEEE Transactons on Comuters, vol. 48, no T. N. Bu and B. R. Moon, Genetc Algorthm and Grah Parttonng, IEEE Transactons on Comuters, vol. 45, no C. Fducca and R. Mattheyses, A Lnear Tme Heurstc for Imrovng Networ Parttons, Proc. 19 th ACM/IEEE Desgn Automaton Conf., Las Vegas, M.R Garey, D.S. Johnson and L. Stocmeyer, Some smlfed NP-comlete grah roblems, Theoretcal Comuter Scence, 1, F. Glover, Tabu Search Part I, ORSA I Comutng, vol1, no3, B. Hendrcson and R. Leland, A multlevel algorthm for arttonng grahs, Proc. Suercomutng 95, ACM. D.S. Johnson, C.R. Aragon, L.A. McGeoch and C. Schevon, Otmzaton by Smulated Annealng: An Exermental Study; Part 1, Grah Parttonng, Oeratons Research, vol. 37, G. Karys and V. Kumar. (1998). A Fast and Hgh Qualty Multlevel Scheme for Parttonng Irregular Grahs, SIAM J. on Scentfc Comutng, to aear. B. Kernghan and S. Ln, An Effcent Heurstc Procedure for Parttonng Grahs, Bell Systems Techncal J., vol. 49, S. Kratrc, C.D. Gellat Jr. and M.P. Vecch, Otmzaton by Smulated Annealng, Scence, vol. 220, no. 4598, B. Monen and R. Demann, A Local Grah Parttonng Heurstc Meetng Bsecton Bounds, Proc. Eghth SIAM Conf. Parallel Processng for Scentfc Comutng. F. Pellegrn and J. Roman, Scotch: A Software Pacage for Statc Mang by Dual Recursve Barttonng of Process and Archtecture Grahs, Proc. HPCN 96 Brussels,