Approximating Clique and Biclique Problems*

Transcription

1 Ž. JOURNAL OF ALGORITHMS 9, ARTICLE NO. AL Approxmatng Clque and Bclque Problems* Dort S. Hochbaum Department of Industral Engneerng and Operatons Research and Walter A. Haas School of Busness, Unersty of Calforna, Berkeley, Calforna 9470 E-mal: Receved August 1, 1997; revsed September 3, 1997 We present here -approxmaton algorthms for several node deleton and edge deleton bclque problems and for an edge deleton clque problem. The bclque problem s to fnd a node nduced subgraph that s bpartte and complete. The obectve s to mnmze the total weght of nodes or edges deleted so that the remanng subgraph s bpartte complete. Several varants of the bclque problem are studed here, where the problem s defned on bpartte graph or on general graphs wth or wthout the requrement that each sde of the bpartton forms an ndependent set. The maxmum clque problem s formulated as maxmzng the number Ž or weght. of edges n the complete subgraph. A -approxmaton algorthm s gven for the mnmum edge deleton verson of ths problem. The approxmaton algorthms gven here are derved as a specal case of an approxmaton technque devsed for a class of formulatons ntroduced by Hochbaum. All approxmaton algorthms descrbed Žand the polynomal algorthms for two versons of the node bclque problem. nvolve calls to a mnmum cut algorthm. One concluson of our analyss of the NP-hard problems here s that all of these problems are MAX SNP-hard and at least as dffcult to approxmate as the vertex cover problem. Another concluson s that the problem of fndng the mnmum node cut-set, the removal of whch leaves two clques n the graph, s NP-hard and -approxmable Academc Press Key Words: Approxmaton algorthm; half ntegralty; node deleton; edge deleton; bclque; maxmum clque 1. INTRODUCTION We present here new approxmaton algorthms based on a technque recently ntroduced by Hochbaum Hoc96. The technque reles on the nteger programmng formulaton of the problem on constrants that nvolve up to three varables per constrant, where one of the three * Research supported n part by Natonal Scence Foundaton award DMI , and by SUN Mcrosystems $5.00 Copyrght 1998 by Academc Press All rghts of reproducton n any form reserved. 174

2 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 175 varables appears only n one constrant. Such problems have approxmaton algorthms easly derved by solvng a certan mnmum cut problem on a related network. The technque may also be used as a tool to dentfy the polynomalty of a problem va the easly recognzed structure of ts constrants, whch we call monotoncty. The collecton of problems explored here s related to the maxmum clque problem and to the bclque problem. A bclque s a complete bpartte graph. The maxmum bclque problem was studed recently by Dawande et al. DKT97. They descrbed nterestng applcatons of fndng the maxmum edge weght subgraph that forms a bclque n bpartte graphs, and proved that the problem s NP-hard. We study ths problem and several other problems of mnmum node deleton or edge deleton so that the remanng subgraph s a bclque. The bclque problems dscussed are lsted n mnmzaton form here: Bpartte edge bclque. Gven a bpartte graph, the problem s to delete the mnmum weght collecton of edges so that the remanng subgraph forms a bclque. General edge bclque. Here the goal s to remove the mnmum weght collecton of edges from a general graph G Ž V, E. so that the remanng subgraph s a bclque. We consder two varants of the problem that are NP-hard. In one varant the edges n each sde of the bclque may reman. In the second varant the nodes on each set of the bpartton must form an ndependent set and be parwse nonadacent n G. Bpartte node bclque. Gven a bpartte graph, the goal s to delete a mnmum weght collecton of nodes, so that the remanng subgraph s a bclque. Ths problem s dentfed as solvable n polynomal tme from the monotoncty of the formulaton. General node bclque. Gven a general graph, the goal s to delete the mnmum weght collecton of nodes so that the remanng subgraph s a bclque. As n the analogue edge problem, the bclque may or may not be requred to have each set n the bpartton ndependent. Wthout ths requrement the problem s shown to be solvable n polynomal tme; wth the requrement t s shown to be NP-hard and -approxmable. Ths latter problem s also equvalent to a problem of a mnmum node separator leavng two clques n a graph. In addton to the bclque problems, we consder an optmzaton-equvalent varant of the maxmum clque problem. Ths problem s to delete mnmum weght collecton of edges so that the remanng subgraph s a clque. The node deleton clque problem s easly seen to be dentcal to a vertex cover problem, and s therefore not dscussed here. The formulaton structure of ths edge deleton clque problem s techncally smlar to that of the edge bclque problems.

3 176 DORIT S. HOCHBAUM Although the bclque problems may seem at frst to be more dffcult than the bpartzaton problem that nvolves deletng nodes or edges so the remanng graph s bpartte, the approxmaton algorthms here are evdence that the opposte s the case: For the edge and node deleton bpartzaton problems, the best approxmaton algorthms known are of factor OŽ log n. ŽGVY96, GVY94., where n s the number of nodes n the graph, whereas all problems dscussed here are -approxmable n polynomal tme. The reader may verfy that n our analyss the completeness restrcton of a bclque plays a role n makng the problem easer. The paper s organzed as follows. We frst revew the relevant technque for dervng approxmaton algorthms for the type of problems dscussed here, IP problems. We then dscuss the node bclque problems, then the edge bclque problems, and fnally the clque problem. We present the full network for several selected problems. One consequence of our analyss here s that snce all NP-hard IP problems are also at least as hard to approxmate as vertex cover Hoc96a, then the problems addressed here are MAX SNP-hard and can be approxmated by a factor better than only f vertex cover has such approxmaton. To date, no such approxmaton s known, and t has been conectured n Hoc83 that -approxmaton s the best possble for the vertex cover problem. Notaton We use ether Ž,. or, 4 to denote an undrected edge. For a graph G Ž V, E. and a vertex V let NŽ. u Ž u,. E 4, the set of neghbors of. Let n NŽ., Nu, Ž. Nu Ž. NŽ.. We refer throughout to a bpartton as the two subsets of nodes that serve on each sde of the bclque or any type of bpartte graph. We wll use sans-serf acronyms to refer to formulatons, and roman letters n the reference to problems.. THE IP ALGORITHM: AN APPROXIMATION TECHNIQUE A class of nteger programmng formulatons wth up to three varables per nequalty, called IP, was analyzed for approxmatons n Hoc96. Whle any lnear optmzaton problem can be wrtten wth at most three varables per nequalty, the dstngushng feature of IP formulatons s that two of the three varables Ž the so-called x-varables. may appear any number of tmes n other constrants, but the thrd one Ž the z-varable.

