B. Bollobas ad A. D. Scott sciece ad the extremal perspective i combiatorics. The extremal problem asks how small a largest bipartite subgraph of a gr

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1 BOLYAI SOCIETY MATHEMATICAL STUDIES, X Bollobas kotet Budapest (Hugary), 1998 Budapest, 000, pp. 1{6. Better Bouds for Max Cut B. BOLLOBAS ad A. D. SCOTT For a multigraph G, let f(g) be the size of a largest cut of G. We dee f(m) to be the miimum of f(g) over graphs of size m, ad fw(m) to be the miimum over multigraphs of size m. For sucietly large, ad 0 k, we determie f(m) for m = + k ad give the extremal graphs. Furthermore, by cosiderig the weighted problem, we determie f(m) ad fw(m) to withi a additive costat for every m, ad d the extremal graphs for may values of m. This exteds idepedet work of Alo ad Halperi. I the secod part of the paper, we tur to the problem of dig eciet algorithms for obtaiig large bipartite subgraphs. We give a liear time algorithm that, for a multigraph G with m edges ad vertices, ds a bipartite subgraph with fw(m) edges. We give a algorithm ruig i time O( ck + m + ) that ds a bipartite subgraph with at least m= + p m=8 + k edges if oe exists ad otherwise provides a optimal partitio. We also provide a liear time weak approximatio algorithm for f(m) m= p m=8. I the al part of the paper, we geeralize our results to the related problems Max k-cut ad Max Directed Cut. 1. Itroductio The well-kow Max Cut problem asks for a largest cut of a graph G. A cut of maximal size clearly correspods to a bipartite subgraph of maximal size, ad we shall use both formulatios. Max Cut is NP-hard ad has bee the focus of extesive study, both from the algorithmic perspective i computer

2 B. Bollobas ad A. D. Scott sciece ad the extremal perspective i combiatorics. The extremal problem asks how small a largest bipartite subgraph of a graph with m edges ca be, ad which graphs achieve this boud. The algorithmic problem asks for eciet algorithms that determie or approximate the maximal size of a bipartite subgraph ad that provide large bipartite subgraphs. A importat survey of the Max Cut problem is give by Poljak ad Tuza [3]; a excellet bibliography from the perspective of combiatorial optimizatio is give by Lauret [6]. This article is a combiatio of survey ad research paper. We shall idicate some recet progress o the Max Cut problem, from both combiatorial ad algorithmic perspectives, ad prove a umber of ew results. The article was origially writte i the autum of 1997 for the Erd}os Workshop i Budapest i the summer of Early i 1998, we became aware of the work of Alo ad Halperi [], who also addressed the extremal Max Cut problem. They determied the recurrece (5) for the extremal fuctio f w (m); however, they did ot determie the extremal graphs or cosider the algorithmic aspects of the problem. Although our approach is similar to Alo ad Halperi's, we give the details for clarity of expositio, ad also so that we ca obtai the extremal graphs for () ad for some cases of (5). I additio, the argumets are used i later sectios o algorithms. For a graph G, let f(g) be the maximal umber of edges i a bipartite subgraph of G. For m > 0, we dee f(m) to be the miimum value of f(g) for graphs G with m edges. As observed by Erd}os, f(m) m=. This ca be see by otig that a radom bipartitio of a graph G gives a cut with expected weight e(g)=. I 1973, aswerig a questio of Erd}os, Edwards ([9], see also [10]) proved that (1) f(m) & r ' m m : 8 We remark that, i fact, we ca demad sigicatly more from a bipartitio: it was show i [6] that every graph G with m edges has a bipartitio V (G) = V 1 [ V such that (1) is satised ad, i additio, for i = 1;, () e(v i ) m + r m :

3 Better Bouds for Max Cut 3 This is a example of a judicious partitioig result, i which we demad that every class i a vertex partitio satises some iequality. We remark l+1 that the result is best possible whe m =, i which case Kl+1 is the uique extremal graph (for l ). We shall ot cosider judicious partitios further i this paper; however, a discussio of related results ad problems ca be foud i [7]. The Edwards boud (1) is exact for complete graphs. Thus if m = the f(m) = f(k ) = b =c. I fact, if 6= the K is the uique extremal graph (for = 3, two additioal extremal graphs are obtaied by takig a edge-disjoit uio of two copies of K 3 ). For other values of m, however, the situatio is less simple. Ideed, Erd}os [11] cojectured that the dierece betwee f(m) ad (1) ca be arbitrarily large. This was proved by Alo [1], who showed that there exist c; c 0 > 0 such that, if is sucietly large ad m =, (3) f(m) + + cp m + r m 8 + c0 m 1= : Alo also showed that, for some c 00 > 0 ad every m > 0, f(m) m + r m 8 + c00 m 1= : Thus the Edwards formula is exact for m = ad out by O(m 1= ) whe m is about halfway betwee ad +1. Our rst aim i this paper is to determie f(m) exactly for a rage of values betwee ad +1. Ideed, suppose m = + k, where ad k k are o-egative itegers with 0. The (two) graphs obtaied by takig a edge-disjoit uio of K ad K k show that k f(m) + +1 while ay graph obtaied by deletig shows that f(m) $( + 1) % : k edges from Km+1

4 B. Bollobas ad A. D. Scott I x we shall prove that, provided m is sucietly large, () f(m) = mi ( k + ; $( + 1) %) : Furthermore, the graphs we have described iclude all the extremal graphs uless k =, whe the graphs that ca be obtaied from a edge-disjoit uio of K ad two copies of K 3 are also extremal. Note that ( + 1) = is smaller tha b =c + bk =c if r k + "(; k) where "(; k) is O(1). If there is some iteger k with k i this rage, the sice f(m) +1 is mootoic icreasig ad f = ( + 1) =, it follows that, for + k m +1, % f(m) = $( + 1) : We obtai the surprisig cosequece that f(m) is costat o itervals of legth up to about p = (m=) 1=. I x3 we tur to the problem of determiig f(m) for arbitrary values of m. Our argumets are similar to those of x, but we are faced with some additioal techical diculties, which make it ecessary to cosider graphs i which the edges are weighted. For a graph G with edge-weightig w, let f(g) be the maximal weight of a cut of G. Let f w (m) be the miimum of f(g) over graphs whose edges are weighted with oegative itegers ad have total weight m (or, equivaletly, over multigraphs with m edges). It is easily see that f w (m) f(m). We prove that, for sucietly large m, ( + 1) (5) f w (m) = mi ; + f w m ; where the iteger is deed by m < +1. This provides a recursive formula for f w (m). (This recurrece was foud idepedetly by Alo ad

