algorithms for graph partitioning problems by means of eigenspace relaxations

Theory and Methodology

Algorithms for graph partitioning problems by means ofeigenspace relaxations

Chih-Chien Tu a, Ce-Kuen Shieh a, Hsuanjen Cheng b,*

a Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan R.O.Cb Department of Mathematics, Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan R.O.C

Received 2 March 1998; accepted 10 November 1998

Abstract

Graph partitioning problems are NP-hard problems and very important in VLSI design. We study relations among

several eigenvalue bounds and algorithms for graph partitioning problems. Also, we design an algorithm for the

problems which performs the following: ®rst it computes the k largest eigenvalues of the a�ne symmetric matrix

function to attain Donath±Ho�man bound; then it calculates a relaxed partition which is an array constant factor of an

eigenspace associated with k eigenvalues; ®nally it generates an actual partition from the relaxed solution of a method

similar to Boppana's algorithm. To compute optimal eigenvalue bounds, one needs to solve eigenvalue optimization

problems which minimize the sum of the k largest eigenvalues of the nonsmooth functions. We use a subgradient

method to compute the Donath±Ho�man eigenvalue bound. Numerical results indicate that although the Donath±

Ho�man bound is not tight for graph partitioning problems, our algorithm can generate optimal partitions. Ó 2000

Elsevier Science B.V. All rights reserved.

Keywords: Optimization; Eigenvalue; Eigenvector; Lower bound; Subgradient method

1. Introduction

Graph partitioning problems are concerned with partitioning the set of vertices of a given graph into kdisjoint subsets. Each subset has a speci®ed size so as to minimize the total weight of edges that connectdistinct subsets. These problems are NP-hard problems (see [11]) and important in VLSI design (see [17]).Several heuristic algorithms perform well in practice (see [16,1,3,13,6,2,19,20]) and have been shown to havegood average case behavior over certain probability distributions on graphs (see [5,4]). We are interested in

European Journal of Operational Research 123 (2000) 86±104www.elsevier.com/locate/orms

* Corresponding author.

E-mail addresses: [email protected] (H. Cheng), [email protected] (C.-C. Tu), [email protected] (C.-K. Shieh).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 0 6 0 - 0

the application of spectral methods which are related to eigenvalue bounds and eigenvector space utili-zations (see [9,1,3±5,20,10]).

In Section 2, we investigate the relations among the Donath±Ho�man eigenvalue bound [9], Barnes'algorithm [3,6], Boppana's eigenvalue bound [4] and Rendl±Wolkowicz's projection model [20]. In Sec-tion 3, we review the Donath±Ho�man eigenvalue bound for graph partitioning problems. In Section 4, wedesign a partitioning algorithm which extends the combination of Barnes' and Boppana's algorithm. For agraph partitioning problem, ®rst, our algorithm computes k eigenvalues to attain the optimal Donath±Ho�man bound; then, it generates a relaxed partition which is the eigenspace associated with k eigenvalues;®nally, an actual partition is computed by using a method, similar to Boppana' algorithm, on the relaxedpartition, i.e. the eigenspace. Again, to compute the optimal Donath±Ho�man eigenvalue bound, one needsto solve an eigenvalue optimization problem which minimizes the sum of the k largest eigenvalues of thea�ne symmetric matrix function. We solve this problem with a subgradient method from Cullum andDonath [7]. In Section 4.1, we test our algorithm with some graphs for bisection problems, i.e. equal-sizedbisection and nonequal-sized bisection problems. In Section 4.2, we solve an equal-sized graph trisectionproblem.

2. Preliminary

Consider a connected, weighted, and undirected graph �V ;E;G� without any self-loop of nodes, where Vis the set of nodes, E is the set of edges of the graph, and G is the associated adjacency matrix. Therefore,Gij � 0 if i � j or fi; jg is not an edge in E and Gij represents the positive weight of the edge fi; jg iffi; jg 2 E. De®ne jV j � n to be the number of nodes, jEj � sum�G�=2 to be the sum of the weights of alledges, where sum�G� is equal to the sum of all n2 entries of the adjacency matrix G. Let m ��m1;m2; . . . ;mk�T 2 Rk be a vector of k integers and m1 P m2 P � � � P mk > 0, where

Pki�1 mi � n. De®ne

hV ;E;G;mi to represent a partitioning problem. Let f�Vi ;Ei� j i � 1; . . . ; kg be k subgraphs, where· Ei � E, i � 1; . . . ; k;· V � Sk

i�1 Vi and Vi \ Vj � ; for i 6� j;· jVi j � mi; i � 1; . . . ; k, and

Pki�1 mi � n.

De®ne ��V1;E1�; �V2;E2�; . . . ; �Vk;Ek�� to represent a partition for the graph �V ;E;G�. The sum of theweights of edges that connect these k subgraphs is jEj ÿPk

i�1 jEij. Thus, the graph partitioning problem,hV ;E;G;mi, is to generate a partition ��V1;E1�; �V2;E2�; . . . ; �Vk;Ek�� such that jEj ÿPk

i�1 jEij has theminimum value among all possible partitions.

Boppana's algorithm solves the equal-sized bisection problem, i.e. m1 � m2 � n=2. Let G be the adja-cency matrix associated with the connected, weighted and undirected graph �V ;E;G�, where jV j � n and nis an even integer. Let U be the set of all n by n diagonal matrices with real matrix elements and denote

K � U 2 U j tr�U�(

�Xn

i�1

Uii � 0

):

For a given partition �V1; V2�, jV1j � jV2j � n=2, a vector x 2 Rn is de®ned by

xj � 1 if node j 2 V1;ÿ1 if node j 2 V2:

��1�

Let un be an n-vector of all 1's. Thus, uTn x � 0 and jjxjj � �Pn

i�1 x2i �1=2 � ��

np

. For any U 2K, a functionf �x;U� in [4] is de®ned by

C.-C. Tu et al. / European Journal of Operational Research 123 (2000) 86±104 87

f �x;U� �Xfi;jg2E

Gij�1ÿ xixj�2

� 1

4

Xi2V

Uii�1ÿ x2i �; �2�

where

1ÿ xixj

2� 1 if edge fi; jg 2 E and nodes i; j belong to different subsets;