4 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 177 may appear only once. An IP problem s formulated as Mn Ý n 1 wx Ýez subect to ax bx k c dz for 1,...,m Ž IP. l x u 1,...,n z nteger 1,...,m x nteger 1,...,n. It s assumed that among the constrants coeffcents, the values of d are ntegers. All other entres may attan arbtrary ratonal values. The range of values of the varables U max Ž u l. 1,...,n s an mportant parameter n the complexty of algorthms for IP. For the bclque and clque problems, the varables assume no more than three values, and the value of U s thus fxed. It s thus assumed throughout the dscusson n ths secton that the value of U s fxed. A crucal property of some IP problems s monotoncty. DEFINITION 1. An nequalty ax by c dz s monotone f a, b 0 and d 1. Indeed, as proved n Hoc96, monotone IP problems are solvable n polynomal tme: THEOREM.1 ŽHoc96.. An IP problem on monotone constrants s solable n ntegers n the tme requred to sole a mnmum cut Žor maxmum flow. problem on a graph wth OŽ n. nodes and OŽ m. edges. A monotone IP wth all constrant coeffcents n 1, 0, 14 s also totally unmodular. That means that all of the subdetermnants of the constrant matrx assume values n 1, 0, 14, and n partcular, that all extreme ponts of the feasble solutons polytope are ntegral. Such IP problems can therefore be solved usng any lnear programmng algorthm, and the optmal Ž basc, or extreme pont. soluton s guaranteed to be nteger. Some of the problems dscussed here are almost monotone, n the sense that the frst part of the monotoncty requrement wth respect to the x-varables apples. We shall call ths form of restrcted monotoncty monotone wth respect to the x-arables. For such problems the volaton of monotoncty s n the z-varables appearng n more than one constrant, or havng coeffcents, d, not equal to 1: COROLLARY.1. x-arables. Consder an IP problem monotone wth respect to the

5 178 DORIT S. HOCHBAUM Ž. If the z-arables appear up to p tmes, then there s a polynomal tme algorthm attanng a superoptmal soluton wth the x-arables ntegral and the z-arables nteger multples of 1p. Ž. If the z-arables appear wth coeffcents d wth D max d 1, then there s a polynomal tme algorthm attanng a superoptmal soluton wth the x-arables ntegral and the z-arables nteger multples of 1D. Wth Theorem.1 the proof s straghtforward. For Ž. each occurrence of a z-varable s nterpreted as a dfferent varable, z Ž., and the cost of each such occurrence s Ž 1p. cz, Ž. where cž z. s the cost coeffcent of z n the obectve functon. The resultng system s monotone and solvable n Ž. p Ž. ntegers where the value of z s then set equal to z 1p Ý1 z. For Ž. the varable z s substtuted by z DZ, and the problem s solved n nteger z as a monotone problem. The value of z s set to z Ž 1D. z for z nteger. Although the general IP s NP-hard, t s solvable n polynomal tme n half ntegers n the x-varables. That soluton s a lower bound to the nteger optmum and thus s a superoptmal soluton. Not only s the bound of better qualty compared to a bound derved from a lnear programmng relaxaton; t s also obtaned by usng a combnatoral algorthm of mnmum cut on graph that runs n strongly polynomal tme and more effcently than a lnear programmng algorthm. Such a superoptmal soluton s useful n approxmaton algorthms. The reader s referred to Hoc96 for addtonal detals. In ths paper we show that for all problems dscussed we can derve a superoptmal half ntegral soluton that can be rounded to a -approxmate soluton. The case of IP problems wth only two varables per nequalty was analyzed n HN94 and n HMNT93. Hochbaum and Naor HN94 devsed a polynomal tme algorthm to solve the monotone problem n ntegers, when the coeffcents a, b n constrant are of opposte sgns. Hochbaum et al. HMNT93 descrbed a polynomal tme -approxmaton algorthm for the nonmonotone verson whch s NP-hard. Problems that are IP wth no more than two varables per nequalty always have a feasble roundng leadng to a -approxmaton, provded that the problem has a feasble nteger soluton. Ths property s not shared wth problems that have three varables per nequalty. But f a feasble roundng exsts, t frequently leads to a more effcent approxmaton algorthm. Several examples of ths type are llustrated n ths paper. In the complexty expressons we let TŽ n, m. be the tme requred to solve a mnmum s, t cut problem on a graph wth m arcs and n nodes. Ž. Ž Ž T n, m may be assumed to be equal to Omnlog n m.., whch s the complexty of the push-relabel algorthm wth dynamc tree data structure

6 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 179 GT88. When we refer to half ntegral solutons, these are feasble solutons wth all components that are nteger multple of half. THEOREM. ŽHoc96.. Assume that n the gen IP problem, d 1 and U. Then, a superoptmal half ntegral soluton to the IP problem s attaned n tme TŽ n, m.. The half ntegral soluton resultng from the soluton s used to derve a -approxmate soluton by roundng ts components to an nteger feasble soluton for each of the problems dscussed. The -approxmaton algorthms presented n ths paper are specal cases of Theorem.. The technque for solvng the IP problem n nteger multples of half nvolves transformng the formulaton to another formulaton where the constrants are monotone and ther coeffcents form a totally unmodular matrx. That, n turn, s solvable n polynomal tme n ntegers. The transformaton s such that only a factor of s lost n the ntegralty of the x-varables f the orgnal formulaton was nonmonotone Žfor monotone formulatons there s no loss of ntegralty.. That s, when the nverse transformaton s appled, every nteger value of the varable s mapped to a half nteger. Ths technque wll be llustrated n detal for some of the problems dscussed here. The constructon of the networks descrbed here follows the method ntroduced n Hoc96. To facltate the decpherng of the networks descrbed, we menton only that each node s assocated wth some bnary choce of values, and the rule of dentfyng a node value s to set t at ts upper bound f and only f t s n the source set of a cut. 3. THE NODE BICLIQUE PROBLEM 3.1. The Bpartte Node Bclque Problem The node bclque problem on a bpartte graph s solvable n polynomal tme. Ths was frst observed by Yannakaks Yan81b. The problem s equvalent to the maxmum ndependent set on bpartte graphs that s known to be solvable by a mnmum cut algorthm. The polynomalty s also evdent from the formulaton that s monotone and thus solvable n polynomal tme n ntegers. To see ths, consder a Ž maxmzaton. formulaton of the problem Žprevously gven n DKT97. gven on the bpartte graph B Ž V, V, E. 1. Let x 1 f node s n the bclque. Ž BNB. Max Ý Vwx subect to x x 1 for edge, 4 E, V 1, V x 0, 14 for all V.