5 Better Bouds for Max Cut 5 Halperi [] ad implies ().) To prove a recursio, ote that if we were to kow f w (m) for every m m 0, where m 0 is sucietly large, the (5) determies f w (m) for all m. I ay case, (5) determies f w (m) to withi a additive costat. Furthermore, by cosiderig K +1 ad graphs of form K [ H with e(h) = m, it is easy to see that f(m) mi ($ ( + 1) % ; + f m ) : Settig C = max m<m0 f(m) fw (m), it follows that for every m > 0, f(m) fw (m) C: It seems likely that f(m) = f w (m) for every m > 0. We use the results of x3 i x, where we retur to the problem of dig extremal graphs for Max Cut. Writig m = k, where the i are oegative itegers with i < i+1 for i < k, we determie f(m) provided k1 is sucietly large: if b i =c + + b k =c < ( i + 1) = P k for each i, the f(m) = i=1 b i =c; we also give the extremal graphs. I the secod part of the paper we cocetrate o algorithmic results. Max Cut is NP-hard (see [], [16]), eve for quite restricted classes of graphs ([1], [5]), ad it is therefore of iterest to d polyomial time algorithms that give large bipartite subgraphs. A umber of authors have give algorithms that yield a bipartite subgraph at least as large as that guarateed by the Edwards boud (1) (see remarks i Sectio, where several algorithms are also discussed; see also [35]). I Sectio 5 we give a liear time algorithm that is optimal i terms of edge-weight: for ay graph with iteger edge-weights ad total weight m, the algorithm yields a bipartite graph of weight at least f w (m). Much recet progress o the Max Cut problem has cocered the existece of good approximatio algorithms. Buildig o results i the theory of probabilistically checkable proofs, Hastad [18] has show that, for ay " > 0, it is NP-hard to approximate Max Cut withi a factor 17=16 ". O the positive side, Goemas ad Williamso [17] have give a 1:1383- approximatio algorithm. (For the Max k-cut problem Ka, Khaa, Lagergre ad Pacoesi [3] have show that it is NP-hard to approximate withi a factor 1 + 1=3k; while Frieze ad Jerrum [15] have give a

6 6 B. Bollobas ad A. D. Scott algorithm that approximates withi a factor (1 1=k + l k=k ) 1.) Furthermore, it is kow that good approximatio algorithms exist for dese graphs (see [], [5], [13]). Note that the diculty for these algorithms lies i recogisig ad partitioig graphs G for which f(g) is large. Graphs for which f(g) is small ca be partitioed by the trivial greedy algorithm that yields a cut of weight at least w(g)=. I Sectios 6 ad 7 we cocetrate o graphs of weight m for which f(g) is close to f w (m). Mahaja ad Rama [9] have show that there are algorithms ruig i time O( 3 + m k ) ad O( ck + m + ) that d a cut of size at least dm=e + k i a graph with m edges ad vertices, if such a cut exists. I Sectio 6, we show that, for ay xed iteger k, there is a algorithm ruig i time O( ck + m + e) th at ds a cut of weight at least m= + p m=8 + k i a graph with iteger edge-weights, e edges ad total weight m if such a cut exists ad otherwise ds a optimal cut. I Sectio 7 we cocetrate o the quatity f(g) m= p m=8: we ote that it is NP-hard to approximate this quatity withi a factor (9=8 "), but provide a liear time algorithm that approximates its logarithm. I the al part of the paper we cosider two related problems. I Sectio 8, we cosider the Max k-cut problem for k >. We prove versios of our results o bipartitios for the k-partite case. Fially, i Sectio 9, we cosider the problem of dig large bipartite subgraphs of a directed graph, ad give some extremal results. Throughout the paper, we use w for a iteger-valued edge-weightig. For disjoit sets of vertices we write E(X; Y ) for the set of edges betwee X ad Y, e(x; Y ) = E(X; Y ) ad w(x; Y ) = PeE(X;Y ) w(e). We will also sometimes write e(x; Y ) ad w(x; Y ) for e fxg; Y ad w fxg; Y. Part I: The Extremal Problem. Max Cut for graphs Our mai aim i this sectio is to d the exact value of f(m) for every sucietly large m of form m = + k +1, ad determie the extremal graphs. The value of f(m) ca also be obtaied from the results of the ext sectio ad from Alo ad Halperi []. However, our aim i

7 Better Bouds for Max Cut 7 this sectio is also to determie the extremal graphs, which tur out to be surprisigly varied. + + r Note that, for ay m, we ca obtai a upper boud for f(m) by writig m = 1, where each i i tur is chose to be as large as possible; the by cosiderig K 1 [ [ K r, it is clear that f(m) b =c b r=c. A straightforward calculatio shows that, for every m, (6) f(m) m + r m 8 + (8m)1= + O(m 1=8 ); while takig k p 1 i the theorem below shows that f(m) m + r m o(1) (8m) 1= for iitely may values of m. Theorem 1. For > , every graph G with e(g) = + k where 0 k 1 satises (7) f(g) mi ( k + $ ; ( + 1) %) : Furthermore, the extremal graphs are the two graphs obtaied by takig a edge disjoit uio of K ad K k if b =c + bk =c b( + 1) =c ad +1 k 6= ; ad all graphs obtaied by deletig k edges from K +1 if b =c + bk =c b( + 1) =c. If k = the the extremal graphs

8 8 B. Bollobas ad A. D. Scott are obtaied by takig a edge-disjoit uio of K ad K (two graphs) or K ad two copies of K 3 (seve graphs). We will make use of several lemmas i our proof of Theorem 1. Lemma is due to Edwards [9]; recetly a short proof was give by Erd}os, Gyarfas ad Kohayakawa [1]. Poljak ad Turzk [33] gave a O( 3 ) algorithm for dig a bipartite subgraph of the type guarateed i the lemma; Ngoc ad Tuza [30] improved upo this by givig a algorithm ruig i time O(m). The proof that we give is similar to the proofs of Erd}os, Gyarfas ad Kohayakawa ad of Ngoc ad Tuza, but is slightly simpler ad also yields a O(m) algorithm. Lemma. For a coected graph G, f(g) e(g) + jgj 1 : Proof. Give a orderig of the vertices of G, we ca partitio V (G) by usig the greedy algorithm: at each step a vertex is added to whichever class cotais fewer of its predecessors (or to either class if both classes cotai the same umber). If we P write e(v) for the umber of predecessors of v that are adjacet to v the vv (G) e(v) = e(g), ad the size of the bipartite graph betwee the two vertex classes is at least X vv (G) e(v)= = e(g) + k where k is the umber of vertices with a odd umber of predecessors. It is therefore eough to d a orderig of V (G) i which at least ( 1)= vertices have a odd umber of predecessors. For jgj 1 this is trivial. If jgj = > 1, the we rst d a set of vertices S such that G[S] is a star ad G S is coected. Let T be a spaig tree of G. If two edvertices of T are adjacet, say v ad w, the let S = fv; wg. Otherwise, let T 0 be the tree obtaied by removig all edvertices of T, let v be ay edvertex of T 0 ad let S cotai v, together with all edvertices of T adjacet to v. I either case, S cotais a vertex v s together with a idepedet set of eighbours of v s ad G S is coected. We repeat the process with G S, cotiuig util at most oe vertex remais.