0 otherwise:

��3�

It is clear that the ®rst sum in the expression for f is equal to the sum of the weights of the edges thatconnect two subgraphs �V1;E1� and �V2;E2�, i.e. jEj ÿ jE1j ÿ jE2j. The second sum is zero because the co-ordinates of the vector x are �1. Combining these two facts implies that f �x;U� represents the sum of theweights of the edges that connect two subgraphs �V1;E1� and �V2;E2�. Moreover, for any U 2K, f �x;U�can be reduced by

f �x;U� �Xfi;jg2E

Gij�1ÿ xixj�2

� 1

4

Xi2V

Uii�1ÿ x2i �

�Xfi;jg2E

Gij

2ÿXfi;jg2E

Gijxixj

2ÿ 1

4

Xi2V

Uiix2i �

1

4

Xi2V

Uii

� sum�G�4

ÿXfi;jg2E

Gijxixj

2ÿ 1

4

Xi2V

Uiix2i

� sum�G�4

ÿ xTGx4ÿ xTUx

4

� sum�G� ÿ xT�G� U�x4

: �4�

Let S � fx 2 RnjuTn x � 0g be a linear space and denote P0 � I ÿ unuT

n =n a projection operator on S,where I is the identity operator on Rn. Let y � x=jjxjj be a unit n-vector and

x � yjjxjj � ��np

y: �5�For any U 2K, a function g�U� is de®ned by

g�U� � minjjxjj� ��np ;x2S

ff �x;U�g

� minjjxjj� ��np ;x2S

sum�G� ÿ xT�G� U�x4

� ��6�

� minjjxjj� ��np sum�G� ÿ xTP0�G� U�P0x

4

� ��7�

� minjjyjj�1

sum�G� ÿ n�yTP0�G� U�P0y�4

� ��8�

� sum�G� ÿ nk1�P0�G� U�P0�4

; �9�

where k1�P0�G� U�P0� is the largest eigenvalue of the matrix P0�G� U�P0 and y is the correspondingeigenvector. Removing a constraint x 2 S from Eq. (6) is equivalent to applying the projection operator P0

on the vector x in Eq. (7). Eq. (8) is obtained by normalizing vector variable x from Eq. (7). Note that the2-norm of any eigenvector is 1.

88 C.-C. Tu et al. / European Journal of Operational Research 123 (2000) 86±104

The Boppana's eigenvalue lower bound of graph bisection width, b1�G�, for adjacency matrix G is

b1�G� � maxU2Kfg�U�g

� maxU2K

sum�G� ÿ nk1�P0�G� U�P0�4

� �: �10�

Notice that g�U� is a concave function in linear space K so b1�G� is the optimal solution of the concavefunction on K. De®ne

F1�G� � maxU2Kfkn�ÿP0�G� U�P0�g; �11�

where k1�ÿP0�G� U�P0� is the largest eigenvalue and kn�ÿP0�G� U�P0� is the smallest eigenvalue of thematrix ÿP0�G� U�P0. Eq. (10) can be further expressed as follows:

b1�G� � maxU2K

sum�G�4

�ÿ n

4k1�P0�G� U�P0�

�� sum�G�

4� n

4maxU2Kfÿk1�P0�G� U�P0�g

� sum�G�4

� n4

maxU2Kfkn�ÿP0�G� U�P0�g

� sum�G�4

� n4

F1�G�: �12�

Let U � Diag�U 11; . . . ; Unn� be a diagonal matrix, where U attains the optimal solution of Eq. (11); that is,

F1�G� � maxU2Kfkn�ÿP0�G� U�P0�g � kn�ÿP0�G� U�P0� � ÿk1�P0�G� U�P0�:

Finally, we compute the eigenvector y corresponding to the largest eigenvalue of the matrix P0�G� U�P0,i.e. k1�P0�G� U�P0�. From the Eq. (5), the eigenvector y is a constant factor of the vector x which is de®nedin Eq. (1). Thus, the graph bisection can be obtained by using either x or y. Output the bisection that hasthe n=2 largest components of the eigenvector y on one side and the n=2 smallest components on the otherside. Certain numerical results of both the equal-sized bisection and the neighborhood of the equal-sizedbisection problems are studied in Tu and Cheng [21].

In [20,1], both indicator matrix X 2 Rn�k and partition matrix P 2 Rn�n are de®ned to represent apartition solution for hV ;E;G;mi. We describe X and P below. The jth column of X , i.e.X �: ; j� � �X1j;X2j; . . . ;Xnj�T, is the indicator vector which represents the subset Vj, where

Xij � 1 if node i 2 Vj;

0 otherwise:

��13�

From the assumptions V � Ski�1 Vi and Vi \ Vj � ;, i 6� j, it has

Pkj�1 Xij � 1 for 16 i6 n and

Pni�1 Xij � mj

for 16 j6 k. Let uk be a k-vector of all 1's. Any indicator matrix X has the following properties:

Xuk � un; X Tun � m; X TX � Diag�m�; X 2 Rn�k is a �0; 1�-matrix: �14�Note that Diag�m� 2 Rk�k is a diagonal matrix with diagonal elements vector m. Let Xm

n denote the set of allindicator matrices, where m � �m1;m2; . . . ;mk�T and jV j � n. That is, for any X 2 Xm

n , X has properties (14).A partition matrix P 2 Rn�n is de®ned by

P � XX T; �15�


where X 2 Xmn . Let Pm

n denote the set of all partition matrices. Let Di 2 Rmi�mi ; i � 1; . . . ; k; denote a squarematrix of all 1's. Let ~P be a block diagonal matrix and denoted as follows:

~P �D1

D2

. ..

Dk

2666437775: �16�

It is easy to see that we can write ~P as ~X ~XT, where ~X 2 Rn�k and

~X ij � 1 ifPiÿ1

k�1

mj � 16 i6Pi

k�1

mj;

0 otherwise:

8<: �17�

From Eqs. (15)±(17), we have the following lemma.