7 180 DORIT S. HOCHBAUM The constrants each nvolve two types of varables, those representng nodes n V1 and those representng nodes n V. Thus multplyng one of these sets, say the varables n V, by 1 gves a formulaton that s monotone. The network constructed for solvng such a formulaton s a bpartte network wth only source and snk-adacent arcs havng fnte capacty. The network s depcted n Fg. 1. The formulaton of BNB s dentcal to the formulaton of the ndependent set problem on the bpartte complement B Ž V, V, E. 1. We sketch for the sake of completeness the basc dea of usng a mnmum cut algorthm for solvng the ndependent set problem on bpartte graphs. The mnmum s, t-cut problem correspondng to the ndependent set problem s defned on a network where all nodes n V1 are lnked to the source wth arcs of capactes us, w, and all nodes n V are lnked to a snk t wth arcs of capactes u, t w. All edges n the bpartte graphs are represented as drected arcs from V1 nodes to V nodes wth nfnte capacty. Gven a fnte capacty cut Ž S, T., the set Ž S V 1. Ž T V. s an ndependent set of weght Ý w Ž V V Ý V T us, 1 1 Ý u. V S, t. Thus the weght of the ndependent set s equal to a constant mnus the weght of the correspondng fnte cut. Mnmzng the capacty of the cut s therefore equvalent to maxmzng the weght of the ndependent set. Although not mmedately evdent, ths algorthm s a specal case of the algorthm for solvng problems on two varables per nequalty of HMNT93, whch generates a network dentcal to that n Fg. 1. The correspondng node deleton problemthe deleton of mnmum weght collecton of nodes so that the remanng bpartte graph s com- FIG. 1. The network used to solve the bpartte node bclque Ž BNB. problem.

8 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 181 pletes obvously solvable n polynomal tme as well. The optmal soluton s the complement of the ndependent set correspondng to the mnmum cut. 3.. General Node Bclque, wthout the Independence Requrement The general node bclque problem s to fnd n a general graph G Ž V,E., a node-nduced subgraph that forms a bclque n that t defnes two dsont subsets of nodes, V 1, V V, that nclude all edges between V1 and V, V1 V. In ths subsecton we consder the verson n whch the bclque s not requred to have the nodes on each sde of the bpartton parwse nonadacent. We provde two dfferent formulatons for ths problem, wth the frst havng two varables per nequalty, and the second havng three varables per nequalty. The second formulaton leads to a more effcent algorthm and s more useful n extensons to other formulatons dscussed here. In the frst formulaton the obectve s to maxmze the weght of the nodes n the bclque. It s shown that the formulaton s monotone, and thus the problem s solvable n polynomal tme Formulaton 1 Let each node have three possble states ndcated by the values 1, 0, 1. The values 1 and 1 mply that the node s n the bpartton, and specfy whch sde of the bpartton t s n. The value 0 mples that node s not n the bclque. A node contrbutes to the obectve functon f t s n the bclque, or f ts value s 1 or 1. The varable y Ž1. s equal to 1 when x 1 and y Ž. 1 when x 1. One trval feasble soluton s a sngle edge, the endponts of whch form a bclque. The formulaton s gven for each possble choce of such edge s, t4 E. The formulaton s gven frst for the maxmzaton problem. It models the maxmum node bclque Ž MNB. problem on general graphs condtoned on a gven par s, t beng n the bclque. Ž MNB( s, t) 1. Ž1. Ž. Max Ý V wy Ý V wy subect to Ž1. 1 x y for all V Ž. 1 x y for all V x x 1 for edge, 4 E x x 1 for edge, 4 E x s 1, xt1. Ž1. Ž. x 1,0,1 4, y, y bnary for all V.

9 18 DORIT S. HOCHBAUM The frst two sets of constrants ensure that nodes contrbute weght to the bclque only when they are selected on ether sde of the bpartton. LEMMA 3.1. The formulaton ŽMNBŽ s, t.. s monotone, and Ž MNB. 1 s Ž Ž Ž n thus solable n ntegers n O m T n,. m... Proof. The formulaton has up to two varables per nequalty, and ther coeffcents are of opposte sgns. To see ths, we replace the varables y Ž1. by ther negatves y Ž1. y Ž1.. The varables y Ž1. thus assume values n 1, 04. Theorem.1 s applcable to ths monotone formulaton. The optmal nteger soluton can therefore be generated n polynomal tme, by constructng a mnmum cut soluton on a certan network. The network for the mnmzaton verson, DNBŽ s, t. 1, s gven n Fg.. The runnng tme for solvng the problem s m tmes the complexty of Ž n mnmum cut on a graph wth n nodes and m. E arcs. The overall Ž Ž Ž n complexty s thus OmT n,. m... To speed up the runnng tme, one could employ deas smlar to those used by Hao and Orln s algorthm for mnmum cut n drected networks HO94. That algorthm nvolves swtchng the dentty of the snk, yet t s necessary also to adapt t to swtchng the dentty of the source. Such an adaptaton was recently presented by Henznger et al. n the context of node connectvty HRG96. Indeed, the node bclque problem s closely related to the node connectvty problem, as we show next. The complementary problem to the maxmum node bclque s the mnmzaton of weght of nodes deleted so that the remanng subgraph s Ž Ž.. FIG.. The network used to solve DNB s, t. 1