9 Better Bouds for Max Cut 9 We order the vertices of G oe star at a time. Give a star S, let R be the set of vertices we have already ordered. Let S + be those vertices of S fv s g with a odd umber of eighbours i R ad let S = S S + [fv s g. After R, take S + (i ay order), followed by v s, followed by S (i ay order). Note that the vertices i S + [ S all have a odd umber of predecessors i this orderig, ad js + [ S j jsj=. Thus i the complete orderig, sice the stars together cotai at least 1 vertices, at least ( 1)= vertices have a odd umber of predecessors. It is easy to see that this proof gives a algorithm that rus i time O(m). Note that the tree T ca be updated ecietly betwee the removal of successive stars. I the proof above, we ca avoid the eed to geerate a star partitio by costructig more directly a orderig of the vertices. Begi with ay orderig v 1 < < v of V (G) i which every vertex except v 1 has at least oe predecessor. For each vertex calculate the umber of predecessors, ad let X be the set of vertices with a eve umber of predecessors. For each x X v 1 d the largest predecessor of x. This ca clearly all be doe i time O(m). Now suppose that two vertices have the same largest predecessor, say v. Reorder V (G) by movig x ad y to immediately before v, leavig x ad y i the same order. The parity of v does ot chage, sice it has gaied two predecessors, while x ad y ow have a odd umber of predecessors. No other vertex has chaged its set of predecessors. However, some vertices that previously had x or y as largest predecessor may ow have v: we ca check this by examiig the eighbours of x ad y. By a similar argumet, if ay x X has largest predecessor y X, the movig x to a positio immediately before y gives a orderig i which x ad y have a odd umber of predecessors ad oly eighbours of x (i the origial orderig) ca have a ew largest predecessor. Repeatig this stage of the algorithm, we ote that each vertex is moved at most oce (whe its parity chages), ad that we examie the eighbours of a vertex oly whe we move it (ote also that we eed oly look at successors of a vertex, ad o vertex gais successors before beig moved). Thus this part of the algorithm rus i time O(m + ) = O(m). Fially, suppose that o two vertices i X have the same largest predecessor ad o vertex i X has largest predecessor i X. Now if x X has o predecessors the either x = v 1, or (sice vertices are oly ever moved dowwards) x must have bee moved at some poit i the algorithm, which implies x 6 X. Thus every x X v 1 has a largest predecessor i V (G) X. It follows that we

10 10 B. Bollobas ad A. D. Scott have a ijectio X v 1! V (G) X, so jxj ( 1)= + 1, ad thus at least jxj ( 1)= vertices have a odd umber of predecessors. The algorithm is ow completed greedily as before. Note that Lemma furishes a quick proof of the Edwards formula (1). Ideed, give a graph G, we may assume G is coected or else idetify oe vertex from each compoet. Let = jgj, so e(g) : it follows from the lemma that f(g) e(g)= + ( 1)=, ad (1) follows by a simple calculatio. Lemma 3 was oted by several authors (see [3], [7], [8], [1]). We prove it here for completeess. Lemma 3. For a oempty graph G, 1 f(g) + 1 e(g): Proof. Fix a colourig of G with t = (G) colours, ad let the colour classes be V 1 ;... ; V t. Let S [ T be a radom partitio of [t] ito a set of size bt=c ad a set of size dt=e. The, writig m = e(g), the expected umber of edges betwee S is V i ad S it V i is t t = t 1 t m = m = Therefore some partitio satises this iequality m: t Lemma 3 also gives a fast proof of the Edwards formula, as observed idepedetly by Alo [1] ad Hofmeister ad Lefma [0]. We will also eed a lemma cocerig partitios of graphs whose edges are weighted with (positive or egative) itegers. Note that as a cosequece of this lemma, for m = we obtai the extremal graphs for the formula of Edwards. Lemma. Let H be a graph whose edges have iteger weights. If is a iteger with w(h)

11 Better Bouds for Max Cut 11 the there is a partitio of H ito two sets such that the total weight of edges betwee the sets is at least b =c. For 6=, the uique extremal graph is K with all edges of weight 1. For = the extremal graphs are K with all edges of weight 1 ad the graphs obtaied by takig the edge sum of two copies of K 3 with all edges of weight 1. Proof. The proof that such a partitio exists is straightforward, sice we may cosider H as a weighted complete graph, by addig a edge of weight 0 betwee every pair of oadjacet vertices. If H cotais a edge with weight at most 0 the cotractig that edge does ot decrease the weight of the graph. Therefore we may assume that H is a complete graph ad all edges have weight at least 1, so jhj. A radom partitio of V (H) ito sets of size jhj= ad jhj= yields a bipartite subgraph of expected weight at least b =c. To derive the extremal graphs, ote rst that we ca assume that all edges have oegative weight, sice cotractig a edge with egative weight icreases the total weight, ad we ca the do better tha b =c i the argumet above (this remark also applies if w(h) > ). If H is ot K with all edges of weight 1, the we ca cotract to a complete graph with at least oe edge of weight greater tha 1. Thus jhj < ad all edges have weight at least 1. Writig h = jhj, a radom bipartitio ito sets of size dh=e ad bh=c yields a bipartite subgraph of expected weight at least (8) bh =c h w(h) = h = h : If h < 1 or h = 1 ad is odd the (8) is strictly larger tha b =c. Otherwise, h = 1 is odd ad, sice all edges have weight at least 1, H is the edge sum of K 1 with all weights 1 ad a graph H with V (H ) = V (H) ad weight 1 = 1. Furthermore, all bipartitios of V (H) ito two sets of size bh=c ad dh=e yield a bipartite subgraph of size exactly b =c, sice otherwise some partitio would exceed the expectatio (8). Sice every bipartitio of K 1 ito two sets of size ( 1)= ad ( 1)= gives a bipartite subgraph of size ( 1) = = = =, it follows that every bipartitio of H ito sets of size ( 1)= ad ( 1)= gives a bipartite subgraph of size =. If H is ot complete, let v be a vertex that is ot adjacet to every other vertex of H. We ca partitio V (H ) v ito two sets V 1 ad V

12 1 B. Bollobas ad A. D. Scott with jv 1 j = jv j = ( )=, such that (v) \ V1 > (v) \ V. The a ddig v to V 1 or V gives two partitios ito sets of size ( 1)= ad ( 1)= that yield bipartite subgraphs with dieret sizes. It follows that H is complete: the oly possibility is =, H = K 3 with all edges of weight 1. Thus for >, H must be K, with all edges of weight 1. For =, a simple case check shows that H ca also be ay graph obtaied by takig the edge sum of two copies of K 3 with all edges of weight 1. Aother boud o f(g) was give by Poljak ad Turzik [31], who showed that every coected graph G with edge-weightig w has a bipartite subgraph of weight at least 1 w(g) + 1 mi T w(t ) where the miimum is take over spaig trees T of G. Poljak ad Turzik show that there is a algorithm ruig i time O( 3 ) that ds the required subgraph; Poljak ad Tuza [3] show that the algorithm rus i time O(m). We show that there is a O(m) algorithm: ote that for uweighted graphs, we obtai a cut of weight at least e(g)= + jgj 1 =, thus givig aother proof of Lemma. Theorem 5. There is a algorithm ruig i time O(m) that ds i every coected graph G with m edges ad edge-weightig w a cut of weight at least 1 w(g) + 1 mi w(t ): T Before provig this theorem we eed a lemma. We say that a collectio of iduced stars or sigle vertices S 1 ;... ; S t i a graph G is a tree-like starcoverig if every vertex i G belogs to some S i ad the graph with vertices S 1 ;... ; S t ad edges betwee S i ad S j i S i \ S j 6= ; is a tree. Lemma 6. There is a algorithm ruig i time O(m) that ds a treelike star-coverig of ay coected graph G with m edges. Proof. Recall that a rooted spaig tree T of a graph G is a depth-rst search (DFS) tree if, for every uv E(G), either the path from the root r

