Bisecting Sparse Random Graphs

Transcription

1 Bisecting Sparse Ranom Graphs Malwina J. Luczak,, Colin McDiarmi Mathematical Institute, University of Oxfor, Oxfor OX 3LB, Unite Kingom; Department of Statistics, University of Oxfor, Oxfor OX 3TG, Unite Kingom; Receie 3 September 999; accepte June 000 ABSTRACT: Consier partitions of the vertex set of a graph G into two sets with sizes iffering by at most : the bisection with of G is the minimum over all such partitions of the number of cross eges between the parts. We are intereste in sparse ranom graphs G with ege probability cn. We show that, if cln 4, then the bisection with is Ž n. n, c n with high probability; while if cln 4, then it is equal to 0 with high probability. There are corresponing threshol results for partitioning into any fixe number of parts. 00 John Wiley & Sons, Inc. Ranom Struct. Alg., 8, 338, 00. INTRODUCTION A leel -partition of a set V is a partition of V into sets with sizes iffering by at most. Given a partition of the vertex set of a graph G, the cross eges are the eges between the parts. The -section with w Ž G. is the minimum number of cross eges over all level -partitions of the vertex set V. When we refer to the bisection with, which has receive much attention, partly because of its relation to very large scale integrate circuit Ž VLSI. esign, 5, 7, 9, 0, 6. For general results on the -section with see 4. It is ifficult Ž NP-har. to etermine the -section with of a graph for any fixe Ž8,., an thus there has been interest in consiering its behavior for Corresponence to: C. McDiarmi. * Supporte by a British Telecommunications stuentship. 00 John Wiley & Sons, Inc. 3

2 3 LUCZAK AND MCDIARMID ranom graphs. We consier the stanar ranom graph G ž/, with vertices n, p,,..., n, an where the n possible eges appear inepenently with probability p. We shall be intereste in the sparse case, where ppž ncn. for some constant c0. This case has been investigate by MacGregor 4, Golberg an Lynch, an Alous. For each constant c, a value fž c. 0 is known such that a.s. the bisection with is at least fž c. n, so that for each c there is a linear lower boun on the bisection with. We use a.s. to mean with probability tening to as n. There are also upper bouns known for the bisection with. In particular, in it is shown that, if 0cln 4 then a.s. the bisection with is on. This follows from the result that, when cln 4, the maximum size of a component in G is about n. Golberg an Lynch n, c n further propose a conjecture which woul imply that the upper boun extens to cover all c, so that if 0c then a.s. the bisection with is on. In this paper we strengthen some of the above results, an isprove the conjecture of Golberg an Lynch: the threshol for linear bisection with is at cln 4, not at c. Our main theorem is as follows. Given an integer, let c ln. Note that c ln 4 an that c ecreases to as. We say that the events A Ž n hol with high probability if there exists 0 such that PA o e.. n Theorem. Let be an integer, an consier the -section with w. For any cc, there exists a constant 0 such that an for any 0cc, w G n with high probability, n, c n w G 0 with high probability. n, c n When pcn the giant component of the ranom graph Gn, p has size about n. Ž We iscuss the giant component in Section.. A main step in proving Theorem is the following result, which is perhaps of inepenent interest. Given, 0, a Ž,. -cut of a graph GŽ V, E. is a partition of V into two sets both of size at least V such that there are at most V cross eges. Lemma. Let c. For any 0, there exists 0 such that, with high probability, the giant component of G has no Ž,. -cut. n, c n n. PROOF Let L Ž G. j enote the jth largest orer of a component of a graph G. Recall the following result concerning the components of ranom graphs ŽEros an Renyi 6, see. 3.