Lemma 1. For any P 2 Pmn , there exist a unique X 2 Xm

n and a permutation matrix Q 2 Rn�n such thatP � XX T and X � QT ~X . Therefore

P � XX T � QT ~X ~XTQ � QT ~PQ:

De®ne tr�AB� �Pni�1�AB�ii �

Pni�1

Pnj�1 AijBji, where A;B 2 Rn�n. Hence, tr�X TGX �=2 represents the sum

of the weights associated with edges that lie in these k subgraphs which are generated by a partition X .Using Lemma 1, we get

tr�X TGX � � tr�GXX T� � tr�GP� � tr�GQT ~PQ�: �18�Thus, hV ;E;G;mi can be solved by one of the following equivalent models:

�GP1�

max1

2tr�X TGX �

s:t: Xuk � un;

X Tun � m;

X TX � Diag�m�;X 2 Rn�k is a �0; 1�-matrix;

8>>>>>>><>>>>>>>:�19�

�GP2�

max1

2tr�GQT ~PQ�

s:t: Qun � un;

QTun � un;

QTQ � I ;

Q 2 Rn�n is a �0; 1�-matrix:

8>>>>>>><>>>>>>>:�20�

The equivalence of �GP1� and �GP2� easily follows from the matrix equation X � QT ~X , where ~X is the givenmatrix in Eq. (17), and the de®nition of permutation matrices. The purpose of solving �GP1� is to ®nd anoptimal indicator matrix X 2 Xm

n . This problem has been exploited by Rendl and Wolkowicz [20]. They usea projection method combined with a diagonal perturbation method to solve �GP1�. Similarly, the purposeof solving �GP2� is to ®nd an optimal permutation matrix Q 2 Rn�n. �GP2� is a special case of the quadraticassignment problem. A general case of quadratic assignment problem has been studied by Hadley et al. [14].

The problem

minP2Pm

n

fjjGÿ P jjg


is to ®nd a partition matrix P 2 Pmn such that P is the best matrix approximation to G in the Frobenius

norm. For any P 2 Pmn , we have

jjGÿ P jj2 � jjGjj2 � jjP jj2 ÿ 2Xn

i�1

Xn

j�1

GijPij

� jjGjj2 � jjP jj2 ÿ 2Xn

i�1

Xn

j�1

GjiPij

� jjGjj2 � jjP jj2 ÿ 2 tr�GP�; �21�where jjGjj is the Frobenius norm of G. In Eq. (21), we have used the important fact that G is a symmetricmatrix. From Lemma 1, there exists a unique X 2 Xm

n such that

jjGÿ P jj2 � jjGjj2 � jjP jj2 ÿ 2 tr�X TGX �; �22�where jjGjj2 � jjP jj2 �� jjGjj2 �Pk

i�1 m2i � is a constant for all P 2 Pm

n .Minimizing both sides of the Eq. (22) gives

ÿmaxX2Xm

n

1

2tr�X TGX �

� �� 1

4minP2Pm

n

jjGn

ÿ P jj2 ÿ jjGjj2 ÿ jjP jj2o; �23�

where

maxX2Xm

n

1

2tr�X TGX �

� �is �GP1�.

Let U be the set of all n by n diagonal matrices with real matrix elements and denote

K � U 2 U j tr�U�(

�Xn

i�1

Uii � 0

): �24�

Diagonal perturbation does not a�ect the result of any graph partition; that is,

maxX2Xm

n

1

2tr�X TGX �

� �� max

X2Xmn

1

2tr�X T�G

�� U�X �

��25�

for any U 2K. This shows that the self-loops of nodes do not a�ect the graph partitioning problem.Applying a diagonal perturbation on Eq. (21) gives

jj�G� U� ÿ P jj2 � jjG� U jj2 � jjP jj2 ÿ 2 tr��G� U�P � �26�� jjG� U jj2 � jjP jj2 ÿ 2 tr�GP � ÿ 2 tr�UP� �27�� jjG� U jj2 � jjP jj2 ÿ 2 tr�X TGX � ÿ 2 tr�UP�: �28�

Since tr�UP� � 0, Eq. (28) can be expressed as follows:

ÿ 1

2tr�X TGX � � 1

4jj�Gn

� U� ÿ P jj2 ÿ jjG� U jj2 ÿ jjP jj2o: �29�

Minimizing both sides of Eq. (29) gives

ÿmaxX2Xm

n

1

2tr�X TGX �

� �� min

P2Pmn

1

4jj�Gn

� U� ÿ P jj2 ÿ jjG� U jj2 ÿ jjP jj2o

�30�

for any U 2K.


The left hand side of Eqs. (23) and (30) are the same. However, the right hand side of Eq. (30) contains adiagonal perturbation variable U , which is used to improve the eigenvalue bound and the approximatesolution. In the following two sections, we focus on studying the Donath±Ho�man eigenvalue bound andsome partitioning algorithms from the right hand side of the Eq. (30).

3. The Donath±Ho�man eigenvalue bound for the graph partitioning problem

We are given m � �m1;m2; . . . ;mk�T 2 Rk the speci®ed size of a partition and G the adjacency matrix ofthe weighted graph �V ;E;G�. Let sum�G� be the sum of all n2 entries of the adjacency matrix G andsum�G�=2 be the sum of the weights of all edges of the graph �V ;E;G�. Let

sum�G�2

ÿmaxX2Xm

n

1

2tr�X TGX �

� �; �31�

be the sum of the weights of edges that connect distinct subsets, Vi , i � 1; . . . ; k, which are represented bythe optimal partition. Let g�m;G;U� be the function of the sum of the weights of edges that connect distinctsubsets, Vi ; i � 1; . . . ; k. Inserting Eq. (30) into Eq. (31) gives

g�m;G;U� � sum�G�2

� minP2Pm

n

1

4jj�Gn

� U� ÿ P jj2 ÿ jjG� U jj2 ÿ jjP jj2o: �32�

We use Eq. (32) to derive a lower bound for g�m;G;U� and some algorithms for our graph partitioningproblem. The inequality Eq. (33) in Theorem 1 and the Donath±Ho�man lower bound of Eq. (42) can bederived from Eq. (32).

