A Branch And Bound Algorithm For The Maximum Clique Problem

Computers Ops Res. Vol. 19, No. 5, pp. 363-375. 1992 03050548/
92 $5.00 + 0.00
Printed in Great Britain. All rights reserved Copyright 0 1992 Pergamon Press Ltd zyxwvuts
A BRANCH AND BOUND ALGORITHM FOR THE
MAXIMUM CLIQUE PROBLEM
PANOS M. PARDALOS’* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
and GREGORY P. R~DGER?~
‘Department of Industrial and Systems Ensnaring, 303 Weil Hall, University of Florida, Gainesville,
FL 32611 and ‘IBM Corporation, Systems T~hnology Division, Burlin~on, VT 05452, U.S.A.
(Received August 1990: in revised form October 1991)
Scope and Purpose-Finding a maximum clique of a graph is a well-known NP-hard problem, equivalent
to finding a maximum independent set of the complement graph. Finding the maximum clique in an
arbitrary graph is a very difficult computational problem. This paper deals primarily with a quadratic
zero-one modeling of the maximum clique problem. A branch and bound algorithm based on this
modeling, and different vertex selection heuristics (the greedy and the nongreedy vertex selection rules),
are used to solve many instances of the maximum clique problem. It is demonstrated that the nongreedy
vertex selection rule and the data structures obtained from the quadratic formulation, together in a branch
and bound algorithm, allow us to solve relatively large graph problems. zyxwvutsrqponmlkjihgfedcbaZYXW
Abstrac t-A method to solve the maximum clique problem based on an unconstmin~ quadratic zero-one
programming fo~ulation is presented. A branch and bound algorithm for unconstrained quadratic
zero-one programming is given that uses a technique to dynamically select variables for the ordering of
the branching tree. Dynamic variable selection is equivalent to vertex selection in a similar branch and
bound algorithm for the maximum clique problem. In this paper we compare two different rules for
selecting a vertex. The first rule selects a variable corresponding to a vertex with high connectivity (a
greedy approach) and the second rule selects a variable corresponding to a vertex with low connectivity
(a nongreedy approach). We demonstrate that the first rule discovers a maximum clique sooner but it
takes significantly longer to verify optimality. Computational results for an efficient vectorizable
implementation on an IBM 3090 are provided for randomly generated graphs with up to 1000 vertices
and I50,OOO
edges.
1. INTRODUCTION
In this paper we present computational results of an algorithm for the maximum clique problem
based on an equivalent quadratic zero-one (QOl) formulation. First we discuss the formulation
of the maximum clique problem, and related graph problems, as an unconstrained QOl program.
Then, the relationships between rules used in a branch and bound algorithm for the QOl program
and rules used in a maximum clique algorithm are shown. For example, the rule for variable
selection used in a QOl algorithm is contrary to the greedy approach for node selection in a clique
algorithm. This fact helps to expose the deficiency of the greedy approach in verifying optimality,
since the size of the generated branch and bound tree for the nongreedy approach is significantly
smaller. On the other hand, since the greedy approach usually discovers a maximum clique as the
incumbent early in the branch and bound process, it serves as a good heuristic procedure for both
the maximum clique use QOl programs.
An unconstrainted QOl program is a problem of the form
minimize f(x) = cTx + +x’Qx, x E(0, 1)“, (1)
where c is a (rational) vector of length n and Q is a (rational) matrix of size n x n. For brevity,
* P. M. Pardalos is a Visiting Associate Professor of Industrial and Systems Engineering at the University of Florida. He
received a B.S. degree in Mathematics from Athens University (Greece) and a Ph.D. degree in Computer Science from
the University of Minnesota. His research interests include mathematical programming, parallel computation and
software development. Dr Pardalos is Editor of the J
ournal ofGlobal Optimization and serves on the editorial boards
of many other optimization journals.
t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
G. P. Rodgers works at IBM Burlington, in the Systems Technology Division. He received a B.S. and a Ph.D. degree in
Computer Science from the Pennsylvania State University. He has published in Annuls of O~r~rio~ Research,
C~~r~~g and other journals. His research interests include mathematical programming, parallel computing and
circuit simulation.
363

364 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
PANOSM. PARDALOS
and GREGORYP. RODCZERS
we drop the term unconstrained. Since for zero-one variables, xi” = xi, one can always bring
problem (I > into the form
minimized = xrAx, x E zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR
(0,1>“, (2)
where A = )Q + D with D = diagonal(c,, . .., c,). Without loss of generality, we may assume that
the matrix A is symmetric.
A clique of an undirected graph G = ( V,E) is a subset of I/whose induced subgraph is a complete
subgraph of G. That is, C is a clique if VUi,VIEC the edge ( ui, Uj) is in zyxwvutsrqponmlkjihgfedcbaZYX
E. A maximum clique of G is
a clique of maximum cardinality. The k-clique problem is the problem of determining if a graph
has a clique with k vertices.
The complement of G is the graph G = (V; E), where E = f(Yi,Uj):Vi,Vj’ K i $j and (ui, vj)$E>.
A uertex packing (also called a stable set or independent set) S of a graph G is defined as a subset
of vertices whose elements are pairwise nonadjacent. That is, if Ui,UjES then (Vi,r.~~)
4 E. A maximum
vertex packing is a vertex packing of maximum cardinality. It is well-known that, S E V is a
maximum clique of G ifI S is a maximum vertex packing of G.
A maximum weighted independent set of an undirected weighted graph G = (V,E) with vertex
weights Wiis a set of independent (nonadjacent) vertices of maximum weight (sum of vertex weights).