13 Better Bouds for Max Cut 13 to u i T cotais v or the path from r to v cotais u. I other words, if we delete v V (T ), the the compoets of T v cotaiig each of the childre of v are ot joied by ay edge. A DFS tree ca be foud i time O(m) (see, for istace, [30]). Let T be a DFS tree i G with root x. For every v V (T ) that is ot a edvertex of T let T v be the iduced star cotaiig v ad its childre. If v is a edvertex of T the let T v = fvg. The T v : v V (T ) is a tree-like star coverig of G. Proof of Theorem 5. Keepig the otatio of the proof of Lemma 6, let T 0 = T v : d T (x; v) 0 mod ad T 1 = T v : d T (x; v) 1 mod : Each of T 0 ad T 1 is a collectio of disjoit iduced stars ad sigle vertices, ad every edge of T is cotaied i some member of T 0 or T 1. We partitio whichever of T 0 ad T 1 has the greater weight, say T i oe star at a time. Suppose we have a partial partitio V 1 [ V ad wish to partitio a star T v. We greedily assig v to oe class ad T v v to the other so that the weight of the partial partitio is icreased by at least w(t v ; V 1 [ V )= + w(t v )=. Repeatig for all stars, we obtai a cut of weight at least 1 w(g) + 1 w(t i) 1 w(g) + 1 w(t ): The algorithm clearly rus i time O(m). Fially, it will be useful to have the followig remark. Lemma 7. If W V (G) ad H = G[W ] the f(g) f(h) + 1 e(g) e(h) :

14 1 B. Bollobas ad A. D. Scott Proof. Partitio H, the add the remaiig vertices from G oe at a time to whichever class has fewer eighbours. The resultig partitio clearly satses the iequality. With these lemmas i had, we tur to the proof of Theorem 1. Proof of Theorem 1. Let 0 = Suppose that > 0 ad G is a graph with (9) m = edges ad (10) f(g) mi + ( k < k ; $( + 1) %) : We shall prove that G is oe of the extremal graphs give i the statemet of the theorem. Note that we may assume that G is coected by idetifyig oe vertex from each compoet. We begi by showig that G cosists of a very large complete graph o O p vertices, together with O p `exceptioal vertices'. Clearly jgj ad k < p + 1. Note also that > 10(3= + k + 1). If k <, we are doe by Lemma ; so we may assume k. Let = (G). Now by (9), (11) b =c + bk =c m + + k ; ad so it follows from (10) that (1) f(g) m + + k : Therefore, by Lemma, (13) jgj + k + 1:

15 Better Bouds for Max Cut 15 Furthermore, Lemma 3 ad (1) imply that ad so by (9) m + k m + k > k 1: Now if G cotais k + idepedet edges, we ca cover G by jgj (k + ) k 1 edges ad vertices, which is equivalet to colourig G with at most k 1 colours. It follows that G cotais at most k + 1 idepedet edges ad therefore that some set Y of at most k + vertices meets all edges of G. Let X = V (G) Y ; the G[X] is complete, ad (1) jxj = jgj jy j k : We have partitioed G ito a large complete graph G[X] ad a small set Y which we shall regard as a set of exceptioal vertices. Note that it follows from (1) that ay partitio of X ito two sets of equal size (or sizes dierig by 1) correspods to a bipartite subgraph of G[X] of size (15) $jxj % e(x) + jxj 1 e(x) + k 3 Throughout the proof, we shall cosider partial partitios of V (G) i which we partitio Y ad some vertices from X, ad the exted these to partitios of V (G) i which X is split as evely as possible. We ow show that every vertex i Y has either very may or very few eighbours i X. Ideed, suppose some v Y has (v) \ X 5k= + ad X (v) 5k= +. We partitio G as follows. Sice > 0 we have > 5k +, so we ca d a partitio X = X 1 [ X with jx 1 j jx j jx 1 j + 1 ad (16) (v) \ X1 (v) \ X 5k + :

16 16 B. Bollobas ad A. D. Scott Addig v to whichever of X 1 ad X cotais fewer of its eighbours, it follows from (1) that we obtai a bipartitio of H = G X [ fvg with at least e(h) + jxj 1 + 5k + 1 > e(h) edges betwee the two classes. Thus, by Lemma 7, + + k (17) f(g) f(h) + 1 m e(g) e(h) > + + k ; which cotradicts (1). We may therefore assume that for every v Y, either (v) \ X < 5k= + or X (v) < 5k= +. Let Y + = v Y : (v) \ X > jxj 5k= Y = v Y : (v) \ X < 5k= + : The Y + [ Y is a partitio of Y. Next we show that the subgraph iduced by X [Y + is early complete. Ideed, we claim that e G[X [ Y + ] < 5k= +. If ot, the let W Y + be miimal such that (18) e G[X [ W ] 5k + : Sice X (v) < 5k= + for every v Y +, we have 5k= + e G[X [ W ] 5k + : Sice > 0, it follows that > (5k +), so we ca d a partitio V 1 [V of X [ W such that jv 1 j jv j jv 1 j + 1 ad all the edges of G[X [ W ] are

17 Better Bouds for Max Cut 17 jx[w j cotaied i V 1. The sice e(x [ W ) + (5k=) +, it follows from (18) ad (1) that f G[X [ W ] jv 1 j jv j $jx [ W j % 1 jxj + jw j 1 e(x [ W ) + 5k= + + > 1 e(x [ W ) + + k ; ad we are doe, as i (17). Thus we may assume that (19) e G[X [ Y + ] < 5k + ; so G[X [ Y + ] is early a complete graph. I particular, (0) f G[X [ Y + ] $jx [ Y + j % Now we show that there are ot too may edges betwee Y ad X [ Y +. Note rst that every vertex v Y has fewer tha 5k= + eighbours i X [ Y + : otherwise, sice e G[X [ Y + ] < 5k= + ad (v) \ X < 5k= + (ad sice jxj > 6(5k= + ) = 15k + 1 which, as > 0, follows from (1)) we ca d a partitio of X [ Y + ito sets W 1 ad W with jw 1 j jw j jw 1 j + 1 such that all edges of G[X [ Y + ] are cotaied i W 1 ad (v) \ W1 (v) \ W 5k= +. Arguig i the same way as from (16) we arrive at a cotradictio. Thus (1) for every v Y. (v) \ (X [ Y + ) < 5k +