Theorem 1. For any U 2K, we have

g�m;G;U�P sum�G�2

ÿ 1

2

Xk

i�1

miki�G(

� U�); �33�

where m � �m1;m2; . . . ;mk�T 2 Rk is the speci®ed size of a partition and k1�G� U�P k2�G� U�P � � � P kk�G� U� are the k largest eigenvalues of the symmetric matrix G� U .

Proof. For any P 2 Pmn , the set of all eigenvalues of P is

mi j m1f P m2 P � � �P mk > 0; mk�1 � � � � � mn � 0g: �34�Applying the Ho�man±Wielandt inequality [15] to jj�G� U� ÿ P jj2 gives

jj�G� U� ÿ P jj2 PXk

i�1

�ki�G� U� ÿ mi�2 �Xn

i�k�1

k2i �G� U�; �35�

where k1�G� U�P � � � P kn�G� U� are n eigenvalues of the matrix G� U . Then, Eq. (32) becomes


� minP2Pm

n

1

4jj�Gn


Psum�G�

2� min

P2Pmn

1

4

Xk

i�1

�ki�G(

� U� ÿ mi�2 �Xn

i�k�1

k2i �G� U� ÿ jjG� U jj2 ÿ jjP jj2

): �36�

Expanding the termPk

i�1�ki�G� U� ÿ mi�2 gives


Xk

i�1

�ki�G� U� ÿ mi�2 �Xk

i�1

k2i �G� U� �

Xk

i�1

m2i ÿ 2

Xk

i�1

miki�G� U�: �37�

Inserting Eq. (37) into Eq. (36), we get


� 1

4minP2Pm

n

Xn

i�1

k2i �G

(� U� �

Xk

i�1

m2i ÿ 2

Xk

i�1

miki�G� U� ÿ jjG� U jj2 ÿ jjP jj2):

�38�Applying two equations,

jjG� U jj2 �Xn

i�1

k2i �G� U� �39�

and

jjP jj2 �Xk

i�1

m2i ; �40�

to Eq. (38), we have


ÿ 1

2

Xk

i�1

miki�G(

� U�)

for any U 2K. �

Let g�m;G� be the Donath±Ho�man lower bound of the sum of the weights of edges which connectdistinct subsets, Vi ; i � 1; . . . ; k. Then, we have

g�m;G� � maxU2K

sum�G�2

(ÿ 1

2

Xk

i�1

miki�G� U�); �41�

� sum�G�2

ÿ 1

2minU2K

Xk

i�1

miki�G(

� U�): �42�

Donath and Ho�man [9] showed thatPl

i�1 ki�G� U� is a convex function of U �2K� for all l, 16 l6 k,by applying the following inequality from Marcus [18].

Xl

i�1

ki�G� U1=2� U2=2�6 1

2

Xl

i�1

ki�G� U1� � 1

2

Xl

i�1

ki�G� U2� for all U1;U2 2K:

Then, they reformulatePk

i�1 miki�G� U� by the summation of a set of nonnegative multiples of convexfunctions of U as follows:

Xk

i�1

miki�G� U� � �m1 ÿ m2�k1�G� U� � �m2 ÿ m3��k1�G� U� � k2�G� U��

� � � � � �mkÿ1 ÿ mk��k1�G� U� � � � � � kkÿ1�G� U�� mk�k1�G� U� � � � � � kk�G� U��: �43�


The summation of a set of nonnegative multiples of convex functions is also a convex function. Therefore,Eq. (43) is a convex function of U �2K�. Theorem 4.1 of Cullum et al. [8] proves that the functionPl

i�1 ki�G� U� has the minimum value on K for all l, 16 l6 k. We can generalize their result to concludethat the function

Pki�1 miki�G� U� also has the minimum value on K.

Theorem 2. The function

Xk

i�1

miki�G� U�

of U has the minimum value on K.

Proof. At ®rst, we want to show that for each l, l � 1; . . . ; k, the convex functionPl

i�1 ki�G� U� isbounded below for U 2K. In fact, we will prove that

Pli�1 ki�G� U�P ÿ 1 for all U 2K and 16 l6 n.

Prove by contradiction. Suppose thatPl

i�1 ki�G� ~U� < ÿ1 for some ~U 2K. Then kl�G� ~U� < ÿ1=laccording to the de®nition of ki�G� ~U�. On the other hand,

Pni�1 ki�G� ~U� � tr�G� ~U� � tr�G� � 0.

However,Xn

i�1

ki�G� ~U�6Xl

i�1

ki�G� ~U� � �nÿ l�kl�G� ~U� < ÿ1ÿ �nÿ l�=l < 0:

Now we reach to a contradiction. ThereforePl

i�1 ki�G� U� is bounded below for U 2K.By Eq. (43), we can express

Pki�1 miki�G� U� as the summation of a set of nonnegative multiples of the

functionsPl

i�1 ki�G� U�, 16 l6 k. A linear combination of a set of functions which are bounded below onK is also bounded below on K. Thus,

Pki�1 miki�G� U� is bounded below on K.

Next we will prove that the set fU 2K jPki�1 miki�G� U�6 ag is a bounded subset of Rn�n for each real

number a. In Theorem 4.1 of Cullum et al. [8], it is shown that the set fU 2K jPki�1 ki�G� U�6 ag is a

bounded subset of Rn�n for each real number a. From Eq. (43) and the fact thatPl

i�1 ki�G� U�P ÿ 1 forall U 2K and all l, 16 l6 n, it follows thatXk

i�1

miki�G� U�P �m1 ÿ m2��ÿ1� � � � � � �mkÿ1 ÿ mk��ÿ1� � mk

Xk

i�1

ki�G� U�

for all U 2K. Therefore

Xk

i�1

ki�G� U�6 1

mk

Xk

i�1

miki�G

� U� � m1 ÿ mk

!

for all U 2K. It implies that fU 2K jPki�1 miki�G� U�6 ag is a subset of the set

fU 2K jPki�1 ki�G� U�6 �a� m1 ÿ mk�=mkg. Because any subset of a bounded set is also bounded, we

can conclude that the set fU 2K jPki�1 miki�G� U�6 ag is also bounded for each a.