The maximum independent set problem is a special case of the maximum weighted independent
set problem with all vertex weights equal to one.
A vertex cover Tof a graph G is a subset of vertices that are connected to all the edges E. That
is, if (Vi, U~)EE then Uior njjf I: A minimum vertex cover is a vertex cover of minimum cardinality.
It is also well-known that, S c I/is a maximum clique of G iff T= V- S is a minimum vertex
cover of G. Hence, the three problems, finding a maximum clique, finding a maximum vertex
packing, and finding a minimum vertex cover are equivalent.
All of the problems described thus far are known to be NP-complete El]. Algorithms for the
maximum clique problem have been studied in detail elsewhere [2-61. Special cases that can be
solved in polynomial time are discussed in Ref. [7]. Recently, there has been a lot of focus on
solving these problems efficiently and extending the range of solvable problems [S-lo]. In this
paper, we present a branch and bound algorithm for the maximum clique problem based on a
QOI formulation. Details of branch and bounds methods can be found in Refs [4,10-143. Related
algorithms and properties for the general problem (2) can be found in Refs [15-191.
In Section 2, it is shown that the maximum clique problem can be formulated as a QOI program,
Formulations of other graph problems as QOl programs are also given. In Section 3, a branch
and bound algorithm for QOf programming is presented. This algorithm features dynamic variable
selection and the ability to force free variables to a specific value by using a rule based on the
ranges of the partial derivative of the objective function with respect to free variables. In Section
4, it is shown how the rules for the QOl algorithm relate to the maximum clique problem. In
Section 5, computational results are given comparing different alternatives for these rules. In
particular, it is shown that the greedy approach generates larger search trees than the nongreedy
approach. The rationale behind this result is the fact that the nongreedy approach tends to promote
the variable forcing rules. However, the greedy approach does have merit as a heuristic since it
tends to discover a maximum clique sooner. Computational results which demonstrate the
electiveness of the greedy heuristic are also presented.
2. EQUIVALENCE OF GRAPH PROBLEMS TO QOI P~OGUAMMIN~
In this section we formulate the maximum clique problem and other related graph problems as
QOl problems.
The maximum clique problem for a graph G = (V; E) with t)i, . .. , o, vertices, is equivalent to
solving the following linear integer program:
minimize f(x) = - $I xi s.t.xi+Xi<l V(u,Yj)E~andxE(O,lf’ (3)

Solving the maximum cliqueproblem 365
and a solution x* to program (3) defines a maximum clique C for G as follows: if x: = 1 then
viEC and if xt = 0 then vi$ C and the cardinality of C is 1C 1= -z = -f(x*). cl
Let 1El denote the number of edges in G. The number of constraints, m, in program (3) is equal
to the number of edges in G. That is,
(4)
Another way of stating the m constraints for program (3) is the quadratic expressions XiXj= 0
V(ci, uj)EE, since for xi, xj E {0, 1)xi + Xj < 1t=rxixj = 0. The clique constraints in program (3) can
be removed by adding the quadratic terms to the objective functon twice. These quadratic terms
represent penalties for violations of xixj = 0. This leads to the following proposition.
Proposition 2
Let G = ( V,E) be a graph with n vertices, let AC be the adjacency matrix of cf, and let I be the
n x n identity matrix. Then, the maximum clique problem for the graph G is equivalent to solving
the following QOl program:
minimizef(x) = - t zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG
Xi + 2 2 XiXj S.t. X E f0, 1}” (5) zyxwvut
i=l (LLq$E
i>J
or equivalently (in a symmetric form),
minimizes = xTAx where A = AC - I, s.t. XE (0,l f”. (6)
A solution, x*, to program (5) or (6) defines a maximum clique C for G as follows: if XT= I then
v,gC and if x: = 0 then a,$C with ICI =--z = -S(x*).
Prooj It is clear that if x* is a optimal solution to program (5), then all quadratic terms xfxt = 0.
If XT= 1 (which corresponds to vertex t’i~C) then .x7 = 0 (which corresponds to vertex
vi$ C)V( vi,uj)# E and vice versa. •I
The off-diagonal elements of the matrix A are the same as the adjacency matrix of G. Hence,
formulations (3) and (6) are advantageous for dense graphs because a sparse data structure can
be used.
A vector x E {O,1)” is a discrete local minimum of the quadratic problem ( 1) ifff(x) <I(y) for
any YE{O,1)” adjacent to x. The next theorem gives an interesting correspondence between discrete
local minima and (maximal) complete subgraphs.
Theorem I
Any zero-one vector x that corresponds to a (maximal) complete subgraph of G is a discrete
local minimum off(x) in formulation (5). Conversely, any discrete local minimum of the function
f(x) corresponds to a (maximal) complete subgraph of G. cl
Similar formulations of the maximum vertex packing problem and the minimum vertex cover
problem as a QOl program are given without proofs. In addition, it can be shown that the k-clique
problem also has an equivalent QOl formulation.
Proposition 3
The maximum vertex packing problem and the minimum vertex cover problem for G(u, E) are
equivalent to solving the following QOl program:
minimizef(x) = x’Ax where A = A, - I, s.t. x E 10, 1)“. (7)
A solution, x*, to program (7) defines a maximum vertex packing, S, for G as follows: if xr = 1
then O,ES and if x: = 0 then v,$S and the cardinality of S is ISI = -z = -f(x*). A solution, x*,
to program (7) also defines a minimum vertex cover, T, for G as follows: if xt = 0 then Vif Tand
if x: $ Tand the cardinality of Tis I7j = n - z = n -.f(x*). El

Proposition 4
PANOS M. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH
PARDALOS
and GREGORY
P. RODGERS
The maximum weighted inde~ndent set problem for a graph G = (V, E) with vertex weights wi
where 1PI = n is equivalent to solving the following QOl program:
where
minimize~~x) = xT.4x, s.t. x E(0, 1)“, (8) zyxwvuts
@ii = -WiVi; t+j - Wi + wj for (ri,rj)EE and i >i; U,j = 0 for (ui,uj)4E.