3 BISECTING SPARSE RANDOM GRAPHS 33 Ž. ct Lemma 3. a For c, the equation te 0 has a unique positie solution Ž c., an Ž c. gies a continuous strictly increasing function from Ž,. to Ž 0,.. Ž b. Let c be a constant, an let pcn. Let Ž n. as n. Then the largest orer of a component satisfies L Gn, p c n n n a.s. an the secon largest orer of a component satisfies where cln c. L G Ž. ln n a.s. n, p By the results above, when c the ranom graph Gn, c n a.s. has a unique component of largest orer, which is much larger than all the other components. We efine the giant component of a ranom graph Gn, c n to be the lexicographi- cally first component of largest orer. In fact, there is a unique component of maximum orer with high probability. We nee a lemma on concentration, which follows irectly from Theorem 3.9 of 3. Ž. Lemma 4. Consier a graph inariant f such that f G f G b wheneer the graphs G an G iffer in only one ege. Let c0 an let pcn. Then for any 0 n, p n, p ž / bcž 3. b ž / ž / P f G E f G n exp n. Lemma 5. Let c0, let pcn, an let k be a positie integer. For any 0, the number Z Z Ž G. of components of orer k satisfies k k n, p Z EZ n with high probability. In particular, the number Z with high probability. k c Z e n n k of isolate ertices satisfies Proof. Apply Lemma 4 with fgz G an b. k Lemma 6. Let c an let pcn. Let Ž c. be as in Lemma 3. For any 0, both L G Ž c. n n an with high probability. n, p LŽ Gn, p. n Proof. The first part is from O Connell 5. For the secon part, we use two results from 3. First, for each k,,..., the number Z Z Ž G. of compok k n, p

4 34 LUCZAK AND MCDIARMID nents that are trees of orer k satisfies as n; an secon, k kc ke Zk n kc e k! k kc Ž c. Ý Ž kc. e k!. k These results together imply that for any 0 there exists k 0 such that k 0 kež Z. Ž Ž c.. n Ý k for all n sufficiently large. Now use the first part an Lemma 5. k Lemma 7. For any 0, there exist an n0 such that the following hols. For all nn 0, an for all connecte graphs G with n ertices, there are at most Ž. n bipartitions of G with at most 0 n cross eges. Proof. Let T be an arbitrary spanning tree of G. Any -partition S, S of T is etermine uniquely by the corresponing set of cross eges, together with the specification for each cross ege of which of its enpoints is in S. For as T is connecte, the cross eges specify a nonempty subset S of S, an then S is the set of vertices such that there is a path from to one of the vertices in S where this path oes not use any of the cross eges. ŽIf S then no path from to S can avoi the cross eges, an if S then any shortest path from to S avois the cross eges.. Hence, since T has n eges, the number of -partitions of T Ž an hence also of G. with at most n cross eges is no more than ž / n n j n Ý j ž Ž. / j n O n, assuming. Now let 0. As 0, ŽŽ... Hence, for sufficiently small an n sufficiently large, there are at most Ž. n partitions with at most n cross eges. Proof of Lemma. Let pcn. The iea of the proof is to construct the ranom graph Gn, p as the union of two new ranom graphs, such that the giant component U of the first new ranom graph Gn, p is close to the giant component of the original ranom graph, there are not too many -partitions of U in Gn, p with few cross eges, an the secon new ranom graph Gn, p puts many eges into each of these -partitions. Let 0. Let c c be such that Ž c. Ž. Ž c., an let ccc. Let pcn, an let ž / pp c p O. p n n

5 BISECTING SPARSE RANDOM GRAPHS 35 We may form the ranom graph Gn, p as the union of two inepenent ranom graphs as follows. We generate inepenent ranom graphs Gn, p an G n, p, an form a thir ranom graph on the same vertex set,...,n4 with ege set the union of the ege sets of these two graphs. This thir graph has exactly the istribution of G, since pž p.ž p. n, p. Let U an U enote the giant components of the ranom graphs Gn, p an G n, p, respectively. It is convenient an shoul cause no confusion to use these symbols to enote also the corresponing vertex sets. We next efine four events A,..., A involving U an U, an efine three Ž small. 4 positive constants,, an. Let A A Ž n. U Ž. Ž c. n, an let 4 4 A A Ž n. U U an U Ž. U. Then by Lemma 6, both the events A an A let 0 satisfy ž / Ž c. c, expž 8., 3 hol with high probability. Let an let 0 be the minimum of an Ž from Lemma 7.. Let an A A Ž n. U has a Ž, n. -cut in G, n, p 4 A A Ž n. U has a Ž,. -cut in G. 4 4 n, p We aim to show that the complementary event A3 hols with high probability. We claim that AA3A 4. Ž. To show this, suppose that A hols an that U has a Ž,. -cut into BC. Let B BU an C CU. Then U has a partition into B C, both B an C are at least U U U U, an the number of cross eges is at most U U. Thus the event A4 hols, an we have prove the claim Ž.. It follows that PŽ A. PŽ A. PŽ A Our remaining task is to show that A4 hols with high probability. By Lemma 7 an our choice of, there are at most Ž. n Ž,. -cuts of U in G n, p. Consier any one such cut, partitioning U into BC say. Conitional on A, in the inepenent ranom graph G, the number of possible eges n, p