10 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 183 a bclque. Ths mnmzaton problem s, of course, also solvable n polynomal tme. The formulaton of the deleton node bclque problem Ž DNB. correspondng to the formulaton ŽMNBŽ s, t.. 1 has one shortcomng the obectve value s not qute the weght of the deleted nodes. Here the value of the nteger obectve s Ý V w Ý x 0 w, whch dffers from the desred obectve by a constant, W Ý V w. The correspondng set of varables for the deletonmnmzaton problem ncludes the varables y Ž1. 1f x 0or1, and y Ž. 1f x 1or0. Ž DNBŽ s, t. 1. Ž1. Ž. Mn Ý V wy Ý V wy subect to Ž1. 1 x y for all V Ž. 1 x y for all V x x 1 for edge, 4 E x x 1 for edge, 4 E x s 1, xt1. Ž1. Ž. x 1,0,1 4, y, y bnary for all V. A descrpton of the network s gven n Fg.. For each node we have four nodes n the network. One, ndcated by x 0, mples that x 0f the node s n the source set of a mnmum cut. Smlarly, the node ndcated wth x 1 mples that x 1 f ths node s n the source set. The two other nodes correspond to y Ž1. and y Ž., and attan ther upper bound, of 0 and 1, respectvely, f the nodes are n the source set. The network has On Ž. nodes and Om Ž n. edges. The DNB problem s Ž Ž Ž n thus solvable n tme OmT n,. m... Although the problem s n P as a consequence of ts monotoncty, t turns out that there s another explanaton for the polynomalty: LEMMA 3.. The deleton node bclque problem, DNB, wthout the ndependence restrcton s equalent to the weghted node connectty problem on the complement graph. 1 Proof. A graph s complete bpartte Ž wthout ndependence restrcton. f and only f ts complement s dsconnectedthe two sdes of the bpartton form unons of connected components n the complement graph. So, the mnmum number of nodes whose deleton leaves a complete bpartte subgraph s equal to the node connectvty of the complement graph. 1 We gratefully acknowledge M. Yannakaks for pontng ths out.

11 184 DORIT S. HOCHBAUM The equvalence of DNB to the node connectvty problem permts the use of the mplementaton descrbed n HRG96 to solve DNB n tme Ž Ž Ž n O T n,. m.. 1, where 1 mn s the unweghted node connectvty of G. The drected verson of the problem s also easy to represent: to formulate the drected node connectvty problem on G Ž V, A., we replace the pars of constrants, x x 1 for edge, 4 E x x 1 for edge, 4 E, by the sngle constrant, x x 1, for Ž,. A. The drected node connectvty s equvalent to a drected node bclque problem wth a complete set of arcs drected from one sde of the bpartton to the other Formulaton Ths alternatve formulaton for the node bclque problem has the advantage of havng an exact obectve. The formulaton reles agan on the argument that the optmal bclque contans at least one par of nodes n the graph and the edge that lnks them. Ths tme, however, we take advantage of the restrcton to nclude nodes s and t n the clque by removng a pror all nodes that cannot be present n the same bclque wth these two nodes. For a par of adacent nodes s, t n the bclque, we consder the subgraph nduced by the neghbors of s, Ns, Ž. that contan t, and the neghbors of t, Nt, Ž. that contan s. Any bclque contanng s and t must have each sde of ts bpartton contaned n Ns Ž. and Nt, Ž. respectvely. Ths constructon appears to reduce the general graph problem to the bpartte verson on Ns, Ž. Nt. Ž. Ths s not the case, however, as there are nodes n Ns, Ž t. Ns Ž. Nt Ž. that are canddates for ether sde of the bpartton. The constructon of the nduced bpartte graph has Ns Ž. for one sde of the bpartton and Nt Ž. for the other. The nodes n Ns, Ž t. appear on both sdes, wth each node duplcated as n Ns Ž. and n Nt, Ž. and each copy havng all edges between and all of the nodes adacent to t on the opposte sde of the bpartton. It s mportant to note that there s no edge between and to prevent both copes from beng present n the bclque. We choose here decson varables x and y that are bnary. The varable x 1 f the node Ns Ž. Nt Ž. Ns, Ž t. s deleted from the graph and the varable y 1 for Ns, Ž t. f the node s deleted from the graph. The determnaton of the sde of the bpartton that belongs

12 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 185 to follows mmedately from ts membershp n ether Ns Ž. or Nt. Ž. The challenge s to make sure that a node that appears on both sdes Žbecause t s n Ns, Ž t.. wll not be charged for unless t appears on nether sde of the selected bclque, and then charged for only once for ts deleton. Ths s acheved frst by settng a constrant x x 1 for Ž,. E, whch apples n partcular to the par, as x x 1, thus ensurng that at least one of the copes t deleted. Second, a node n Ns, Ž t. s consdered deleted only f both copes of the node are deleted and the correspondng varables values are 1, n whch case the value of the correspondng y varable s 1. Let V Ns Ž. Nt, Ž. s, t and Es, t be the set of edges wth both endponts n V s, t. Ž Ž.. DNB s, t zs, t Mn Ý V NŽ s, t. wx Ý NŽ s, t. wy s, t subect to x x 1 y for Ns, Ž t. x 4 x 1 for edge, Es, t Nt, Ž. Ns Ž. x 0, 14, for V. Remark 3.1. It s optonal but not essental to nclude here the condton that x s xt 0. If one of these two nodes s deleted n the optmal soluton, then the edge Ž s, t. s not a part of an optmal bclque. LEMMA 3.3. The formulaton ŽDNBŽ s, t.. s monotone, and the lnear programmng relaxaton s basc solutons are ntegral. Proof. To see that the formulaton s equvalent to a monotone one, we multply the varables x for n Ns Ž. by 1 so that they attan values n 1, 04. The resultng formulaton s ŽDNBŽ s, t.. zs, t Mn Ý V N wx Ý NŽ s, t. wy s, t Ž s, t.. subect to x. for Ns, Ž t. 1 Ž x y x x 1 for edge, 4 E, Nt, Ž. Ns Ž. x 4 1, 0, for Ns Ž. x 4 0, 1, for Nt Ž. y 0, 14, for Ns, Ž t.. Ths formulaton s now monotone. Therefore a procedure nvolvng mnmum cut s delverng an nteger soluton. Furthermore, the constrant s, t

13 186 DORIT S. HOCHBAUM matrx has all coeffcents n 0, 1, 14 and s monotone and therefore totally unmodular, as shown n Hoc96, and as dscussed n the Introducton, t s a cut polytope. Hence the lnear programmng optmal soluton and all basc solutons are also nteger. Soluton method for general node bclque. We solve for each s, t V such that Ž s, t. E the formulaton ŽDNBŽ s, t... We then choose among the values Ý V V w z, t the smallest value of the relaxaton. Snce s, t the formulaton s monotone, the optmal soluton delvered s n ntegers. The network s gven n Fg. 3. Note that a node n Ns Ž. s n the source set f and only f ts value s 0, and n the snk set f and only f ts value s 1. Ž n Complexty. For each s, t par there are up to. m Ns, Ž t. On Ž. edges n the constructed graph. The complexty s thus m tmes the complexty of solvng a mnmum cut problem on a graph wth On Ž. nodes and On Ž. edges, Om Ž TŽn, n... Ths runnng tme s a constant factor faster than for formulaton 1. The dfference n runnng tme s attrbuted to havng a formulaton wth three varables per nequalty versus the two varables per nequalty nterpretaton n the prevous formulaton. As we shall see, for edge bclque formulatons ths dfferent nterpretaton may Ž Ž.. FIG. 3. The network used to solve DNB s, t.