A solution, x*, to program (8) defines a maximum weighted independent set, S, for G as follows:
ifxf = 1then uiES and ifx,* = 0 then ui$ S and the maximum weight for the set S is -z = - f(x*).
Cl
We have seen that a class of graph problems can be formulated as QOl programs. This fact is
of practical interest if a QOI algorithm can solve any of these reformulations efficiently. In the next
section we present an efficient branch and bound algorithm for solving the QOl program. Then,
an efficient algorithm for solving the maximum clique problem as a QOl program is presented.
3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
A BRANCH AND BOUND ALGORITHM FOR QOl PROGRAMMING
Branch and bound is a general method which has been applied to various problems in
combinato~al optimization [4, 10, 1I]. The main idea of a branch and bound algorithm is to
decompose the given problem into several partial subproblems of smaller size. Each of these
subproblems is further decomposed until it can be proved that the resulting subproblems cannot
yield an optimal solution or it can no longer be decomposed. The search strategy defines the order
in which partial problems are tested or decomposed. Such strategies include the depth-first search,
the breadth-first search, the best-bound search and search strategies based on some heuristic search,
For an excellent formal description of various search techniques see Ref. 111J.
The objective of branch and bound algorithms is to find a global optimum by searching the
entire branch and bound tree. However, a complete search of the branch and bound tree may be
impractical. Thus, many branch and bound implementations have provisions for stopping execution
after some specified time limit. As a result, there exist two primary objectives of branch and bound
algorithms. The first is to limit the search space in order to implicitly enumerate all possible
solutions. The second is to find the best possible solution from among the space of solutions that
is searched. Later, it will be shown that these two objectives may conflict,
The QOl program (2) can be decomposed into two subproblems by selecting a variable, xi*and
fixing it to zero for one subproblem and to one for the other subproblem, where xi is a variable
chosen from a list of remaining free variables. Once a variable is chosen it is removed from the list
of free variables called thefree list and it is placed in the fixed list. When the free list is empty, all
variables are fixed and the subproblem represents a compiete assignment of values.
The branch and bound tree has a potential size of 2”+’ - 1 nodes. This is prohibitively large
for even moderate size problems. May subproblems can be ignored because it can be determined
that further decomposition would result in a subopti~al solution. This procedure is called pruning.
There are two categories of pruning rules used: the lower bound rule and forcing rules. According
to the lower bound rule, if the value of a lower bound function g, for a given subproblem, exceeds
a known optimal, then that subproblem can only yield a suboptimal solution. Any lower bound
function must satisfy the following three rules in relation to the objective function f for the
subproblem P,:
g(Pi) <f( Pi), where f( Pi) is the objective function value for any complete
assignment of zero-one values for the subproblem P,.
g( Pi) =f( Pi), where PI is a subproblem that represents a complete assignment of
zero-one values denoted by x. This says that lower bound function must have
the same value as the objective function when the subproblem can no longer be
decomposed.
g( Pi) 3 g( Pi) if Pj is a subproblem that represents a further decomposition of the

Solving the maximum clique problem 367
subproblem Pi( Pi is a son of Pi in the branch and bound tree). This says that the
lower bound function is nondecreasing in the descent of the tree.
For problem (2) we choose an easy to compute lower bound function. Let ieo be the level in the
search tree (the number of fixed variables). Initially, lea = 0. Let zyxwvutsrqponmlkjihgfedcbaZYXWVUT
ieu be the level in the search tree
(the number of fixed variables). Initially, lea = 0. Let zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
p,, . . . , plovbe the indices of the fixed variables
andp,,,,,,..., p. be the indices of the free variables, then the lower bound g is defined as follows:
g== i &ii-
fev II lC?V zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO
le a zyxwvutsrqponmlk
lee
2 t: c & # - XP,) + c & A 1 - XPJ + c r: U~~X~,X~~ (9 zyxwvutsr
I=1 jfl [ i=l j-t+1 i=1 1i-1 j=i
where ai; = min (0, aij) and a; = max {O,Uijf are the negative and positive coefficients of A,
respectively. The lower bound pruning rule says that if g B OP’f; where OPTis the known incumbent
objective function value, then the subproblem should not be decomposed further.
Forcing rules can be used to generate only one branch for a given variable if certain conditions
exist. This is also known as preprocessingthe subproblem. A variable may be forced if it can be
shown that the alternate value can only yield suboptimal solutions. For problem (2), variables may
be forced by examining the range of the gradient of the continuous objective function. It can be
shown that if the continuous objective function is always increasing for a free variable Xi in the
co~f~~~o~s range XiE[O, 11, then xi may be forced to zero. Likewise, if the function is always
decreasing then the variable may be forced to one. We implement this rule by calculating the range
of continuous partial derivatives of free variables in the unit hypercube determined by the fixed
variables. The lower and upper bounds of the continous partial derivatives of the free variables
are given by lb, and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
ub,,,:
and
ubP*
= t: ~P,P,XP,
+ ii a& + uAP, for i = leu + 1,. . . , n.
i=l j=kT+ 1
(11)
j#i
Actually, equations ( IO) and ( 11) give the range for the partial derivatives of an e~#i~u~~ffr
function
to problem (2) (with the same zero-one solutions), whose range of the partial derivatives is
minimized [ 14J. This is done by linearizing the quadratic term aiiXi”since xf = xi, Now, the two
rules used to force variables in the QOl algorithm are as follows:
if Ib,>Othenxz=Ofori=lev+ I,...,n (Rule 1)
ifubp,,<Othenx;,=Ofori==leu+l ,..., n. (Rule 2)
Forcing variables is extremely desirable since it considerably reduces the size of the search tree.
