A class of spectral bounds for Max k-cut

Transcription

1 A class of spectral bouds for Max k-cut Miguel F. Ajos, José Neto December 07 Abstract Let G be a udirected ad edge-weighted simple graph. I this paper we itroduce a class of bouds for the maximum k-cut problem i G. Their expressio otably ivolves eigevalues of the weight matrix together with some other geometrical parameters (distaces betwee a discrete poit set ad a liear subspace). This exteds a boud recetly itroduced by Nikiforov. We also show cases whe the provided bouds strictly improve over other eigevalue bouds from the literature. Keywords: Max k-cut, Adjacecy matrix eigevalues, Adjacecy matrix eigevectors Itroductio Let G = (V, E) be a udirected simple graph havig ode set V = {,,..., }, edge set E, ad let w R E deote a weight fuctio o the edges. Let k deote a positive iteger. Give ay partitio (V, V,..., V k ) of V ito k subsets V, V,..., V k (some of which may be empty), the k-cut defied by this partitio is the set δ(v, V,..., V k ) of edges i E havig their edpoits i differet sets of the partitio. Ad the weight of a k-cut is the sum of the weights of the edges it cotais. Give this, the maximum k-cut problem cosists i fidig mc k (G, W ): the maximum weight of a k-cut i G. I what follows, let W R deote the weighted adjacecy matrix whose etries are defied by W ij = w ij if ij E ad W ij = 0 otherwise. So i particular, it is a symmetric matrix with a zero diagoal. Give two disjoit ode subsets A, B, let w[a, B] deote the sum of the weights of the edges havig oe edpoit i A ad the other i B: w[a, B] = (i,j) A B : ij E w ij. Similarly, w[a] deotes the sum of the weights of the edges with both edpoits i A: w[a] = (i,j) A : w ij. Now, let λ λ... λ deote ij E,i<j NSERC-Hydro-Québec-Scheider Electric Idustrial Research Chair, GERAD & Ecole Polytechique de Motréal, QC, Caada H3C 3A7. ajos@stafordalumi.org Samovar, CNRS, Telecom SudParis, Uiversité Paris-Saclay, 9 rue Charles Fourier, 90 Evry, Frace. Jose.Neto@telecom-sudparis.eu

2 the eigevalues of W ad let ν, ν,..., ν be the correspodig uit ad pairwise orthogoal eigevectors. For ay positive iteger q, let q stad for the q-dimesioal all oes vector. Give ay vector x R, Diag(x) stads for the square diagoal matrix of order, havig x for diagoal. The Laplacia matrix is L = Diag(W ) W. Its maximum eigevalue is deoted by λ (L). I this paper we are iterested i bouds for the maximum k-cut problem that ivolve eigevalues of the Laplacia L or of the weight matrix W. For the particular case whe k =, Mohar ad Poljak [5] proved the iequality mc (G, W ) 4 λ (L). More recetly, va Dam ad Sotirov [4] proved the followig upper boud o mc k (G, W ), still makig use of the largest eigevalue of the Laplacia ad providig i the same referece several graphs for which this boud is tight together with some comparisos with other bouds stemmig from semidefiite formulatios. Theorem.. [4] mc k (G, W ) (k ) λ (L). () k Also recetly, Nikiforov [6] itroduced a upper boud for the maximum cardiality of a k-cut i G (i.e. the maximum k-cut problem with w e =, e E), which may be easily exteded to the weighted case ad ca be formulated as follows. Theorem.. [6] mc k (G, W ) k k ( w[v ] λ ) () As he otes, the bouds from Theorems. ad. are equivalet for regular graphs but they are icomparable i geeral. I this paper we show the boud from Theorem. ca be still further improved by makig use of the whole spectrum (i.e. all eigevalues ad eigevectors) of the matrix W i lieu of its smallest eigevalue oly. This is achieved by itroducig a atural extesio of a earlier work doe for the maxcut problem (i.e. the maximum k-cut problem for the particular case whe k = ) []. We metio some additioal otatio to be used. Give a positive iteger q, [q] stads for the set of itegers {,,..., q}. The ier scalar product is deoted by,, ad the Euclidea orm by. Spectral bouds With o loss of geerality, we assume the graph G is complete (settig zero weights o o existig edges). Give r R \ {0, }, let d j,r deote the distace