14 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 187 result n a more sgnfcant gap n the complexty of the approxmaton algorthm General Node Bclque, wth the Independence Requrement Addng the ndependence requrement lends the prevous formulatons nonmonotone: t s necessary to nclude constrants of the type x x 1 1 for, Ns, Ž. or for, Nt. Ž. 1 1 Such constrants are no longer monotone, snce the varables cannot be parttoned nto two dstnct sets so that one set s coeffcents can be made negatve. We verfy that such a partton s mpossble by demonstratng that the general node bclque problem s an NP-hard problem. LEMMA 3.4. The general node bclque problem wth ndependence requrement s NP-hard. Proof. We reduce the ndependent set problem to ths general node bclque problem. Gven an ndependent set problem on a graph G Ž. V, E, we construct a graph G by duplcatng the set of nodes V as V and V and the edges as E and E. Now on every node n V wth every node n V. A bclque n G s any par of ndependent sets n V and V. In partcular, the weght of the nodes n the bclque s maxmzed f the ndependent set n V and the one n V are of maxmum weght. For an alternatve proof that the problem s NP-hard, observe that the bclque subgraph property s heredtary, and as such the complexty argument of Yannakaks Yan81b mples t s NP-hard. To formulate the problem we employ the choce of varables x as bnary varables equal to 1 and only f node s deleted. As before, we construct the bpartte graph on Ns, Ž. Nt Ž. for any choce of adacent nodes s and t. Ths tme, snce each sde of the bpartton must be ndependent, all nodes of Ns, Ž t. are removed from the graph, as they are adacent to both s and t and thus cannot be on ether sde of the bpartton. We let NŽ. s Ns Ž. Ns, Ž t. and NŽ. t Nt Ž. Ns, Ž t., and Vs, t NŽ s. NŽ t.. NBIŽ s, t. Ž. zs, t Mn Ý V wx s, t subect to x x 1 for edge, 4 E, NŽ t., NŽ s. x x 1 for edge, 4 E, NŽ t. x x 1 for edge, 4 E, NŽ s. x 0, 14, for V. s, t

15 188 DORIT S. HOCHBAUM The frst set of constrants says that for any edge mssng n the bpartton, at least one endpont s deleted so as not to volate the complete bpartte requrement. The two other sets of constrants say that for any edge wthn Ns Ž. or Nt, Ž. at least one endpont s not n the bclque, as otherwse the ndependence requrement wll be volated. Complexty and soluton method. There are a couple of alternatve ways of solvng ths problem. In one we monotonze and solve m mnmum cut problems on a graph wth Ž n n. nodes and On Ž. s t edges for a total complexty of Om Ž TŽn, n... Alternatvely, observe that the formulaton ŽNBIŽ s, t.. s that of a vertex cover on a graph contanng the set of edges nduced by NŽ s. and NŽ t. and the complement of the edge set n the bpartton. Each of these m vertex cover problems s -approxmable n polynomal tme. The -approxmaton for the general node bclque problem s the mnmum of Ý V V w zs, t for all pars s and t. It s s, t possble to use Bar-Yehuda and Even s -approxmaton algorthm for vertex cover BYE81, whch runs lnear n tme n the number of elementsedges that need to be covered. Here ths number s On Ž.. The procedure has to be run for each selected edge Ž s, t., and thus the overall complexty s Omn Ž.. The approprate network s depcted n Fg. 4. FIG. 4. The network used to solve the node bclque problem wth ndependence requrement ŽNBIŽ s, t... Here, 4 E, or, NŽ t., or, NŽ s.. The set Vs, t s assumed to contan n nodes. V 1,..., n 4. s, t

16 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 189 Therefore we have a polynomal -approxmaton algorthm for mnmzng node bclque on general graphs. Remark 3.. Consder the clque vertex connectvty problem, whch s to fnd a mnmum weght node separator, the removal of whch leaves two dsconnected clques. That problem s dentcal to the general node bclque wth the ndependence requrement Ž NBI. on the complement graph. Ž To see ths, apply the same arguments as n Lemma 3... The clque vertex connectvty problem s hence NP-hard and -approxmable, as a consequence of the dscusson above. Ths s remarkable n that the node deleton problem that leaves a sngle clque n a graph s equvalent to the vertex cover problem and thus s -approxmable. Here we requre that the deleted node set leaves two clques, and yet the problem s stll -approxmable wthout an ncrease n complexty. In contrast, the node deleton problem to two clques that are not requred to be fully dsconnected s the bpartzaton node deleton problem. For ths problem the best approxmaton factor known to date s Ž. O log n GVY THE EDGE BICLIQUE PROBLEM ON BIPARTITE GRAPHS The edge-weghted bclque problem s to delete from a bpartte graph B Ž V, V, E. 1 a mnmum weght collecton of edges so that the remanng edges nduce a complete bpartte grapha bclque. We refer to ths problem wth the acronym BEB Ž bpartte edge bclque.. Dawande et al. proved that the weghted verson of ths problem s NP-complete by reducton from maxmum clque DKT97. For the sake of completeness, we sketch ths reducton. LEMMA 4.1 ŽDKT97.. Edge bclque on bpartte graph s an NP-hard problem. Proof. The reducton s from the maxmum clque problem defned on a graph G Ž V, E.. Construct a bpartte graph Ž V, V, E. wth the set of edges E Ž u,.ž u,. E, oru 4. The edges of the form Ž u, u. get the weght of 1, and the others the weght of 0. A maxmum weght bclque corresponds to a maxmum clque wth a number of nodes equal to the weght of the bclque. We present two alternatve formulatons of the problem. There s a trade-off between the two formulatons, wth one leadng to a superoptmal half ntegral soluton faster than the other. Yet the slower formulaton provdes a tghter lower bound but the same approxmaton factor.