In fact, the rule that is used to select a variable to branch on (to fix to zero and one) when none
can be forced is to choose the variable that is least likely to be forced in subsequent levels of the
search tree, thus leaving the other variables to ~tentially be forced at a lower level. By using the
permutation vector p, it is possible to select variables in any order. The variable that is (heuristicaly)
least likely to be forced is chosen first, according to the next rule:
branch on xp, where Si = max (min ( -tb,, ub,), k = lea+ 1,. zyxwvutsrqponmlkjihgfedcba
.., a>.
L
(Rule 3)
In Section 4, it will be shown that this rule is the opposite of the greedy method in the transformed
maximum clique problem. When a branch occurs, two subproblems are generated, one for xp, = 1
and one for xp, = 0. The subproblem to expand first is decided by which assignment of values
causes the lower bound 9 to increase the least. This is a partial best-first strategy.
We now consider the search strategy. For our particular problem, we are able to evaluate a
single branch and bound vertex very quickly [in O(n) time]. Since large-scale problems will have
CMR
19:5-E

368 PANOSM. PARDALOS
and GREGORY
P. RODGERS
many vertices, it is impractical to store all the vertices that may be required with a best-first or
breadth-first search strategy. We therefore impose the depth-first strategy. This also frees more
storage to be use d for computational efficiencies. Depth-first search still gives the capability to
choose the value of the branch (zero or one) if the value canot be fixed. We can still use a partial
best-first strategy to choose the value that increases g the least. However, if the subproblem results
in no change to the incumbent then the choice would have been irrelevant.
The algorithm to solve problem (2) is given as Algorithm 1. The expanded subproblems (S) are
stored on a stack that has a maximum depth of n + 1, where n is the dimension of the problem.
Note that the only value that is required to be saved on the stack is the level where a branch
occurs. As the branch and bound tree is descended, ku is changed and indices are swapped in p
to represent the order that variable are selected. Thus, the free and fixed variable lists are implicitly
changed.
The first part of Algorithm 1 is the initialization of the branch and bound procedure. As a
stopping criterion, - 1 is initially placed on the stack. The maximum number of subproblems to
solve, MAX& is initialized to some limit based on CPU resources at line 5. The size of the branch
and bound tree is characterized by the number of subproblems, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR
NSUBP. Hence NSUBS > MAXS
is an additional stopping criterion which can be used to control the amount of computer time
available to the problem.
It should be assumed that a good heuristic is used to initialize the values for OPTand x* at
lines 1 and 2. For the implementation described in this paper two heuristics were used, the gradient
midpoint method and a greedy method. The gradient midpoint method examines the range of the
gradient in the unit hypercube. If the midpoint of the range of a partial derivative is positive, then
the corresponding zero-one variable is set to zero. If it is negative, then the zero-one variable is
set to one. This zero-one vector is then used as a starting point for a discrete local min search.
For a detailed discussion of the gradient midpoint method see Ref. [ 131. The greedy method will
be discussed in Section 4.
At each vertex which is represented by an iteration of the while-loop at line 8, the objective
function lower bound is calculated (line 9) according to equation (9). Line 10 states that if pruning
is required (g 2 OPT) or a leaf node has been reached (Ieev= n), then the algorithm is at a terminal
node. If the test at line 11 (g < OP7) is true then it is implied that lev = n and a new minimizer
has been discovered. The incumbent value OPTand the minimizer are updated in lines 12 and 13.
The next node to search is obtained by popping a new level from the stack (line 15) and changing
the value of the binary variable associated with that level (line 16). The change to the free and
fixed variable lists in p are implicit. The statistic, NSUBP, is updated at line 17 to reflect the fact
that another subproblem has been solved.
Algorithm 1. Depth-first branch and boundalgorithm for a QOl program
Procedure QOl (A, x*)
1 OPT+ best known minimum from heuristic
2 x* + best known minimizer from heuristic
3 P[l,n]+ c1,nl
4 push{- 1,stack)
5 MAXS + value based on CPU resource limit
6 ZeutO
7 NSUBPt 0,
8 white teu # - 1 and NSUBP < MAXS do
9 Calculate lower bound g
10 if 9 2 OPTor fee = n tben
11 if 9 < OPT then
12 OPT+ g
13 x~+xi,i=lr...,n
14 endif
15 pop(leu,stack)
16 if(Ieu#-1)thenx .+-l-xpln
17 NSUBP +- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
NSUBP 4’ 1
18 else
19
af
[lb,, ub,] +- range of ax over x E[O, 13”for i = Zeo+ 1,. ..,n
PI
20 if~bp~~Oor~bpi~O,forsome~,~=~eu+l,...,~t~n
21 ifubp,~Othenx,iclelsex,cO

22 else
23 i + j where dj = max {min ( zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
-lb,, ub,,), k = leu + 1,. ,n}
t
24 x,,,+ 0 or 1 depending on value that increases g least
25 push zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(lev + 1, stack)
26 endif
27 lev + lev + 1
28 Pk”t* Pi
29 endif
30 endwhile
As the algorithm descends depth-first, the range of partial derivatives of free variables is calculated
(line 19) according to equations ( 10) and (11). Line 20 represents the test to see if any free variables
can be forced by the gradient rule, while line 21 is the selection of the forced value. If a variable
can be forced by the gradient zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
i is set to the index into p that contains the index of the variable to
be forced. If a branch is required, i.e. preprocessing is complete, the least likely variable to force
is chosen by Rule 3 (line 23). It is then determined which branch, xP, = 1 or xP,= 0, results in a
lesser lower bound, g. The alternate subproblem is saved by putting the level associated with the
branch variable on the stack (line 25). Whether a variable was branched on or forced, subsequent
levels of the current subproblem must see that variable as fixed. Lines 27 and 28 fix a variable for
the next subproblem by increasing lev by one and swapping the appropriate index in the permutation
vector p.