6 36 LUCZAK AND MCDIARMID between B an C is ž / ž / 3 B C U c n. Thus, conitional on A, the number X of eges in Gn, p between B an C has a binomial istribution with mean B C p nn. But if Y has a binomial istribution with mean m then PYm expž m8. see, for example Theorem A3, or 3 Theorem.3Ž. c. Hence an so PŽ X na. expž 8. expž n8. Ž., / PŽ A4A. ž. Finally, PŽ A. PŽ A A. PŽ A. 4 4, an it follows that A4 hols with high probability, as require. We nee two more lemmas in orer to complete the proof of the theorem. Lemma 8. Let be an integer, let GŽ V, E. be a graph with n ertices, an let UV be such that U n n, where 0Ž.. Suppose that there is a leel -partition of G with at most x cross eges. Then there is a bipartition of G with at most x cross eges, such that both parts contain at least n ertices in U. Proof. Let S,...,S be a -partition of V such that t S U n for each i,..., an t t. Now t t U t n. i i Thus, if t n then we may partition V into S an S S. So we may suppose that t n. Let j,...,4 be such that Note that tj t n, an so t t nt t. j j / ž / t tj ž n n n. Hence we may partition V into S Sj an Sj S. Lemma 9. Gien a list of positie integers aa ak summing to n, if a n an the number z of inices i such that ai is at least Ž.Ž a., then there is a partition of a,...,ak into sets, with sums iffering by at most. n n

7 BISECTING SPARSE RANDOM GRAPHS 37 Proof. First we put a into the ith set for i,...,. Ž If k we are one.. i We then a the remaining ai one by one to the sets arbitrarily, so that the sum of the numbers in each set is at most n. This can never fail, since when we try to a a number x, the total resiual capacity, namely n less the sum allocate, is at least zxž x.. Finally, we a the s. Proof of Theorem. First consier the case cc. For some 0Ž., L Ž G. n n, p n with high probability, by Lemma 6, an so the esire result follows from Lemmas an 8. Now let cc. Then by Lemmas 5 an 6, each of the events L Ž G. n, p n, c c L Gn, p e n, an Z e n hol with high probability, an the esire result follows from Lemma 9. Theorem above establishes the first-orer behavior of the -section with of G n, c n, which is what is important for algorithmic implications. The finer behavior exactly on the threshol is not aresse above. Let us simply make the following observations here. Fix an integer, an let pc n. Then, by 4 Ž. P w Gn, p 0 as n. Also, by Lemma 3, if n as n, then ' w G n n a.s. 3 n, p Perhaps this last inequality can be strengthene? ACKNOWLEDGMENT We woul like to thank the referees for helpful comments. REFERENCES D. Alous, The harmonic mean formula for probabilities of unions: applications to sparse ranom graphs, Discrete Math 76 Ž 989., N. Alon an J. Spencer, The probabilistic metho, Wiley, New York, B. Bollobas, Ranom graphs, Acaemic Press, New York, D. Barraez, S. Boucheron, an W. Fernanez e la Vega, The giant component is normal, Combin Probab Comput Ž to appear.. 5 T.N. Biu, S. Chauhuri, F.T. Leighton, an M. Sipser, Graph bisection algorithms with goo average case behaviour, Combinatorica 7 Ž 987., P. Eros an A. Renyi, On the evolution of ranom graphs, Publ Math Inst Hungar Aca Sci 5 Ž 960., A. Frieze an M. Jerrum, Improve approximation algorithms for MAX k-cut an MAX BISECTION, Algorithmica 8 Ž 997., 677.

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