17 190 DORIT S. HOCHBAUM 4.1. Formulaton 1 Let a node varable x be 1 f node s n the bclque and 0 otherwse. Let varable z be defned for each edge Ž,. E as equal to 1 f the edge s deleted or 0 otherwse. Equvalently stated, z 1 f and only f x x. The followng formulaton of bpartte bclque s an IP: Ž BEB1. Mn ÝŽ,. E cz subect to Ž x x. z for Ž,. E x x 1 for Ž,. E V, V. 1 x, z bnary for all,. The frst set of constrants guarantees that unless an edge has both endponts n the bclque, t must be deleted. The second set ensures that every par of nodes ncluded n the bclque, on opposte sdes of the bpartton, must have an edge between them. Together these constrants say that the set of nodes selected s a bclque, and that edges not n the bclque are deleted. In each constrant there s one node varable that belongs to V1 and one to V. These can thus be made to appear wth opposte sgns. Only the coeffcent of z destroys the total unmodular- ty of the constrant matrx: all entres n a totally unmodular matrx must be 0, 1 or 1. We can thus solve n polynomal tme the problem n nteger x-varables and half ntegral z-varables as n Corollary.1 Ž.. The network and ts constructon are dscussed later n Subsecton Formulaton Usng the same varables as n formulaton 1, we state the problem n an equvalent dsaggregate formulaton: Ž BEB. Mn ÝŽ,. E cz subect to 1 x z for Ž,. E x x 1 for Ž,. E V, V. 1 x, z bnary for all,. Ths formulaton s dentcal to Ž BEB1., except that the frst set of constrants s splt nto twce as many equvalent constrants, each enforcng the requrement that f an endpont of an edge s not n the bclque, then the edge must be deleted. The formulaton s tghter than Ž BEB1., snce a fractonal feasble soluton to Ž BEB. s also feasble for Ž BEB1.,

18 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 191 but not the other way around. Ž BEB. s n general slower to solve. If we cast t as a problem n two varables per nequalty, then the number of nodes n the network created s Om Ž n., as opposed to On Ž. nodes n Ž BEB1.. The number of arcs n the networks s the same for both formulatons: Om Ž n.. However, we can treat Ž BEB. as a formulaton wth up to three varables per nequalty, whle the double appearance of the varable z can be consdered as two dfferent varables as n Corollary.1Ž.. The resultng network correspondng to ths formulaton would then be equvalent to that of Ž BEB1., whch wll be dscussed n detal later. As all coeffcents are n 1, 0, 14, t s not obvous that the constrants of Ž BEB. cannot be wrtten as a monotone system. Ths would be a consequence of the NP-hardness of the problem. We settle ths drectly n the followng lemma: LEMMA 4.. The constrant matrx of ( BEB) s not totally unmodular. Proof. The followng subset of constrants creates a 6-cycle wth correspondng determnant equal to. The constrants nvolve the nodes, V,, V, and the edges Ž,., Ž, E, the edges Ž,., Ž,. E. The sx nequaltes creatng a 6-cycle are x z 1 1 x z 1 x x 1 1 x z 1 1 x z 1 1 x x The determnant of the 6 6 submatrx defned by the coeffcents of these constrants s, and thus the matrx s not totally unmodular Solng BEB1 Ether formulaton leads to a -approxmaton algorthm. We show how ths s done for formulaton 1. To transform the constrants nto a monotone system, we apply a transformaton on the varables: x x for V 1 x x for V q z.

19 19 DORIT S. HOCHBAUM Wth ths transformaton q 0, 4. Substtutng ths requrement by 0 q and nteger, the constrants are of the form Ž relaxed BEB1. Ž. x x 1 for, E V, V. x x q for, E Ž. 1 x 0, 14, x 1, 04 q 0, 1, 4, for all,. The transformed constrants form a relaxaton of BEB1, relaxed BEB1. The constrants coeffcents consttute a totally unmodular matrx: a matrx wth one 1 and one 1 n each row, appended wth the dentty matrx. All extreme ponts of such polytopes are ntegral. Therefore ths problem s solvable n ntegers usng lnear programmng. Havng nteger extreme ponts means that q assumes values n 0, 1, 4 rather than n 0, 4. The nteger optmal soluton to relaxed-beb1 provdes a superoptmal half ntegral soluton to the problem BEB, n whch the x varables assume 1 nteger values and z assume values n 0,, 14. We now show how to solve relaxed BEB1 as a mnmum cut problem on a certan network The Network for Relaxed BEB1 As prevously mentoned, a node n the network belongs n the source set f and only f the correspondng varable value s at the upper bound, whch s 1 for x and 0 for x. Wth ths nterpretaton, a constrant from the second set x x 1s represented by an arc gong from x to x. The arc has nfnte capacty, enforcng the requrement that f x 1, then x 0. A constrant from the frst set, x x q, s represented by an arc from the source 1 to x 1 of capacty c and an arc from x 0 to the snk of the same 1 capacty, c. The gadget used to represent such a constrant, wth cq Ž. representng the cost per unt of q n the obectve, s x x q, q 0,1,4 cž q. cž q. x x " t " s # #

20 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 193 FIG. 5. problem. The network uses to solve the relaxed bpartte edge bclque, relaxed BEB1 All arcs Ž,. adacent to node are consoldated nto one arc from s to x and from x to t. Fgure 5 llustrates the entre bpartte network n whch a mnmum cut corresponds to an optmal soluton to the relaxed BEB1. Ths s proved n the next lemma, whch s a specal case of Theorem.. Ž. LEMMA 4.3. Any fnte cut S, S corresponds to a feasble soluton to relaxed BEB1, and the mnmum cut corresponds to an optmal soluton to relaxed BEB1. Proof. Recall that varables are n the source set f and only f they are at ther upper bound. Namely, x ½ 1 0 x S x S ½ 1 x S x 0 x S. An arc s charged to the cut f t s an arc Ž. Ž., t and x S. s, and x S, or f t s an arc