The calculation of the gradient bounds and the objective lower bound dominates the
computational requirements. In practice, the calculation of the bounds are done more efficiently
than suggested by equations (9- 11). These formulas require O(n2) operations per vertex. In our
implementation we are able to update all of the bounds in <2n additions per vertex. This is done
by using the bounds at the previous level. For more details regarding implementation efficiencies
see Refs [ 12-141.
The loop at statement 8 was written so that it can easily be restarted if no modification is done
to the arguments stack, p, lev and x. This is helpful if NSUBP exceeds the threshold MAXS and
the user would like to apply more resources to the same problem. Also this feature allows this
algorithm to be easily modified for execution on a parallel processor. For a discussion of a parallel
version of this algorithm see Refs [ 12, 133.
4. A CLIQUE EQUIVALENT ALGORITHM
In this section the behavior of Algorithm 1 is considered when it is used to solve a maximum
clique problem using the formulation given by equation (6). It will be shown how the decision
rules in Algorithm 1 relate to a maximum clique problem.
Consider Algorithm 2 for finding a clique in a graph G = (K e). The control of this algorithm
is identical to that of Algorithm 1. The primary differences are the data structures and the meaning
of the control variables. The input consists of a graph represented as a list of vertices and edges
denoted Vand E, respectively. The output is a subset of the vertices, C*, that define the maximum
clique if MAXS is not exceeded. Throughout the algorithm set sizes are denoted by vertical bars
(i.e. 1XI is the size of set X). Since the algorithm finds a clique of maximum cardinality, the variable
g is an upper bound on the clique size for a given subproblem and pruning occurs when g < OPT,
where OPT= IC* 1is the size of the current incumbent clique.
Each subproblem of the branch and bound tree is denoted by the arrangement of the vertices
from I/into the three sets V’, C and D. Initially, v’ = Vand C and D are empty. At each level of
the tree a new vertex, vi, is taken from Vand put into either C or D. Hence, the following condition
always holds: 1VI + (C I + ID I = IVj = n. The vertices in C represent a clique in the input graph
G. The set D is the set of discarded vertices. The induced subgraph, G’, for a subproblem with vertex
sets C, D and V’ is defined as the graph G’ = (v’, E’), where E’ = {(vi, vj)lvi, vjc v’ and (vi, V~)E
E}.
The three important characteristics of the algorithm, the upper bound, the forcing rules, and the
branching rule, are discussed below.
This algorithm uses a simple upper bound g = ICl + IV’ I = n - ID I. Initially, g = n = IVI, and

PANOSM. PARDALOSand GREGORYP. RODGERS
if V’ = 0 then g = 1Cl. This simple upper bound is sufficient to prove that
equivalent to Algorithm 1. Later we will see how this bound can be improved.
Algorithm 2. Branch and boundalgorithm for a maximum clique program
Procedure MCLIQUE( KE, C*)
1 OPT+ largest known clique size
2 C* + set of vertices for largest known clique
3 CtD+0
4 v’+V
5 push zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(stack-empty-marker, stack)
6 MAXS + value based on CPU resource limit
7 NSUBP +- 0,
8 while stack-not-empty and NSUBP < MAXS do
9 g+n-IDI
10
11
ifg<OPTorV’=@then
if g > OPTtheo
OPT+- g
c* + c
eadif
this algorithm is
12
13
14
15 pop ({ v’, C, D), stack)
16 NSUBP t NSCJ
BP + 1
17 else
18 ditjXil where X, = {(ui,uj)~(uiruj)~E and u~EV’}VU,EV
19 die lXil where Xi = {(ui,uj)~(uiruj)~E and u~EC}VU~EV
20 ifd,<ICord,=IV’I-lforsomeuiEV’then
21 ifd,<ICIthenD+Duu,; V’tV’-uielseC+Cuui; V-Y’-ui
22 else
23 i + j where 6, = min dk
U&G”
24 VtV’-vi
25 push ({ V’, C, D u vi}, stack)
26 c+cuui
27 endif
28 endif
29 endwhile
A vertex is forced when it is placed in set C or set D and no alternate subproblem is stacked
(line 21). The forcing rules use the vectors d and 1 which vary for each subproblem. The elements zyxwvutsr
di and di are associated with each vertex USE
I”. The variable, di, is defined as the number of edges
in E’, or equivalently, di is the degree of vertex i in the induced subgraph G’. The variable di, is the
number of edges in E from ui to vertices in C. With these definitions the forcing rules at line 20
become clear. The first rule,
ifdi<ICIthenDtDuvi;‘VcV-vi foruiEI”, (Rule 4)
states that if a vertex is not connected to all vertices in C, then that vertex must be discarded for
that particular subproblem. The second rule,
ifdi=IV’I-1 thenCcCut.+;VcV-ui foruieV’, (Rule 5)
states that if the vertex is connected to all other vertices in I”, then that vertex must be in a
maximum clique for that subproblem.