18 18 B. Bollobas ad A. D. Scott Now suppose that e(y ; X [ Y + ) > 70 3= : Let Y 0 Y be miimal such that e(y 0 ; X [ Y + ) > 70 3=, ad let U = (Y 0 ) \ (X [ Y + ). Note that the miimality of Y 0 ad (1) imply that juj < 70 3= + 5k= +. Sice > 0 it follows that juj < jxj=. Let Y 0 = Y 1 [ Y be a radom partitio, where each vertex of Y 0 is i Y 1 or Y idepedetly with probability 1=. Let U 1 = u U : (u) \ Y1 > (u) \ Y ad let U = U U 1. For u U, let d u = (u) \ Y0 ad dee (u) = (u) \ Y1 (u) \ Y. The E (u) = E S(du ) ; where we write S(d u ) for the positio after d u steps of a simple symmetric radom walk o Z startig from 0. It is easily checked that E S(d) p d=, ad so E e(y 1 ; U ) + e(y ; U 1 ) = 1 e(u; Y 0) + 1 X E (u) 1 e(u; Y 0) + 1 uu X uu d 1= u = 1 e(u; Y 0) + e(u; Y 0 )=1 1= ; sice d u jy 0 j jy j k + < 9 p ad so d 1= u > d u =3 1= ; also E e(y 1 ; Y ) = 1 e(y 0):

19 Better Bouds for Max Cut 19 Sice juj < jxj=, we ca exted the partitio U 1 [ U of U to a partitio T 1 [ T of X [ Y + with jt 1 j jt j jt 1 j + 1. The, by (19) ad (1), e(t 1 ; T ) $jx [ Y + j % e G[X [ Y + ] 1 e(x [ Y + ) + jx [ Y + j 1 > 1 e(x [ Y + ) + 1k 11 : 5 k Thus, partitioig X [ Y + [ Y 0 ito T 1 [ Y ad T [ Y 1, we see that f G[X [ Y + [ Y 0 ] E e(t 1 [ Y ; T [ Y 1 ) = E e(t 1 ; T ) + e(y 1 ; Y ) + e(y 1 ; U ) + e(y ; U 1 ) > 1 e(x [ Y + [ Y 0 ) + 1k 11 > 1 e(x [ Y + [ Y 0 ) + + k ; + e(u; Y 0) 1 1= provided e(u; Y 0 ) > (5k + 33) 1=, which follows from e(u; Y 0 ) > 70 3=, sice > 0. Thus it follows from Lemma 7 ad (1) that we may assume () e(y ; X [ Y + ) 70 3= : Next we prove that jx[y + j = or +1. Now sice jy j jy j k+, we have k + e(y ) = 8k + 6k + 1:

20 0 B. Bollobas ad A. D. Scott So, sice > 0, it follows that e(x [ Y + ) e(g) e(y ) e(y ; X [ Y + ) > + k (8k + 6k + 1) 70 3= 15 k 13 k 1 703= 16: So i order to have eough vertices for the edges, we must have jx [ Y + j 16, ad thus by (1), (3) jy j k + 17: Now if jx [ Y + j 1, the e(x [ Y + ) 1 e(y ) > + ad so, sice > 0, k e(x [ Y + ) e(y ; X [ Y + ) k = k + ; which cotradicts (3). So jx[y + j. Similarly, we have jx[y + j +1, sice, by (19), + + e G[X [ Y + ] > 5k > + k :

21 Better Bouds for Max Cut 1 We have show that jx [ Y + j = or jx [ Y + j = + 1. If jx [ Y + j = + 1 the, by (0), f(g) $( + 1) % with equality i Y +1 = ;, i which case G cosists of K +1 with m edges deleted. Otherwise jx [ Y + j =. Now let H be the weighted graph cosistig of all edges of E(Y ) [ E(Y ; X [ Y + ) with weight +1 ad all edges of E G[X [ Y + k ] with weight 1, so H has total weight. It follows from Lemma that H has a cut of weight at least bk =c. Note that sice > 0 it follows from (1), (19) ad () that jhj jy j + e(y ; X [ Y + ) k = + 5k= + < jxj=. We ca therefore exted a partitio of H to a partitio of G i which X [ Y + is evely partitioed, so f(g) + f(h) + k with equality i H = K k, with all edges of weight 1 (or H = K or H = K 3 whe k = ). It follows immediately that G[X [ Y + ] is complete, ad the extremal graphs are as described i the statemet of the theorem. What prevets us from extedig the argumet i the proof of Theorem 1 to graphs with + k + l edges? The problem is that whe we remove the copy of K i the argumet above, we are left with a graph with k weighted edges. If we have edges the Lemma gives us the uique extremal graph, whereas to deal with + l k edges we would eed a versio of Theorem 1 for weighted graphs. Our aim i the ext sectio is to prove such a theorem. I particular, it will eable us to determie f(m) exactly for a much wider rage of m, ad to withi a additive costat for every value of m. It will ot, however, yield all the extremal graphs.

22 B. Bollobas ad A. D. Scott 3. Max cut for multigraphs Our aim i this sectio is to determie f(m) to withi a additive costat for every iteger m, ad to determie f(m) exactly for a larger rage of values of m. I order to do this, we cosider graphs with iteger edgeweightigs: ote that, if all weights are positive, we ca thik of these as multigraphs, where the weight of a edge idicates its multiplicity. As i the uweighted case, for a graph G with edge-weightig w, we write f(g) for the maximal weight of a bipartite subgraph of G. We P dee f w (m) to be the miimum of f(g) over graphs G with w(g) = w(e) = m ee(g) ad all weights o-egative itegers. Note that the restrictio to positive itegers meas that f w (m) is the miimum of f(g) over a ite set of multigraphs. Thus there is a (very slow) algorithm to determie f(m). However, we do ot lose aythig by allowig egative weights: if w(g) = m ad G has a edge xy with egative weight, the cosider the graph H = G=xy obtaied by cotractig the edge xy to a sigle vertex z ad deig w(vz) = w(vx) + w(vy) for v 6= x; y. Repeatig the process util we obtai a graph H with o edges of egative weight, it is clear that w(h) w(g). Sice f w (m) is mootoe icreasig, it follows that f(g) f(h) f w (m). What ca we say about f w (m)? Clearly it is subadditive: sice f(g [ H) = f(g) + f(h) for ay graphs G ad H, it follows that f w (m + r) f w (m) + f w (r) ad, similarly, f(m + r) f(m) + f(r). Furthermore, it follows from Lemma that, for m =, we have fw (m) = f(m) = b =c. All the work i this sectio will go ito provig a lower boud for f w (m) for other values of m. We begi with Theorem 8, which provides a recursive lower boud o f w (m) ad hece f(m) (as oted i the itroductio, this was proved idepedetly by Alo ad Halperi []). The approach we use i provig the theorem is similar to that used i the proof of Theorem 1. However sice we are dealig with weighted graphs the details are rather dieret. Theorem 8. Let G be a graph with iteger-valued edge-weightig w. Suppose w(g) = m. The, provided m > m 0, () f(g) mi 1 + f w m ;