21 194 DORIT S. HOCHBAUM Consder the four possble cases for the values of the x varables assocated wth nodes and and ther correspondng membershp n source or snk sets. Ž. x x q Arcs n cut S, S Cost of cut arcs 1, S 1, S 0 0 1, S 0, S 1 1 Ž x, t. c 0, S 1, S 1 1 Ž s, x. c 0, S 0, S Ž. Ž s, x, x, t. c Thus the value of the cut s the same as the value of the feasble soluton, and vce versa. The mnmum cut thus provdes the optmal nteger soluton to the problem wth the varables q. The mnmum cut soluton provdes a superoptmal half ntegral soluton n whch only z may be fractonal, whenever q 1. A feasble roundng s acheved by roundng z up. The number of nodes n the network used to derve the half ntegral soluton s n n V V 1 1. The number of arcs s m n1 n, where m s the number of arcs n the complement graph m nn 1 m. We thus have a -approxmaton algorthm of complexty TŽn1 n, n1 n m. for ths NP-hard problem. The readers famlar wth the vertex cover problem may notce that the bpartte network used to solve the bpartte edge bclque approxmately s the same network as would be used to solve the vertex cover problem on a 1 bpartte graph where the weght of node s ÝŽ, k. E c k, half the sum of weghts of the adacent edges. The nonedges Ž those n E. are the ones to be covered. Indeed, ths vertex cover problem s a factor of relaxaton of the edge bclque problem on bpartte graphs: for every nonedge Ž,. E, delete the set of edges adacent to ether or, so that the total cost of the deleted edges s mnmum. If the edges are deleted because of only one endpont, then the cost charged s half of the cost of the edge. Ths renders the soluton a lower bound that s also wthn a factor of of a value of a feasble soluton that s an upper bound. The network for the dsaggregate formulaton Ž BEB. has a node for each varable, for a total of m n nodes, and two arcs for each constrant, for a total of m m arcs. The detaled descrpton of the network s omtted. Interpretng the varables z n the formulaton as a thrd z-varable results n exactly the same network as the one for Ž BEB1. n Fg. 5.

22 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS EDGE BICLIQUE ON GENERAL GRAPHS The am n the edge bclque problem s to delete a mnmum weght collecton of edges from a graph G Ž V, E. so that the remanng edge-nduced subgraph Ž V, V, E. 1 1, forms a bclque. When defned on general graphs the problem has two possble nterpretatons, mentoned earler n the Introducton: Verson 1: Nodes n V and V must form an ndependent set n the 1 Ž. graph G V, E. Verson : V and V may form any subgraph n G, and the edges 1 wthn them need not be elmnated to create a bclque. The two versons are NP-hard, snce the general graph problem generalzes the bpartte case Ž the bpartte case s reducble to the general case.. Verson 1 s also NP-hard because of the ndependence requrement for each sde of the bpartton. To see ths we construct a new graph formed by duplcatng G twce, wth nodes V and V and all edge weghts set to zero, and placng all possble edges between all nodes of V and V wth edge weghts equal to 1. Clearly, any par of subsets of V and V forms a bclque. If each sde of the bclque s requred to be ndependent, the problem s equvalent to maxmzng the sze of an ndependent set n G Formulaton and approxmaton of Verson 1 An optmal edge bclque contans at least one edge. The formulatons are therefore gven for each possble guess of such an edge, Ž s, t. E. The presence of Ž s, t. n the bclque mples that the nodes n one sde of the bclque are n the set of neghbors of s, Ns, Ž. and the nodes on the other sde are n the set of the neghbors of t, Nt. Ž. Let Ns, Ž t. Ns Ž. Nt Ž. as before. Here each sde of the bpartton must form an ndependent set n G Ž V, E.. Nodes of Ns, Ž t. are adacent to both s and t and thus cannot be on the same bclque wth ths par of nodes on opposte sdes. Therefore the canddate nodes for one sde of the bpartton are n the set NŽ s. Ns Ž. Ns, Ž t., and for the other sde the nodes are n NŽ t. Nt Ž. Ns, Ž t.. The edge bclque problem wth the requrement that Ž s, t. s n the bclque s denoted by EBs, t and the optmal value by z s, t. Ths problem s defned on the set of nodes V NŽ s. NŽ t. s, t and the set of edges Es, t that have both endponts n V s, t. All other edges are deleted. The optmal soluton to the edge bclque wll be mnž s, t. E ÝŽ,. E E c z s, t. s, t Let x 1 f node s deleted, s and t be on opposte sdes of the bclque, and z be a bnary edge varable equal to 1 f and only f edge

23 196 DORIT S. HOCHBAUM Ž,. s deleted: Ž (1) EB s,t. zs, t Mn ÝŽ,. E cz s, t subect to x z for Ž,. E s, t, Vs, t x Ž. x 1, Es, t and, NŽ s. x Ž. x 1, Es, t and, NŽ t. x Ž. x 1 for, Es, t NŽ t., NŽ s.. x 0, 14 for Vs, t z 0, 14 for Ž,. E. The frst set of constrants says that all edges adacent to a node not n the bclque are deleted. The second and thrd sets ensure the ndependence of the nodes on each sde of the bclque. The fourth set of constrants ensures that the bclque contans all possble edges and s a complete bpartte graph. Ths formulaton has no more than two varables per nequalty. A -approxmaton algorthm thus follows mmedately by solvng a mnmum cut on a graph on On Ž m. nodes, and OŽŽ n.. edges as n HMNT93. 1 The half ntegral optmal soluton s rounded up for z and for 1 x. Once each nstance for a par Ž s, t. E has been -approxmated wth an obectve value z s, t, and the weght of all deleted edges n E Es, t s added to ths value, the mnmum s selected across all s, t pars. Formally, Ž1. gven a -approxmaton to EB s, t of value z s, t z s, t, the mn Ý c z s a -approxmaton to verson 1 of the Ž s, t. E Ž,. E E s, t s, t edge bclque problem. Remark 5.1. If the frst set of constrants s nterpreted as three varables per nequalty, then the complexty of solvng the problem s mproved, but snce each varable z appears n two constrants, ths leads to a 4-approxmaton algorthm. 5.. Formulaton and Approxmaton of Verson Here the sets Ns, Ž. Nt Ž. are agan potental two sdes of the bclque for a gven adacent par of nodes s, t. In ths problem we permt adacent nodes on each sde of the bpartton. Therefore, unlke the case for verson 1, the nodes n the set Ns, Ž t. are also canddates for ncluson n the bclque. Consequently, t s essental to make sure that only one of the two copes of such nodes s selected. Moreover, some of the edges Žthose that connect nodes on Ns, Ž t.. are duplcated, but should not be charged for twce. There s a charge for such deleted edge only f both copes are deleted. s, t