When no more vertices can be forced the branching rule at line 23 determines which vertex to
branch on. The branching rule is as follows:
branch on vi where di = mind,.
keV
(Rule 6)
The intuition behind this rule is not as obvious as the forcing rules. It chooses the vertex of lowest
connectivity in the induced subgraph G’. One might expect to choose the vertex of
highest connectivity since the algorithm is searching for a maximum clique. Choosing the vertex
of highest connectivity would be the typical greedy approach. However, the nongreedy approach is
equivalent to the decision rule used in the QOl formulation (Rule 3). This rule results in the
generation of a smaller branch and bound tree for reasons that will be discussed later. It is relatively
easy to show that the decision rules of the two algorithms are equivalent. Hence, the branch and
bound tree will have the same structure which results in the same number of subproblems.

Solving the maximum clique problem 371 zyxwvuts
lheorem 2
Algorithm 2 solves the maximum clique probiem in the same number of subproblems as Algorithm
1 using the formulation given by equation (6). 0
Next, we analyze the two equivalent Algorithms 1 and 2 for the maximum clique problem. In
terms of computational efficiency, Algorithm 1 is especially suited for dense graphs. The sets C, D
and V’ are implicitly stored as a permutation vector p, a logical vector x and a level indicator lev.
That is, the vertices up,,. . . , v,,,,,are in sets C and D while the vertices r+,,~,+,,
. . ., op. are in the set
v’. The zero-one variables, xP,, . . ., xp,_ define to which set, C or D, the fixed vertices belong. For
dense graphs G, the primary storage requirements are for the sparse adjacency matrix G. However,
the storage of the nonzero coefficients, 1 and - 1, need not ibe explicit. As mentioned earlier, the
computational requirements are dominated by the calculation of g, tb and ub. This is done efficiently
in O(n) additions per node of the branch and bound tree. When a sparse data structure is used a
tighter bound is O(ei) additions per node, where ei is the number of edges incident on vertex vi in
G and ui is the vertex that was fixed in the previous level of the branch and bound tree.
The upper bound and the forcing rules (Rules 4 and 5) for Algorithm 2 offer little insight into
a good algorithm for the maximum clique algorithm. The use of this information is the least that
should be done in a branch and bound algorithm for the maximum clique problem. However, the
branching rule (Rule 6) does offer some nontrivial intuition.
This branching rule helps to achieve a smaller branch and bound tree in an indirect way. Earlier,
it was mentioned that Rule 6 is a ~~~g~ee~y approach to choosing a branching variable, since it
chooses a vertex with smallest degree from the induced subgraph. It has been argued in Ref. C33
that a greedy approach helps to achieve the actual maximum clique as an incumbent earlier and,
thus, assists the bounding process to reduce the size of the branch and bound tree. Our computational
results verify this argument. However, overall tree reduction is better accomplished by trying to
encourage the activation of the forcing rules. In other words, the motivation behind Rules 3 and
6 is to choose a variable or vertex that has little chance of fixing in subsequent levels of the branch
and bound tree or will help to cause other variables to be forced in subsequent levels. For example,
removing a vertex with low connectivity from V’ leaves vertices with high connectivity in the new
subproblem. This increases the potential of activating Rules 4 and 5. Rule 4 is activated when there
is no edge from a vertex left in v’ and a vertex in C; Rule 5 is activated when an edge in I/’ is
connected to all other edges in V’. In Section 5, we give empirical evidence that the nongreedy
approach is significantly better in reducing the size of the branch and bound tree.
As stated earlier, the upper bound g in Algorithm 2 is crude at best. It is the number of vertices
in C pluss the number of vertices in v’ or g = 1C / + 1VI. For random graphs an improved upper
bound would be
g = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJ
ICI + 1 + maxd,,
WV
(12)
because a clique in the induced subgraph could be no larger than 1+ the largest degree. Moreover,
this idea could be extended to a new forcing rule (the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
degr ee-incumbent rule) to avoid the generation
of suboptimal subproblems:
if di < OPT- IC/
- 1then D + I) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP
u v<; F’+- V- Ui for UiEV’. zyxwvutsrqponmlkjihgfe
(Rule 7)
The above analysis motivated us to develop an algorithm s~cifi~lly for the maximum clique
problem using an implementation of algorithm 1 as a base. We were able to eliminate all of the
negative aspects mentioned above. For sparse graphs our revised algorithm updates the vectors d
and 1 from the actual sparse representation of A,. We also assumed the values of the coefficients
and added the degree-incumbent rule. The computational results from this implementation are
presented in the next section.