3 betwee the set of vectors {r, } ad the subspace V ect(ν, ν,..., ν j ) that is geerated by the first j eigevectors of W : d j,r = mi { z y : z {r, }, y V ect(ν, ν,..., ν j )}. (3) Theorem.. For ay r R \ {0, }, mc k (G, W ) (r + k )(w[v ] λ ) k (r ) l [ ] (λ l+ λ l )d l,r (4) Proof. Let (V, V,..., V k ) deote a partitio of V correspodig to a optimal solutio of the maximum k-cut problem. For all i [k], let the vector y i {r, } be defied as follows: y i l = r if l V i ad otherwise. We have: y i, W y i = r w[v i ] + j [k]\{i} w[v j] + r j [k]\{i} w[v i, V j ]+ (j,l) ([k]\{i}) : w[v j, V l ] (5) j<l Let us ow compute the sum of each term occurrig i the right-had-side of (5) over all i [k]. i [k] r w[v i ] = r (w[v ] mc k (G, W )), i [k] j [k]\{i} w[v j] = (k ) (w[v ] mc k (G, W )), i [k] r j [k]\{i} w[v i, V j ] = 4r mc k (G, W ), i [k] (j,l) ([k]\{i}) : j<l w[v j, V l ] = (k )mc k (G, W ). Thus, we deduce y i, W y i = mc k (G, W )( r + r ) + w[v ](r + k ). (6) i [k] We ow derive a lower boud o y i, W y i makig use of the spectrum of W. For, we metio some prelimiary properties. Note that sice W is symmetric we may assume (ν, ν,..., ν ) forms a orthoormal basis, ad cosider the expressio of y i i this basis: y i = l [] α lν l with α R. The, we have y i = l [] α l = + V i (r ). From the defiitio of the distace defied above we deduce d j,r l=j+ α l, j [ ]. Thus, we have y i, W y i = l [] ( λ lαl = λ + Vi (r ) ) ( l= α l + l= λ lαl = λ + Vi (r ) ) + l= (λ l λ )αl The, iteratively makig use of the iequality αj j =,...,, we deduce y i, W y i ( λ + Vi (r ) ) + l [ ] d j,r l=j+ α j (λ l+ λ l )d l,r. for 3

4 Ad summig these iequalities for all i [k] we obtai y i, W y i λ ( k + r ) + k (λ l+ λ l )d l,r (7) i [k] Fially combiig (6) ad (7), the result follows. l [ ] Note that all the terms occurig i the last sum of the iequality (4) are oegative, so that removig from the right-had side some or all of the terms ivolved i this sum, the expressio obtaied still provides a upper boud o mc k (G, W ). I particular, iequality () follows as a corollary of Theorem. takig r = k ad removig the last sum from the right-had side of iequality (4). Remark Eforcig the value amog the two possible values for the compoets of the vectors used i the defiitio of the distaces (3) is doe just to slightly simplify the presetatio. We are basically iterested i the distace betwee V ect(ν, ν,..., ν j ) ad a set of vectors whose compoets are restricted to take ay of two ozero values. If we deote by d j,r,r the distace betwee V ect(ν, ν,..., ν j ) ad the set of vectors {r, r } with (r, r ) (R \ {0}), the d j,r,r = r d r j, r, j [], ad the results we get by usig such vectors are equivalet to the oes preseted. Remark I view of the boud (4) o mc k (G, W ), oe may ask for the best choice for the parameter r. If we cosider the trucated boud obtaied from (4) by removig the last sum, we ca show the ratio r +k (r ) is miimized for r = k, which is the value used by Nikiforov [6] ad leads to formula (). Oe may ask for the best such choice by cosiderig the whole expressio of the boud i (4). Prelimiary computatioal experimets show that other values of r may lead to strictly better bouds, depedig o the istace. The approach udertake to prove Theorem. ca also be used to obtai lower bouds o the weight of ay k-cut. Let lc k (G, W ) deote the miimum weight of a k-cut i G ad let d j,r deote the distace betwee the set of vectors {r, } ad the subspace V ect(ν j, ν j+,..., ν ) that is geerated by the last j + eigevectors of W : Theorem.. lc k (G, W ) d j,r = mi { z y : z {r, }, y V ect(ν j, ν j+,..., ν )}. (8) (r ) (r + k )(w[v ] λ ) + k l [ ] (λ l+ λ l )d l+,r Proof. Similar to that of Theorem.. Or we ca also use Theorem. with the weight matrix W istead of W, which gives a upper boud o lc k (G, W ). (9) 4