23 Better Bouds for Max Cut 3 where we dee f w (r) = 0 for r < 0. Proof. Let G be a graph with w(g) = m ad f(g) = f w (m) that does ot satisfy (). We ca cosider G as a weighted complete graph by deig w(xy) = 0 if x ad y are oadjacet. If G cotais a edge xy with opositive weight the replace G with G=xy. Clearly w(g=xy) w(g) ad f(g=xy) f(g). Repeatig this process, we may assume that G is a weighted complete graph with all edges of positive weight. Let m = w(g) ad dee the iteger by + 1 (5) m < : Sice G is complete ad every edge has weight at least 1, we have jgj. We will use the fact that, as i (6), for some c; c 0 > 0 ad every iteger m, (6) f(m) m + r m 8 + cm1= < m + + c 0p : We will use c 1 ; c ;... to refer to costats i the proof below; suitable costats ca easily be determied. I several places, we shall assume that is larger tha some xed costat. Note that the proof of Lemma 3 carries over straightforwardly to the weighted case (see also Sectio 5). I particular, sice G is complete, it follows from the weighted versio of Lemma 3 that 1 f(g) + 1 m jgj ad so by (6), Sice m, we have m jgj + c p 0 : (7) jgj ( 1) + c 0 p

24 B. Bollobas ad A. D. Scott = c 0 = p > c 1 p : We ow d a small set of vertices that meets all edges with weight greater tha 1. Let M be a matchig of maximal weight i G. Cosider the radom partitio V (G) = V 1 [ V, where for each edge xy M we idepedetly assig x V 1 ad y V or x V ad y V 1 with equal probability; if there is a vertex ot covered by M, we assig it to V 1 or V with equal probability. The expected weight of edges betwee V 1 ad V is w(m) w(g) w(m) = w(g) + 1 w(m): It follows from (6) that ay matchig i G has weight at most (8) + c p 0 : Now let e 1 ;... ; e k be a maximal set of idepedet edges of G with w(e i ) > 1 for i = 1;... ; k. Exted this arbitrarily to a maximal matchig M. The jmj jgj 1 =, so by (7), w(m) jgj 1 + k > c p k ad so by (8) we have k 1 c1 p c0 p c p. Let Y be the set of vertices spaed by e 1 ;... ; e k. The (9) jy j = k c p ad ay edge of weight greater tha 1 is icidet with Y. Let X = V (G)Y : the by (7) ad (9) (30) jxj jgj jy j (c 1 + c ) p : Note that X iduces a complete graph with all edges of weight 1.

25 Better Bouds for Max Cut 5 Now for y Y, cosider the edges betwee y ad X, ad order them i icreasig order of weight (order edges of the same weight arbitrarily). Let Z 1 be the vertices i X icidet with the rst jxj= edges ad let Z = X Z 1. Cosider the partitio of X [ fyg ito Z 1 [ fyg ad Z : sice this partitios X ito sets of size jxj= ad jxj=, we see that f X [ fyg $jxj % + w(y; Z ) 1 jxj 1 w(x) w(y; X) + 1 (w(y; Z ) w(y; Z 1 )) = 1 w X [ fyg + jxj 1 Hece, by Lemma 7 ad (30), f(g) 1 w(g) + (c 1 + c ) p 1 ad so, by (6), + w(y; Z ) w(y; Z ) : + 1 w(y; Z ) w(y; Z 1 ) (31) w(y; Z ) w(y; Z 1 ) (c 0 + c + c 1 =) p + 1 < c 3 p : It follows, i particular, that for each y Y there is a iteger t(y) such that all but at most c 3 p of the edges betwee y ad X have the same weight t(y). We have deed t(v) for v Y ; set t(v) = 1 for v X. The t(v) deotes the \typical" weight of edges icidet with a vertex v. We could obtai a graph H with the same vertex weights as G from a complete graph o P vv (G) t(v) vertices by partitioig its vertices ito sets T v : v V (G), where jt v j = t(v): cotractig each set T v to a sigle vertex v gives a graph i which all but O p edges from each vertex v have weight t(v) (sice all but O p vertices v i H correspod to vertices v i X for which t(v) = 1). We shall show that i fact G is ot too far

26 6 B. Bollobas ad A. D. Scott from H. Note that the edge i H betwee vertices v ad w has weight t(v)t(w). With this i mid, for a edge xy i G we dee u(xy) = t(x)t(y) w(xy): Thus u deotes the weight we have to add to each edge of G i order to obtai the graph H. We kow that u(e) = 0 for every edge e i G[X]. Suppose that for some y Y we have X xx u(xy) c p t(y)+u(xy), we where we dee c = c 0 +c +c 1 =+1. The sice w(xy) = t(y) for all but at p most c 3 edges betwee y ad X, ad w(y; X) = PxX ca partitio X ito Z 1 [ Z as before (except that we order edges betwee y ad X with icreasig u-weight). The total weight (with weightig w) of edges betwee Z 1 [ fyg ad Z is the at least (3) $jxj % + w(y; X) + 1 X u(yx) w X [ fyg > xx + jxj 1 + c 1= ; ad hece by Lemma 7 ad (30) f(g) 1 w(g) + 1 (c1 + c ) p c p > m + + c 0p ; which cotradicts (6). (33) Thus we may assume that, for every y Y, X xx u(xy) < c p :

27 Better Bouds for Max Cut 7 Suppose that X X u(xy) > c5 3= xx yy where c 5 = c 3=. It follows from (33) that we ca pick a subset Y 0 of Y such that (3) c 5 3= < X xx X yy 0 u(xy) < c5 3= + c p : The the umber of vertices x X such that u(xy) 6= 0 for some y Y 0 is at most c 5 3= + c p < =, provided is sucietly large. We costruct a partitio of Y 0 [ X as follows. Let Y 0 = Y 1 [ Y be a radom partitio of Y 0, where each y Y 0 is i Y 1 or Y idepedetly with probability 1=. Let X 0 be the set of vertices x X such that w(xy) 6= 0 for some y Y 0. Let Z 1 = Y 1 [ x X 0 : u(x; Y 1 ) < u(x; Y ) ad Z = Y [ x X 0 : u(x; Y ) u(x; Y 1 ) : Fially, exted the partial partitio Z 1 [ Z to a partitio of X [ Y 0 by addig the remaiig members of X arbitrarily so that the al partitio W 1 [ W satises (35) X X t(v) vw 1 vw t(v) X vw 1 t(v) + 1: (Note that this is possible sice Z 1 [Z cotais at most = elemets of X, provided is sucietly large.) Now by Lemma 9 below, for x X 0 we have E u(x; Y1 ) u(x; Y ) E P U i=1 1 1 p U, where U = P yy 0 u(xy).