5. COMPUTATIONAL RESULTS
In this section we present a variety of computational results with random graphs. It is well-known
that test problems with random graphs represent, on average, difficult instances of the maximum

372 PANOS M. PARDALOS and GREGORY P. RODGERS
Procedure MGRAPH(n, HU, KU)
I n-1000
2 DENSIlT+ 0.1005865
3 DSEED + 6551667.0
4 NEDGEStO
5 for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF
i = 1 to n
6 K Ui .- NEDGES + 1
1 forj=i+l ton
if GGUBFS(DSEED) < DENSI7Ytbm
9 NEDGES + NEDGES + 1
H~NEDCES+~
11 endif
12 endfor
13 endfor
14 Ku,+, + NEDGES + 1
Fig. I. Benchmark IOOOA using the IMSL routine GGUBFS
Table 1. Computational results for FORTRAN code on an IBM 3090-3OOE,50 random graphs per experiment
Experiment No. of fknsity
no. vertices of graph zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
AV. Heuristic % Of problems Av. CPU time
maximum av. maximum solved by (see) for
clique size clique size heuristic heuristic
2
4
6
8
9
IO
II
I2
13
14
I5
I6
I7
18
50
50
50
50
50
50
50
50
50
loo
IO0
100
100
100
100
100
100
IO0
IO
20
30
40
50
60
70
80
90
IO
20
30
40
50
60
70
80
90
3.26
4.14
5.10
6.22
7.46
9.08
Il.38
14.80
21.54
3.96
5.00
6.10
7.58
9.18
Il.44
14.66
20.02
30.74
3.12
3.94
4.78
5.74
6.80
8.46
10.58
13.88
20.34
3.32
4.34
5.38
6.72
8.24
IO.18
13.34
17.94
28.48
86
82
68
58
44
52
40
38
40
36
34
36
32
30
I6
IO
4
IO
0.011
0.010
0.010
0.009
0.008
0.007
0.054
0.052
0.049
0.046
0.042
0.037
0.032
0.026
0.020
clique problem. A subroutine that generates the random graphs used in all experiments is described
in Fig. 1. The first computational results are used to show the effectiveness of a greedy
heuristic and to compare the nongreedy branching rule and the greedy branching rule. The efficient
implementation was primarily due to the data structure used and an updating technique. These
efficiencies are also discussed in this section.
Table 1 demonstrates the effectiveness of the greedy heuristic. Our goal in using such a heuristic
was to find a close value to the optimal in as little time as possible. Our implementation of the
greedy heuristic chooses the node of highest connectivity and then chooses the neighbor with
highest connectivity, and so on, until no more vertices can be chosen while still maintaining a
completely connected graph. The greedy heuristic finds a graph that may or may not be the
maximum clique. This is not to be confused with the greedy rule in a branch and bound algorithm.
The final output of a branch and bound algorithm, whether the greedy rule is used or not, is an
actual maximum clique. Table 1 shows that the greedy heuristic is more effective for low density
graphs in terms of finding the maximum clique (see the column labeled “% of problems solved by
heuristic’*). This is not surprising, since the maximum clique size is smaller and thus there are more
maximum cliques, which results in a higher chance for the greedy heuristic to build a maximum
clique. Perhaps a more interesting statistic in terms of the effect on bounding capabilities in a
branch and bound algorithm is the “heuristic av. maximum clique size”. The heuristic finds a clique
which is usually within 10% of the maximum clique size.
Table 2 gives additional statistics on the same test cases given in Table 1. This table compares
the greedy rule with the nongreedy rule. Notice that for large problems the nongreedy rule is

Solving the maximum clique problem
Table 2. Comparison of greedy and nongreedy algorithms
373
Experiment
"0.
AV.
no. of
subproblems
Greedy results
Av. % of tree
searched before
solution*
Av. AV.
CPU time , no.of
(set) subproblems
Nongreedy results
Av. % of tree
searched before
solutiona
AV.
CPU time
(set)
d!
3
4
5
6
7
8
9
f0
11
I2
13
14
I5
16
17
18
6 54 0.014 24
16 18 0.017 42
27 20 0.020 44
53 is 0.026 61
124 21 0.047 181
333 20 0.095 309
1150 8 0.257 584
8058 12 t.393 1566
124,818 2 17.961 342 I
29
59
172
503
1778
8807
73,167
1,754,349
-
15 0.081
13 0.104
12 0.158
I5 0.317
9 1.035
12 3.979
I1 25.991
16 SOS.635
-
86
94
235
1006
1660
7033
24,545
171,034
2,976,732
85
28
4i
47
60
43
51
51
61
44
21
38
:;:
::
45
55
0.017
0.022
0.024
0.02?
0.046
0.067
0.103
0.217
0.391
0.104
0.119
0.159
0.382
0.710
2.015
6.035
35.609
539.923
“Only includes problems where the heuristic did not find the solution.
substantially faster than the greedy approach. However, for problems where the heuristic did not
find the solution, the use of the greedy rule in a branch and bound algorithm discovers a maximum
clique sooner than the use of the nongreedy rule. This early discovery does help the bounding
process somewhat. However, this is insignificant compared with the benefit received from using
the nongreedy rule to improve the activation of the forcing rules in larger branch and bound trees.
In our implementation the adjacency matrix was stored using a standard sparse data structure.
For each vertex a list of the adjacent vertices is stored. These n lists are stored sequentially in one
large array HA. The pointers to the beginning of the lists are stored in an array KA. For example,
vertex zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
i is connected to the vertices HAKA, through HAKAi+,_i.
The majority of the computation in an iteration of a branch and bound algorithm deals with
the calculation of the vectors d and 2 (see lines 18 and 19 of Algorithm 2). This can be done
efficiently by only calculating d and 4 once and ~p~u~~~~them between iterations of the branch
and bound algorithm. This updating can be done as follows:
dHRh
+ &iAk - 1 for k = KA,, . .., KA,g +i - 1
if xg,,,= 1 then &Ax + &AI, + 1 for k = KAp,e,.,. .. , KA,,sc+l - 1.