Finally, we prove thatPk

i�1 miki�G� U� has the minimum value on K. Let m0 be the greatest lowerbound of the function

Pki�1 miki�G� U� on K. The number m0 exists because

Pki�1 miki�G� U� is bounded

below on K. We need to show that there exists a U 2K such that m0 �Pk

i�1 miki�G� U�; that is, m0 is theminimum value. By the de®nition of greatest lower bound, for each positive integer N there exists a UN 2Ksuch that

m06Xk

i�1

miki�G� UN �6m0 � 1=N : �44�


Therefore UN 2 fU 2K jPki�1 miki�G� U�6m0 � 1g for all N . Because the set fU 2K jPk

i�1 miki�G�U�6m0 � 1g is a bounded set, there exists a subsequence fVNg of the sequence fUNg such that fVNg is aconvergent sequence. Let us say that fVNg converges to U in Rn�n. U 2K because VN 2K and K is aclosed subspace of Rn�n. We observe that

Pki�1 miki�G� U� is a continuous function on K due to its

convexity on K. From the continuity ofPk

i�1 miki�G� U� and Eq. (44), it is easy to see thatPki�1 miki�G� U� � m0. �

4. Algorithms for the graph partitioning problem

Our graph partitioning algorithm is to ®nd a P 2 Pmn such that

jj�G� U� ÿ P jj2 � minP2Pm

n

jj�G� U� ÿ P jj2; �45�

where U is the matrix in K which attains the optimal Donath±Ho�man lower bound in Eq. (42). That is,U 2K and


ÿ 1

2minU2K

Xk

i�1

miki�G(

� U�); �46�

� sum�G�2

ÿ 1

2

Xk

i�1

miki�G(

� U�): �47�

Note that U can be computed by either semide®nite programming algorithms or subgradient algorithms.The problem of ®nding a P 2 Pm

n with a given U 2K has been studied by Barnes et al. [6]. Here, wepropose another approach to compute a P 2 Pm

n (P � X XT, X 2 Xm

n ) such that

jjG� U ÿ P jj2 � minP2Pm

n

jjG� U ÿ P jj2: �48�

Combining Eqs. (31)±(33) and Lemma 1 gives

sum�G�2

ÿmaxX2Xm

n

1

2tr�X TGX �

� �� sum�G�

2ÿmax

X2Xmn

1

2tr�X T�G

�� U�X �

��49�

� sum�G�2

ÿmaxX2Xm

n

1

2tr�X T�G

�� U�X �

��50�

� sum�G�2

� minP2Pm

n

1

4jj�Gn


�51�

� sum�G�2

� minP2Pm

n

1

4jj�Gn


�52�

Psum�G�

2ÿ 1

2

Xk

i�1

miki�G(

� U�)

�53�

Psum�G�

2ÿ 1

2

Xk

i�1

miki�G(

� U�): �54�


For ®nding a P 2 Pmn to satisfy Eq. (48), we ®rst generate a relaxed solution ~P which satis®es

sum�G�2

� 1

4jj�Gn

� U� ÿ ~P jj2 ÿ jjG� U jj2 ÿ jj ~P jj2o� sum�G�

2ÿ 1

2

Xk

i�1

miki�G(

� U�): �55�

Then, ~P is used to approximate the optimal solution P 2 Pmn . Notice that ~P is not always in Pm

n . Since bothjjG� U jj2 and jjP jj2 in Eq. (52) are constants, we focus on studying the term jj�G� U� ÿ P jj2 in Eq. (52)and citing the following results from Barnes and Ho�man [3] for ®nding a ~P to satisfy Eq. (55). Let G�U � Q1R1QT

1 and P � Q2R2Q2 be the symmetric Schur factorizations (see Golub and Van Loan [12]) of thematrices G� U and P , where Q1QT

1 � I and Q2QT2 � I . This gives

jj�G� U� ÿ P jj2 � jjQ1R1QT1 ÿ Q2R2QT

2 jj2 � jjR1 ÿ QT1 Q2R2QT

2 Q1jj2: �56�

If we apply inequality Eq. (35) to U � U , then we have

jj�G� U� ÿ P jj2 PXk

i�1


i�k�1

k2i �G� U�: �57�

We observe that the equality of Eq. (57) as well as the equality of Eqs. (52) and (53) can be attained byplugging the following three equations into Eq. (56).· QT

1 Q2 � I , i.e. Q1 � Q2, where I is the identity matrix of order n;· R1 � Diag�k1�G� U�; . . . ; kn�G� U��, where k1�G� U�P � � � P kn�G� U�;· R2 � Diag�m1; . . . ;mk; 0; . . . ; 0� 2 Rn�n, where m1 P m2 P � � � P mk.The ®rst condition indicates that both matrices G� U and P share the same eigenspaces. The second andthird conditions formulate the right hand side of the inequality Eq. (57), i.e.

jjG� U ÿ P jj2 � jjQ1R1QT1 ÿ Q2R2QT

2 jj2

� jjR1 ÿ QT1 Q2R2QT

2 Q1jj2

� jjR1 ÿ R2jj2

�Xk

i�1


i�k�1

k2i �G� U�:

The approximate partition matrix ~P can be obtained by computing ~P � Q1R2QT1 . (Note that if P 2 Pm

nthen all entries of P are either 0 or 1.) In spite of ~P 62 Pm

n , the rank of ~P is k because ~P contains exactly k non-zero eigenvalues; i.e., m1; . . . ;mk. The relaxed indicator matrix ~X can be computed from ~P . LetQ1 � �q1; . . . ; qn�, where qj is the jth column of the matrix Q1 and qj is the eigenvector associated withkj�G� U�. De®ne ~Q � �q1; . . . ; qk� and D � Diag�m1; . . . ;mk�. Thus, we have

~P � Q1R2QT1

� ~QD ~QT

� ~QD1=2� ~QD1=2�T

� ~X ~XT: �58�

The approximate indicator matrix ~X can be computed by

~X � ~QD1=2: �59�


Let ~X � �x1; . . . ; xk�, where xj is the jth column of ~X . From Eq. (59), the jth column of ~X is equal to aconstant factor, i.e.