28 8 B. Bollobas ad A. D. Scott So by (33), (3) ad (35), the expected weight of edges joiig W 1 ad W is at least E t(x [ Y 0 ) = + u(y 0 )= + X xx max u(x; Y 1 ); u(x; Y ) 1 w(x [ Y 0 ) + jxj 1 1 w(x [ Y 0 ) + jxj 1 + E X xx + 1 X u(x; Y1 ) u(x; Y ) xx X yy 0 u(xy) 1= 1 w(x [ Y 0 ) + jxj 1 + c 5 3= p c 1= 1 w(x [ Y 0 ) + jxj 1 + c p : As i (3), this yields a cotradictio, so we may assume that (36) X xx X u(xy) < c5 3= : yy It follows that there are at most c 5 3= vertices of X which are icidet to a edge e with u(e) 6= 0. Let G 0 be the graph with edges e E(G) : u(e) 6= 0 with edgeweightig u ad vertices v V (G) : u(vw) 6= 0 for some w V (G). The, by (9) ad (36), jg 0 j c 5 3= + jy j c 6 3= which by (7) is smaller tha = for sucietly large. Fially, let W 1 [ W be a partitio of G 0 such that the total weight of edges betwee W 1 ad W is at least f(g 0 ). Sice jg 0 j < = provided

29 Better Bouds for Max Cut 9 is sucietly large, it follows from (30) that we ca exted W 1 [ W to a partitio V 1 [ V of V (G) such that t(v1 ) t(v ) 1: P Let t = vv (G) t(v). The w(g) = X vwe(g) t(v)t(w) + t + u(g 0 ); X vwe(g) u(vw) while the weight of edges betwee V 1 ad V is at least t t(v 1 )t(v ) + u(v 1 ; V ) + f(g 0 ): Therefore t f(g) t + f(g 0 ) + f w w(g) t : We have used a estimate i the proof above that is a immediate cosequece P of the followig trivial lemma. We are iterested i radom sums " i a i, where the a i are idepedet Beroulli radom variables takig values +1 ad 1 with probability 1=. We shall write istead of " i. Lemma 9. Let s 1 + +s k be a partitio of ad t 1 + +t l a reemet of s s k. The E kx s i E i=1 lx t i : i=1

30 30 B. Bollobas ad A. D. Scott Proof. It is eough to cosider the simple reemet whe l = k + 1, P P s i = t i for i < k ad s k = t k + t k+1. We may couple the sums s i ad ti so that s i ad t i have the P same sig P for i < k, while s k, t k ad k1 t k+1 are idepedet. Let S = s k1 i=1 i = t i=1 i. We must show EjS s k j EjS t k t k+1 j: Now for real umbers 0 ad L 0, jl + j + jl j L + + jl j L + + jl j + = jl + j + jl j; so i geeral for jj jj ad ay L, jl + j + jl j jl + j + jl j: Coditioig o the value of S, we see that sice jt k t k+1 j jt k + t k+1 j, we have S + (tk t k+1 ) + S (tk t k+1 ) js + tk + t k+1 j + js t k t k+1 j ad so EjS t k t k+1 j 1 js + t k + t k+1 j + 1 js t k t k+1 j = EjS s k j: The result follows immediately. For what value of is the quatity i Theorem 8 miimized? Suppose 0 m < 0 +1, say m = 0 + r. Clearly we must have 0 + 1, ad it follows from (30) that we may assume > 0 c p 0. We claim that () is miimized with = 0 or = Now sice the argumet

31 Better Bouds for Max Cut 31 of Lemma 3 applies to multigraphs as well as graphs, we ca deduce the Edwards formula for multigraphs: f w (m) m + r m 8 + O(1): Sice f w (m) f(m), it follows from (6) that (37) f w (m) = m + r m 8 + O(m1= ): If = 0 t, with 0 t c p, the so by (37), + f w m m = t( 0 t) + t + r = m + + t( 0 t) + t 8 + r! 1= + O( 0 t + r + 1) 1= : Provided is sucietly large, ad 0 t < c p 0, this is miimal whe t = 0. We coclude the followig. Theorem 10. For every sucietly large positive iteger m, (38) f w (m) = mi % ($( + 1) ; + f w m ) ; where is deed by m < + 1 :

32 3 B. Bollobas ad A. D. Scott As remarked i the itroductio, probably f w (m) = f(m) for every m. Eve if this is ot true, it seems likely that (38) holds with f(m) i place of f w (m) whe m is sucietly large.. Extremal graphs for Max Cut We ca apply Theorem 10 to obtai extremal graphs ad multigraphs i more cases tha Theorem 1 ad Lemma. Let us ote rst that ay iteger m ca be writte i the form 1 m = + k + + for some k > 0, where 1 > > k ad each i i tur is chose to be as large as possible. For 1 i < k, let M i = i1 + $( i + 1) % ad dee M = k the it follows by repeated applicatio of Theorem 10 that, provided k1 is sucietly large, f w (m) = mi fm 1 ;... ; M k1 ; Mg: For 1 i < k, we ca obtai a graph G with m edges ad f(g) = M i by deletig i +1 i k edges from the graph (39) K 1 [ [ K mi1 [ K i +1;

33 Better Bouds for Max Cut 33 while the graph (0) K 1 [ [ K k has m edges ad o bipartite subgraph with more tha M edges. Note that i both (39) ad (0), we could istead take ay edge-disjoit uio of the complete graphs. Thus there may be may possible extremal graphs. Recall that i the case k =, Theorem 1 asserts that for sucietly large m, the extremal graphs are precisely the graphs (39) ad (0) ad their variats obtaied by takig dieret edge-disjoit uios (ote that the case k = is special, sice we ca take two copies of K 3 istead of K ). Keepig the otatio of the last few paragraphs, we ca exted Theorem 10 for graphs as follows. Theorem 11. Let m be a positive iteger ad dee k, 1 ;... ; k, ad M 1 ;... ; M k1 ; M as above. Suppose that (1) M < mi fm 1 ;... ; M k1 g: The, provided k1 is sucietly large, f(m) = M ad the extremal graphs are obtaied by takig a edge-disjoit uio of K 1 ;... ; K k, uless k =, i which case there is a additioal set of extremal graphs obtaied by takig a edge-disjoit uio of K 1 ;... ; K k1 ; K 3; K 3. Proof. We argue by iductio o k. For k = 1, the result follows immediately from Lemma. For k, we kow from Theorem 10 ad example (0) that f(m) = f w (m) = M. Let G be a graph with m edges ad f(g) = f(m). As i the proof of Theorem 1, we ca decompose G as the edge-sum of K ad H, where H is a weighted graph i which all edges are weighted 1. Furthermore, ay partitio of H ca be exteded to a optimal partitio of K, so f(g) = f(k ) + f(h) ad we must therefore

34 3 B. Bollobas ad A. D. Scott have = 1 ad so w(h) = + + k. If H has a edge xy with egative weight, the cotractig xy gives a weighted graph H 0 with w(h 0 ) > k + + : It follows from Theorem 10 ad the iductive hypothesis that f(h 0 ) k + 1 ad so, sice f(h) f(h 0 ), we obtai f(g) f(k 1 ) + f(h) > M, which is a cotradictio. + + k Thus all edges of H must have weight +1, so we ca cosider H as a uweighted graph, with edges. For i < k, let Mi 0 = + + i1 + $( i + 1) % ad let M 0 = + + k it follows from (1) that M 0 < mi fm 0 ;... ; M 0 k1 g: Thus we may apply the iductive hypothesis to H: immediately. the result follows A similar argumet gives the followig result for weighted graphs. Theorem 1. Uder the coditios of Theorem 11, f w (m) = M