Recall that xp,, represents the vertex that was just fixed in the last iteration of the depth-first branch
and bound algorithm. Since it has been removed from I-“,any vertex to which it was connected
should have the degree, di, reduced by one. If xp,, was set to one (put in the set C), then a must
be updated to reflect the connectivity to vertices in C. An additional requirement to implement
this efficiency is to put the arrays zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
d and il on the stack and remove them when a new subproblem
is popped. If storage is a premium, d and 8 can be completely recalculated when a new subproblem
is popped from the stack. Then, the updating can continue as described in subsequent levels of the
subproblem.
Our implementation was done with VS FORTRAN on an IBM 3090-300E with a vectorizing
facility. It is difficult for a compiler to safely vectorize the two loops used to update d and iI because
of the potential for recurrent values in the indexing vector HA. However, since no recurrence exist
vectorization can be forced with compiler directives. By vectorizing these two loops we achieved
a reduction in CPU time between 20 and 30%.
Figure 1 gives the exact formulation for the generation of a random graph using the IMSL
random number generator GGUBFS. The generated graph has 1000 vertices and 50,000 edges
which is roughly 10% dense. Two larger problems with exactly 100,000 and 150,000 edges are
generated by specifying DENSiTY== 0.2001455 and DENSiZY= 0.300115, respectively. These three

374 PANOSM.PARDALOS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML
a nd GREGSXYP, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ
RWGERS
Table 3. Computa tiona l re sults for be nc hmarks IOOOA,IOWE a nd IoooC
usinn VS FORTRAN ananIBM 309@3OOE
A B c
No. of vertices 1000 loo0 1000
No. of edges 50,ooO lOO,C@O 150,090
No. of branch and bound subproble~s 12733 88171 2227723
CPU time (secf (unv~to~zed) 40 373
CPU time (set) (vectorized) 31 264 3962
Did the heuristicfind the solution? NO YES NO
Maximum cliquesize 6 7 10
Solution A = (aIT
B, “225, bu. L
’4G.r
0493.% 44)
Solution B = (Q,. uT
3. Use ,, vSQlrvsJ6. uT
e 6. vT
y7)
SolutionC = fcaa, ~1~~.
uIJJ. buss,vJO6.
U5ts.L‘Jff.@sw%lJI %wf
problems are referred to as benchmark lOOOA,1OOOB
and lOOOC,respectively, They are explicitly
given here for future comparisons.
Procedure MGRAPH (see Fig. I) generates the upper triangle of the adjacency matrix in the
arrays HU and KU. The complete adjacency matrix can easily be created from these arrays to get
the form required by our ~mplementatiou (HA and KA). The computational results (of the nongreedy
approach) for these problems are given in Table 3. Note that Problem C is not solved without
vectorization (if takes too long).
6. CUN~~USIUN
In this paper we present a method to solve the maximum clique problem as a special case of
the QOl problem. We demonstrate that for large problems, the nongreedy vertex selection rule is
better than a greedy vertex selection rule in the branch and bound algorithm. This is because the
nongreedy rule facilitates the activation of preprocessing rules. However, the greedy selection rule
has merit as a heuristic since it tends to discover the optimal solution sooner. An efficient
implementation allows us to solve relatively large graph problems.
The techniques described herein are relatively simple. The primary contributions are the use of
the nongreedy vertex selection rule for large problems and the data structures obtained from
unconstrained QOl programming. More sophisticated fathoming algorithms do exist [ 8, 201. For
example, a technique due to Balas and Yu [8] tests if the induced subgraph is chordal for which
it is easy to find a maximum clique. A ~ombi~a~on of these ideas may lead to even faster algorithms,
As a result, we provide exact specifications for benchmarks to facilitate future comparisons.
~~~now~~dge~e~rs-we are indebted to the IBM Corporation for a grant under the IBM Research Support Program to
use the IBM 3090 at the Palo Alto Scientific Center in Palo Alto California. Richard Blaine, Ronald Grodevant and Kelly
McCormick, all from IBM, provided invaluable assistance to us during this program. Research by the second author is
funded by IBM through the IBM resident study program.
1. M, R. Ciarey and S. J. Johnson, Computers and Intractability, A Guide to the Theory of NP-Gompieteness. Freeman,
New York (1979).
2,
3.
C. Bran and J. Kerboscb, F~~~~~a~i cliques ofan undirected graph Coals. Ass. Come. Mach. 16,575-577( 1973).
M. Gendreau, J-C. Picard and L, Zubieta, An E@cirnr frn~~je~r
~n~~~rurion A~gor~tbrn~rtke ,~uxjrn~rnC&gateFrob~em.
Lectwe Notes in Economics and ~uthemarieu~ Systems 304 (Edited by A. Kurzhanski et al.), pp. 79-91. Springer-Verlag,
New York (1988).
4.
5.
6.
7.
L. G. Mitten, Branch and bound method: general formulation and properties. Ops Res. IS, 24-34 ( 1970).
1. M. Robson, Algorithms for maximom inde~ndent sets. f. Algorithms 7,425~440 (1986).
R. E. Tarjan and A. E. Trojanowski, Finding a maximum independent set. SIAM J1 Comput. 6, 537-546 (1977).
M. Gratschel, L. Lovasz and A. Schrijver, Geometric Algor~rhmsand Combinatoriuf Optimization. Springer-Vetlag, New
York (1988).
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9.
E. Balas and C. S. Yu, Finding the maximum clique in an arbitrary graph. SIAM JI Cornput. f5, ~0~-!~8 (1986).
L. Gerhards and W. Lindenberg, Clique detection for nondirected graphs: two new algorithms. Comparing 21,295-322
(1979).
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11. T. Ibaraki, Enumerative approaches to combinatorial optimization. Arm. Ops Res. lO/ll ( 1987).
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programming. Computing 45, 131- 144 (1990).
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