5 Theorems. ad. lead to the defiitio of the spectral boud gap, which is the differece betwee the upper ad lower spectral bouds: ( (r ) r + k ) (λ λ ) k ( l,r) (λ l+ λ l ) d l+,r + d. l [ ] 3 O some particular cases Geerally, computig the distaces (d j,r ) j= ivolved i the expressio of the boud (4) is N P-hard (see Propositio 4.4 i []). I this sectio we provide a upper boud o mc k (G, W ) for the particular case whe is a eigevector of W. (This is otably the case whe cosiderig the Max k-cut problem i regular graphs with uit edge weights). Its expressio does ot ivolve distaces ad leads to a upper boud o mc k (G, W ) that is lower tha or equal to the bouds of Theorems.-.. We start with a auxiliary result o the miimum squared distace betwee ay vector i {, r} ad the subspace i R that is orthogoal to V ect( ), deoted by V ect( ). Propositio 3.. { mi y z : y {, r}, z V ect( ) } = { if r, mi( (s+r ), s ) otherwise, with s mod ( r), 0 s < r, for the case whe r <. Proof. Let p {0,,..., } ad ŷ {r, } such that ŷ has exactly p etries with value r. Let ˆd deote the squared distace betwee ŷ ad V ect( ), that is, the quatity ˆd = ŷ, = (p (r ) + ). For p {0,,..., }, the miimum of ˆd is obtaied for p = 0 if r ad for p = or p =, otherwise. r r Usig Propositio 3. together with the fact that d j,r d j+,r, j [ ] the ext result follows. Corollary 3.. If is a eigevector of W associated with the eigevalue λ q, the mc k (G, W ) ((r + k )(w[v ] λ (r ) ) k mi((s + r ), s ) ) l [q ] (λ l+ λ l ) with r < ad s mod ( r), 0 s < r. (0) 5

6 Takig r = k (as is doe i Nikiforov s proof [6] of Theorem.) leads to the followig simpler expressio. Corollary 3.3. If G is a complete graph ad W is its adjacecy matrix, the mc k (G, W ) ( (k ) mi((s k), s ) ), () k with s mod k, 0 s < k. Proof. The eigevalues of the adjacecy matrix of the complete graph K are with multiplicity ad with multiplicity. The vector is a eigevector associated with the eigevalue λ =. The result follows from (0) with q = ad r = k. Corollary 3.3 gives a ifiite class of graphs (complete graphs such that mi((s k), s ) > 0) where our ew boud (4) strictly improves over Nikiforov s boud (). The boud () has also the feature of coicidig with the optimal objective value of Max k-cut for some cases. Ideed, by Turá s Theorem, the maximum cardiality of a k-cut i the complete graph K is, ad this (k ) k correspods to the boud () if s = (k ) mod k, where s mod k, 0 s < k. For k =, it follows that the boud () coicides with the optimal objective value of mc k (G, W ) for all complete graphs (see also Propositio 4.4 i []), whereas this fails for the bouds of Theorems. ad. for complete graphs havig a odd umber of vertices. Refereces [] Be-Ameur, W., Neto, J.: Spectral bouds for the maximum cut problem. Networks 5 (008) 8 3 [] Be-Ameur, W., Neto, J.: Spectral bouds for ucostraied (, )- quadratic optimizatio problems. Europea Joural of Operatioal Research 07 (00) 5-4 [3] Be-Ameur, W., Mahjoub, A.R., Neto, J.: The Maximum Cut Problem, i Paradigms of Combiatorial Optimizatio, d Editio (ed V. Th. Paschos), Joh Wiley & Sos, Ic., Hoboke, NJ, USA (04) 3-7 doi: 0.00/ ch6 [4] va Dam, E.R., Sotirov, R.: New bouds for the max-k-cut ad chromatic umber of a graph. Liear Algebra ad its Applicatios 488 (06) 6-34 [5] Mohar, B., Poljak, S.: Eigevalues ad the max-cut problem. Czechoslovak Mathematical Joural 40: (990) [6] Nikiforov, V.: Max k-cut ad the smallest eigevalue. Liear Algebra ad its Applicatios 504 (06)