35 Better Bouds for Max Cut 35 ad the extremal weighted graphs are the edge sums of K 1 ;... ; K k, uless k =, i which case the edge sums of K 1 ;... ; K ; K k1 3; K 3 are also extremal. Proof. We argue as i the proof of Theorem 11, except that we use the decompositio of Theorem 1 below. Note that i the decompositio Kt H if f(kt ) = bt =c the Kt must ot have bee cotracted What happes whe M mi fm 1 ;... ; M k g? We cojecture that the atural extesio of Theorem 1 should hold: the extremal grahs are obtaied by deletig edges i (39). The weighted case seems more complicated. Part II: Algorithms for Max Cut 5. A extremal algorithm for Max Cut I this secod part of the paper, we tur from extremal questios to the problem of dig polyomial time algorithms that give large bipartite subgraphs of a graph or edge-weighted graph. I this sectio we describe a liear time algorithm that, give a graph with total edge weight m, gives a bipartite subgraph of weight at least f(m). I subsequet sectios, we give a liear time algorithm that, for graphs G of weight m, ds a cut of weight at least m= + p m=8 + k if such a cut exists a d otherwise ds a optimal cut, ad a algorithm that approximates the order of magitude of f(g) m= p m=8. We remark that it ofte appears to be easier to d eciet algorithms for partitioig uweighted graphs tha it is for partitioig weighted graphs. We shall assume below that we are dealig with graphs that have iteger edge-weightigs, where we al low both positive ad egative weights. We may also assume that our graphs are coected: give a graph G with vertices ad e edges, we ca idetify a vertex from each compoet i time O(e + ) to obtai a graph H with f(g) = f(h); ay biparti tio of H yields a equivalet bipartitio of G i time O(). (We should also ote that we have assumed that all arithmetical operatios ca be performed i uit time, regardless of the magitude of edge-weights.)

36 36 B. Bollobas ad A. D. Scott The mai result of this sectio is the followig liear time algorithm. Theorem 13. There is a algorithm that, give a graph G with e edges, iteger-valued edge-weightig w ad total weight m ds a cut of weight at least f w (m) i time O e + jgj. By takig w 1, we obtai the followig immediate corollary for uweighted graphs. Corollary 1. There is a algorithm that, give a multigraph G with m edges ad vertices, ds i time O(m+) a bipartite subgraph of G with at least f w (m) edges. May of the results from Sectios ad 3 have eciet correspodig algorithms. Let us ote rst that Theorem 5 has the followig immediate corollary. Lemma 15. There is a algorithm that, give a coected graph G with vertices ad e edges, ad a edge-weightig w with positive itegers ad total weight m, ds i time O(e+) a bipartite subgraph of G with weight at least w(g) + jgj 1 : We shall d it useful to have the followig lemma. Lemma 16. There is a algorithm that, give a graph G with e edges ad edge-weightig w, ds i time O(e + ) a cotractio of G to a complete graph i which all edges have positive weight. Proof. Begi by deletig all edges with weight 0. We the take a greedy colourig of G (which takes time O(+e)) ad cotract each colour class to a sigle vertex to obtai a weighted complete graph H with jhj = O p e. We ow repeated ly cotract edges of opositive weight util we obtai a graph with all edge weights positive. Sice each cotractio takes time O p e ad there are at most O p e cotractios, the algorithm termiates i time O(e) (ote that we ca deal with all edges with opositive weight i time O(e) by processig oe vertex at a time).

37 Better Bouds for Max Cut 37 We shall also eed algorithmic versios of Lemmas 3, ad 7, ad a result cocerig weighted matchigs. Most of the followig lemma ca be foud i Hofmeister ad Lefma [19]. Lemma 17. We cosider graphs G with vertices, e edges ad edgeweightig w. (i) There is a algorithm ruig i time O(e+) that, give a matchig M i G, ds a cut with weight at least 1 w(g) + 1 w(m): (ii) There is a algorithm ruig i time O(e + ) that, give a proper k-colourig of G, ds a cut with weight at least w(g): k I particular, there is a O(e + ) algorithm that ds a cut with weight at least w(g): jgj (iii) Give a weighted graph G ad a partial partitio V 1 [ V of V (G), we ca d i time O(e + ) a cut of weight at least w(v 1 ; V ) + 1 w(g) w(v1 [ V ) : (iv) There is a algorithm ruig i time O(e+) that ds a matchig of weight at least w(g)=. Proof. Parts (i), (ii) ad (iv) are obtaied by applyig algorithms from [19]. Note that if jgj is odd, we may add i a isolated vertex. Part (iii) follows by usig the greedy algorithm: add each vertex of V (G) (V 1 [ V ) i tur to whichever side of the partitio gives the heavier cut.

38 38 B. Bollobas ad A. D. Scott We ow prove the mai result of this sectio. m < +1, fw (m) = mi ( + 1) = ; b =c + Proof of Theorem 13. Let m 0 be large eough for (38) to apply: that is, for m m 0, ad f w m m. By (1) (for weighted graphs) ad (6), there is m 1 such that, provided m m 1, () m + r m 8 1 f w(m) m + r m 8 + m1= : The mai part of our algorithm will apply to edge-weighted graphs with weight at least M = max fm 0 ; m 1 ; Kg, where K is a large xed costat; we deal separately with graphs of smaller weight. We begi by cotractig G to a complete weighted graph with the algorithm from Lemma 16. We may thus assume that G is a complete weighted graph with e edges, all of positive weight, ad that w(g) = m. We shall show that we ca d a cut of weight at least f w (m) i time O(e). If w(h) M, the sice there are oly itely may graphs with positive edge weights ad total weight at most M (that is, multigraphs with at most M edges), we ca examie all partitios of H i xed time, or else store all optimal partitios as a look-up table. Note that this may itroduce a large costat ito the time or space complexity of the algorithm: we retur to this poit after the proof. We may therefore assume that G has weight m M. Our algorithm follows parallel to the proof of Theorem p 8. Note that if at ay time we d a cut with weight at least m= + m=8 + m 1= the we ca halt the algorithm immediately. (3) Dee the iteger by m < + 1 : The, by (), () f w (m) m + + 3p :

39 Better Bouds for Max Cut 39 Sice G is complete we have jgj : O the other had, by Lemma 17(ii) we ca d i time O e + jgj a bipartite subgraph of G with weight at least jgj m. Thus we ca halt the algorithm if (5) r m > m m jgj m1= ; which is true for sucietly large m uless (6) jgj > 8 p Note that this implies m is O(e). Now we d a small set of edges that meets all edges i G of weight more tha 1. We ca d a maximal matchig M i G i time O(e), by choosig greedily edges of weight more tha 1 ad the llig out with edges of weight 1. By Lemma 17(i) we ca d i time O(e) a bipartite subgraph of G with weight at least m + w(m) =. Thus we are doe if w(m) > p m= + m 1=. Otherwise, provided m is sucietly large, usig (3) we see that M cotais at most (7) r m + m1= jgj < 7 p edges of weight greater tha 1. We obtai either a bipartite subgraph with weight at least f w (m), i which case we halt the algorithm, or else a set of at most 7 p edges, ad hece a set Y of at most 1 p vertices of G, meetig all edges with weight more tha 1. Let X = V (G) Y, so by (6) ad (7), jxj = jgj jy j p : Note that G[X] is a complete graph i which all edges have weight 1. For y Y, let us cosider the edges betwee y ad X. As i the proof of