24 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 197 Let x 1 when node s deleted from the bclque and z 1 when edge Ž,. s deleted: Ž () EB s,t. zs, t Mn ÝŽ,. E Ž NŽ s.nžt.. cz s, t subect to x z for Ž,. E V Ns, Ž t. s, t s, t x x 1 z for Ns, Ž t., Vs, t x x 1 for Ž,. E s, t, Nt, Ž. Ns Ž. x 0, 14 for Vs, t z 0, 14 for all,. The frst set of constrants enforces the deleton of edges adacent to deleted nodes not n Ns, Ž t.. The edges adacent to nodes n Ns, Ž t. are deleted only f both copes of the node are deleted, as n the second set of constrants. The thrd set of constrants ensures the completeness of the bclque. Ths formulaton has up to three varables per nequalty. In the next lemma we show that the formulaton s almost monotone, except that the varables z may appear twce. Ž Ž. LEMMA 5.1. The formulaton EB. s, t has monotone nequaltes wth respect to the x-arables, wth the z-arables appearng n two constrants each. Proof. To show the monotoncty of the constrants, we let the varables for nodes n Nt Ž. assume values n 0, 14 wth x 1 f node Nt Ž. s deleted. The formulaton now has the two varables appearng wth opposte sgns n every constrant: Ž () monotone EB s,t. zs, t Mn ÝŽ,. E Ž NŽ s.nžt.. cz s, t subect to x z for Ž,. E s, t, Ns Ž. Ns, Ž t., Nt Ž. x z for Ž,. E, Nt Ž. Ns, Ž t., Ns Ž. x x 1 for Ž,. E s, t, Ns, Ž. Nt Ž. x x 1 z for Ns, Ž t., Vs, t x 4 0, 1 for Ns Ž. x 4 1, 0 for Nt Ž. z 0, 14 for all,.

25 198 DORIT S. HOCHBAUM Ths s a monotone formulaton, except that each z varable appears twce: once for each endpont constrant. To recover the monotoncty of constrants, we substtute for each edge the two occurrences of z by z Ž1. and z Ž., where z z Ž1. and z z Ž.. As n Corollary.1Ž., Ž. 1 Ž1. Ž. z z z. Ths monotone formulaton s solvable optmally n ntegers. When the 1 values of z are recovered, they are n 0,, 14 Ž the values of the x-vara- bles are ntegers.. Roundng the values up provdes a feasble soluton that s at most twce the optmum. Notce that the constructed network here s the same as n Fg. 3, except 1 that the weghts w are replaced by ÝŽ, k. E c k. The complexty of the resultng -approxmaton algorthm s Om Ž TŽn, n A -APPROXIMATION FOR A PROBLEM EQUIVALENT TO MAXIMUM CLIQUE The maxmum clque problem s a well-known optmzaton problem that s notorously hard to approxmate, as shown by Hastad Ha96. The problem s to fnd n a graph G Ž V, E. the largest set of nodes that form a clquea complete graph. An equvalent statement of the clque problem s to fnd the complete subgraph that maxmzes the number Ž or sum of weghts. of the edges n the subgraph. There s a clque of sze k f and only f there s a clque on Ž k. edges. The complement of ths edge-varant of the maxmum clque problem s to fnd a mnmum weght set of edges to delete so the remanng edge-nduced subgraph s a clque. Let x be a varable that s 1 f node s not n the clque, and 0 otherwse. Let z be 1 f edge Ž,. E s deleted. The frst set of constrants ensures that f an edge has an endpont not n the clque, then t must be deleted. The second set of constrants says that the set of nodes selected forms a clque by requrng that f an edge s not n the graph, then both of ts endponts cannot be n the clque: Ž. Clque Mn ÝŽ,. E cz subect to x z for Ž,. E, V x x 1 for Ž,. E x, z bnary for all,.

26 APPROXIMATING CLIQUE AND BICLIQUE PROBLEMS 199 Wth ths formulaton each nequalty has no more than two varables. Thus the problem s -approxmable, snce the results of HMNT93 apply drectly. In the network we have Ž m n. nodes Žone for each varable n the monotonzed verson. and 4m m edges. The resultng complexty of the -approxmaton algorthm s therefore TŽŽ m n.,4mm., Ž whch s Omnlog n.. The formulaton Ž Clque. has, lke all problems n two varables per nequalty, an nterpretaton as SAT wth the clauses Ž x, x. for each Ž,. E and Ž x, z. for every node and Ž,. E. Furthermore, t s reducble to a vertex cover problem by usng the transformaton descrbed n Hoc96a, p. 13. The resultng bpartte Ž monotonzed. vertex cover problem has the same number of nodes Om Ž n. as above. The number of edges of ths vertex cover problem s quadratc n the number above,.e., On Ž 4.. EPILOGUE The orgnal verson of the paper contaned a 4-approxmaton algorthm to the clque problem. Followng a presentaton of ths result, I receved a number of suggestons regardng the mprovements of the approxmaton factor of the Clque problem from 4 to. Among these, Reuven Bar- Yehuda was the frst to pont ths fact out by restatng the problem: for each nonedge and each par of edges adacent to the nonedge, at least one edge of the par must be deleted. That problem s a vertex cover problem n whch the edges of E play the role of the vertces that must cover each nonedge. The set of constrants s thus z z 1 for Ž,. E, Ž, p., Ž, k. E. p k The number of varables s m E, and the number of constrants s mn Žfor m E.. The runnng tme requred for the -approxmaton of ths Ž vertex cover problem s thus Omn.. The SODA98 program commttee provded the SAT nterpretaton, whch motvated the formulaton presented here. A formulaton dentcal to ours, was proposed ndependently by the referee. ACKNOWLEDGMENT Ths s to express my grattude to an anonymous referee on ths paper. Hs nsghtful comments led to sgnfcant mprovements n the scope and content of the results presented.

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