��mjp

, of the jth column of ~Q. That is, xj � ��mjp

qj for j � 1; . . . ; k. Thus, we can con-struct the following algorithm.

Algorithm 1 (Version 1). Graph partition1. Apply a subgradient algorithm to compute a U 2K for g�m;G� in Eq. (42) such that

g�m;G� � 12sum�G� ÿ 1

2

Pki�1 miki�G� U�;

2. Apply a symmetric QR algorithm to generate Q1 � �q1; . . . ; qn� such that

G� U � Q1R1QT1 ;Q1QT

1 � I and R1 � Diag�k1�G� U�; . . . ; kn�G� U��;3. Partition a graph with �q1; . . . ; qk� as follows:

for i � 1: k,Choose the mi largest components of qi to the set Vi ;

end

In Step 3, it may occur that one vertex may be assigned to multiple subsets (subgraphs). ConsultingBoppana's algorithm [4], the above algorithm is modi®ed for the graph bisection problem as follows.

Algorithm 2. Graph bisection1. Apply a subgradient algorithm to compute a U 2K for g�m;G� in Eq. (42) such that

g�m;G� � 12sum�G� ÿ 1

2�m1k1�G� U� � m2k2�G� U��;

2. Apply a symmetric QR algorithm to generate Q1 � �q1; . . . ; qn� such that

G� U � Q1R1QT1 ;Q1QT

1 � I ; and R1 � Diag�k1�G� U�; . . . ; kn�G� U��;3. Compute a vector x � ��

m1

pq1 ÿ ��

m2

pq2 2 Rn;

4. Output the bisection that has the m1 largest components of x on one side and the m2 smallest componentson the other side;

Likewise, we have a modi®ed algorithm for graph partitioning problem as follows.

Algorithm 3 (Version 2). Graph partition1. Apply a subgradient algorithm to compute a U 2K for g�m;G� in (42) such that

g�m;G� � 12sum�G� ÿ 1

2

Pki�1 miki�G� U�;

2. Apply a symmetric QR algorithm to generate Q1 � �q1; . . . ; qn� such that G� U � Q1R1QT1 , Q1QT

1 � I ,and R1 � Diag�k1�G� U�; . . . ; kn�G� U��;

3. Compute ~X � �~x1; . . . ;~xk� � � ��m1

pq1; . . . ;

��mkp

qk� 2 Rn�k;4. Generate �X � ��x1; . . . ;�xk� 2 Rn�k using ~X as follows:

For i � 1: k,�xi � �k ÿ 1�~xi ÿ

Pkj�1;j 6�i ~xj;

end5. Compute the indicator matrix Y 2 Rn�k using �X as follows:

Initially, set Yi;j � 0 for all i; j.For j � 1: k,

a. �uj; Indexj� � sort��xj�, where ui in descending order associated with their original array locations,i.e. Indexj;b. Output ®rst mj vertices to the set Vj; that is,

For i � 1: mj

Y ��Indexj�i; j� � 1;end


c. Put the rest nÿ mj elements of �uj; Indexj� to a queue;end

f Step 5(b) above may cause some j associated with conditionPk

j�1 Yi;j > 1. g6. For i � 1: n;

If (Pk

j�1 Yi;j > 1) Then

Find an a such that ��xa�i � maxf��x1�i; . . . ; ��xk�ig and Yi;a � 1;Set Yi;c � 0 for all c, 16 c6 k and a 6� c;

end

end

fStep 6 above may cause some j associated with conditionPn

i�1 Yi;j < mj.Let S � fVt1 ; . . . ; Vtkg � fV1; . . . ; Vkg, where tj 2 f1; . . . ; kg and

Pni�1 Yi;tj < mtj .g

While (S2 6� ;) do

7. Dequeue a component from �utj ; Indextj�, if head�utj�P head�uti� for all Vti 2 S2;Let �u; ind� � �head�utj�; head�Indextj�� be the dequeued element;

8. Set Yind;tj to 1, ifPk

j�1 Yind;j � 0;9. S2 � S2 ÿ fVtjg, if

Pni�1 Yi;tj � mtj ;

end

De®ne

F2�G� � minU2K

Xk

i�1

miki�G(

� U�): �60�

F2�G� is a nonsmooth function. Eq. (42) can be further expressed as follows:

g�m;G� � sum�G�2

ÿ 1

2minU2K

Xk

i�1

miki�G(

� U�)

� sum�G�2

ÿ 1

2F2�U�: �61�

According to Theorem 2,Pk

i�1 miki�G� U� is a convex function of U and is bounded below on the set K.We solve F2�G� with a subgradient algorithm similar to Cullum and Donath [7]. Thus, the ®rst statement ofthe Algorithm 1, 2 and 3 can be solved. We demonstrate our algorithm for graph bisection problems andgraph trisection problems in the following two sections.

4.1. Numerical results for graph bisection problems with Algorithm 2

Two graph bisection problems, with a graph in Table 1, are discussed: nonequal-sized bisection andequal-sized bisection problem. Outputs of our program for a bisection problem, m1 � 13 and m2 � 7, arelisted in Tables 2 and 3 and for a bisection problem, m1 � m2 � 10, are listed in Tables 4 and 5. The bi-section width for the bisection problem, m1 � 13 and m2 � 7, coincides with the result from [3].

Consider U in Table 4 for the graph bisection problem, m1 � m2 � 10, with a graph in Table 1. The fourlargest eigenvalues of matrix G� U are· k1�G� U� � 5:89175696983242,· k2�G� U� � 3:05603354686931,· k3�G� U� � 3:05603354685185,· k4�G� U� � 2:58605850478356.


Table 2

U is the output of our program for a bisection problem, m1 � 13 and m2 � 7, with a graph from Table 1

U 1;1 0.66751496406330

U 2;2 ÿ0.34904736200109

U 3;3 0.41794804548016

U 4;4 ÿ0.42888087322814

U 5;5 ÿ0.75890012651343

U 6;6 1.76554574124930

U 7;7 ÿ0.45249517808302

U 8;8 ÿ1.54192843622922

U 9;9 ÿ0.05582133471487

U 10;10 ÿ1.01565676561578

U 11;11 0.72661574437735

U 12;12 0.30148946029352

U 13;13 0.05340256266996

U 14;14 ÿ1.93898794976448

U 15;15 ÿ0.27485719685519

U 16;16 ÿ2.08532238837842

U 17;17 1.56323893706142

U 18;18 1.46698198670610

U 19;19 1.11705223011346

U 20;20 0.82210793936907

Table 1

An edge set from Barnes and Ho�man [3]

Node Connections to

1 2,3,4,7,8,17

2 3,10,14,15,16

3 8,12,16

4 7,9,11,17

5 6,9,11,15,16,20

6 7

7 9,15,16

8 10,12,14,16,18

9 12,20

10 12,14,16,19

11 18,19,20

12 13,15

13 14,16,18,19

14 16,18,19

15 16,17,19

17 18

Table 3

A graph bisection is obtained by using Algorithm 2, where n � 20 jV1j � m1 � 13 and jV2j � m2 � 7 (the bisection has 10 edges between

two subgraphs �V1;E1� and �V2;E2�, i.e. jEcj � 10, the lower bound of bisection width is g�m;G� � 6:449)

Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

V1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 1 0

V2 0 0 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1


Table 6

U is the output of Boppana's bisection algorithm for a bisection problem, m1 � m2 � 10, with the graph in Table 1 (Boppana's lower

bound for bisection width is 12.143, i.e. b1�G� � 12:143)

U 1;1 1.27102564435746

U 2;2 0.28727716144961

U 3;3 0.10497732591099

U 4;4 ÿ1.90971677568034

U 5;5 ÿ0.70451835868563

U 6;6 0.46762249207462

U 7;7 ÿ0.61646622322411

U 8;8 ÿ0.76152306184472

U 9;9 ÿ0.50088724979201

U 10;10 ÿ1.61437691605311

U 11;11 ÿ0.39711265479085

U 12;12 1.20336669433984

U 13;13 ÿ0.40277222096750

U 14;14 ÿ1.97381483196651

U 15;15 2.34979142578078

U 16;16 0.17668122384536

U 17;17 0.84615964409596

U 18;18 1.33843208195500

U 19;19 1.36350828387177

U 20;20 ÿ0.52765368467661

Table 4

U is the output of our program for a bisection problem, m1 � m2 � 10, with the graph from Table 1

U 1;1 0.53285693889255

U 2;2 0.15028834214240

U 3;3 0.62508954197516

U 4;4 ÿ1.05907875150326

U 5;5 ÿ0.96304151854414

U 6;6 1.10033865929436

U 7;7 ÿ0.91575433002474

U 8;8 ÿ0.70763620544659

U 9;9 ÿ0.68412458497585

U 10;10 ÿ0.77734013713616

U 11;11 0.24902166001814

U 12;12 0.70928691504805

U 13;13 0.24456579473111

U 14;14 ÿ1.32975903451539

U 15;15 0.10011226102746

U 16;16 ÿ0.92382515142651

U 17;17 0.82015591826952

U 18;18 1.48356483820223

U 19;19 1.12131607617513

U 20;20 0.22396276779651

Table 5

A graph bisection is generated by Algorithm 2 for n � 20 and m1 � m2 � n=2 (although g�m;G� � 10:261, the above bisection is

optimal and has jEcj � 13)

Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

V1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1

V2 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 0


The lower bound of bisection width is g�m;G� � 10:26104741649135. Comparing with the bisection inTables 6 and 7 which is generated by Boppana's bisection algorithm, the graph bisection in Table 5 is anoptimal bisection. Let· b1�G� be the Boppana's eigenvalue lower bound of bisection width;· jEcj be the actual bisection width;· bop be the optimal bisection width.Then, the following inequality always holds:

b1�G�6bop6 jEcj: �62�Also, if jEcj ÿ b1�G� < 1 then jEcj is optimal. Nevertheless, the gap between g�m;G� �� d10:26e � 11� andjEcj (� 13 in Table 5) is 2. Therefore, the Donath±Ho�man eigenvalue lower bound for this problem is nottight. Considering our purpose and the accuracy of the computational output, we slightly change thede®nition of the multiplicity of eigenvalues of the matrix G� U by using the assumption that for a small �,

ki�G� U� � kj�G� U� () jki�G� U� ÿ kj�G� U�j < �

for all i 6� j. Let � � 10ÿ8. The multiplicity of k2�G� U� becomes 2 because k2�G� U� ÿ k3�G� U� �1:7456� 10ÿ11.

Table 7

An actual bisection with the eigenvector, y, corresponding to k1�P0�G� U�P0� (the bisection has jEcj � 13 and b1�G� � 12:143, since

jEcj ÿ b1�G� � 13ÿ 12:143 < 1, this bisection is optimal)

Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

V1 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 1

V2 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 1 0

Table 8

U is the output of our program for a trisection problem, m1 � m2 � m3 � 7, with a graph: three complete subgraphs of 7 nodes �V1;E1�,�V2;E2�, and �V3;E3� with weight 1 for all edges such that �V1;E1� and �V2;E2� are connected by one edge with weight 1, and �V2;E2� and

�V3;E3� are connected by one edge with weight 1

U 1;1 0.02726969886716

U 2;2 0.02726969886715

U 3;3 0.02726969886713

U 4;4 0.02726969886713

U 5;5 0.02726969886713

U 6;6 0.02726969886715

U 7;7 ÿ0.12320386683589

U 8;8 ÿ0.11900132985539

U 9;9 0.03143480139498

U 10;10 0.03143480139500

U 11;11 0.03143480139495

U 12;12 0.03143480139496

U 13;13 0.03143480139500

U 14;14 ÿ0.11900132985535

U 15;15 ÿ0.12320386683544

U 16;16 0.02726969886761

U 17;17 0.02726969886760

U 18;18 0.02726969886761

U 19;19 0.02726969886761

U 20;20 0.02726969886761

U 21;21 0.02726969886761


4.2. Numerical results for graph trisection problems with Algorithm 3

We test Algorithm 3 with three graphs for the equal-sized graph trisection problem. Three graphs are1. Three complete subgraphs of 7 nodes �V1;E1�, �V2;E2�, and �V3;E3� with weight 1 for all edges such that�V1;E1� and �V2;E2� are connected by one edge with weight 1, and �V2;E2� and �V3;E3� are connected byone edge with weight 1.

2. Three cycle subgraphs of 4 nodes �V1;E1�, �V2;E2�, and �V3;E3� with weight 1 for all edges such that�V1;E1� and �V2;E2� are connected by one edge with weight 1, and �V2;E2� and �V3;E3� are connectedby one edge with weight 1.

3. Three complete subgraphs of 7 nodes �V1;E1�, �V2;E2�, and �V3;E3� with weight 1 for all edges such that�V1;E1� and �V2;E2� are connected by one edge with weight 3, and �V2;E2� and �V3;E3� are connected byone edge with weight 5.

Table 9

A graph trisection is computed by using Algorithm 3 for n � 21 and m1 � m2 � m3 � 7 (since jEcj � 2 and g�m;G� � 1:759, the above

graph trisection is optimal)

Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

V1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

V2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1

V3 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0

Table 10

U is the output of our program for a trisection problem, m1 � m2 � m3 � 4, with a graph: three cycle subgraphs of 4 nodes �V1;E1�,�V2;E2�, and �V3;E3� with weight 1 for all edges such that �V1;E1� and �V2;E2� are connected by one edge with weight 1, and �V2;E2� and


U 1;1 0.13857480366746

U 2;2 0.03738868976077

U 3;3 0.13857480366746

U 4;4 ÿ0.21288312873796

U 5;5 ÿ0.17683753733999

U 6;6 0.07518236898224

U 7;7 0.07518236898223

U 8;8 ÿ0.17683753733999

U 9;9 ÿ0.21288312873795

U 10;10 0.13857480366748

U 11;11 0.03738868976077

U 12;12 0.13857480366747

Table 11

A graph trisection is computed by using Algorithm 3 for n � 21 and m1 � m2 � m3 � 7 (since jEcj � 2 and g�m;G;U� � 1:419, the

above graph trisection is optimal)

Node no. 1 2 3 4 5 6 7 8 9 10 11 12

V1 1 1 1 1 0 0 0 0 0 0 0 0

V2 0 0 0 0 1 1 1 1 0 0 0 0

V3 0 0 0 0 0 0 0 0 1 1 1 1


Outputs for the ®rst graph are listed in Tables 8 and 9. Outputs for the second graph are listed in Tables 10and 11. Outputs for the third graph are listed in Tables 12 and 13. The three trisections in Tables 9, 11 and13 are optimal trisections. Notice that the lower bound for the weighted graph, i.e. the third graph, isg�m;G� � 4:649 and the actual trisection is 8. Thus, g�m;G� is not tight in this case.

5. Conclusions

By applying Lemma 1 and Eq. (18), one can solve a quadratic assignment problem by mapping theproblem itself into a graph partitioning problem. Algorithm 3 solves a general graph partitioning problem.Numerical results in previous sections indicate the following:· Boppana's algorithm solves only equal-sized graph bisection problem and produces optimal bisections

with high probability for certain connected, unweighted, undirected graphs;· Boppana's bound is better than the Donath±Ho�man bound for equal-sized graph bisection problem on

most connected, unweighted, undirected graphs;

Table 12

U is the output of our program for a trisection problem, m1 � m2 � m3 � 7, with a graph: three complete subgraphs of 7 nodes �V1;E1�,�V2;E2�, and �V3;E3� with weight 1 for all edges such that �V1;E1� and �V2;E2� are connected by one edge with weight 3, and �V2;E2� and


U 1;1 0.35522433791818

U 2;2 0.35522433791818

U 3;3 0.35522433791818

U 4;4 0.35522433791819

U 5;5 0.35522433791818

U 6;6 0.35522433791818

U 7;7 ÿ0.85743415337017

U 8;8 ÿ0.79912378904672

U 9;9 0.40937567865678

U 10;10 0.40937567865678

U 11;11 0.40937567865678

U 12;12 0.40937567865678

U 13;13 0.40937567865678

U 14;14 ÿ2.39369340079021

U 15;15 ÿ2.42462052087802

U 16;16 0.38277457388189

U 17;17 0.38277457388189

U 18;18 0.38277457388190

U 19;19 0.38277457388189

U 20;20 0.38277457388189

U 21;21 0.38277457388189

Table 13

A graph trisection is computed by using Algorithm 3 for n � 21 and m1 � m2 � m3 � 7 (although g�m;G;U� � 4:649, the above graph

trisection is optimal and has two edges of weight 8 among three subgraphs �V1;E1�, �V2;E2� and �V3;E3�)Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

V1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

V2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1

V3 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0


· Although the Donath±Ho�man bound is not tight for graph partitioning problems, Algorithms 2 and 3can generate optimal partitions.

For those graphs having sparse adjacency matrices, subgradient method can be accelerated with theLanczos block algorithm [7] for computing a few eigenvalues. This problem has been solved by Falkneret al. [10]. Furthermore, subgradient method can be accelerated with parallel computation.

References

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algorithms for graph partitioning problems by means of eigenspace relaxations

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