Average Polynomial Time Complexity Of Some NP-Complete Problems

Theoretical Computer Science 46 (1986) 219-327
North-Holland
219
AVERAGE POLYNOMIAL TIME COMPLEXITY OF SOME NP-
COMPLETE PROBLEMS
PHAN DINH Dieu
Institute of Computer Science, Lie giai, Ba dinh, Hanoi, Vietnam
LE CONG Thanh and LE TUAN Hoa
Institute of Mathematics, Bo Ho, Hanoi, Vietnam
Communicated by M. Nivat
Received September 1984
Revised December 1985
Abstract. We define CLIQUEN(n) to be the problem CLIQUErestricted to the set of graphs
G = ( V, E) such that [El <~N(] V]). In this paper it is proved that when N(n) = [n~], e a rational
number, 0< e<2, CLIQUEN(n) is an NP-complete problem, and, under the assumption of a
uniform distribution on the set of instances, CLIQUEN(n) can be solved by a deterministic
algorithm in average (and almost everywhere) polynomial time.
Some cases of nonuniform distributions on the set of instances, for which the usual problem
CLIQUE is solved in average polynomial time, are also considered.
1. Introduction
This paper deals with the average time complexity of algorithms solving NP-
complete problems. The main result of this paper shows that a family of subproblems
of the CLIQUE problem which are still NP-complete can simultaneously be solved
by a deterministic algorithm in average polynomial time, and even in polynomial
time almost everywhere. This result is closely related to other ones about the average
polynomial time complexity of NP-complete problems, e.g., to the result of Apolloni
and DiGreigerio in [2] for the SAT problem.
It is well-known that, for solving NP-complete problems, the up to now known
algorithms require in general an exponential time in the worst case. This exponential
time complexity still remains even for recently improved algorithms. E.g., an
algorithm introduced by Shroeppel and Shamir in [15] for certain NP-complete
problems works in 0(2 "/2) time and O(2 n/4) space, the improved algorithm given
by Tarjan and Trojanowski in [17] for finding a maximum independent set in an
n-vertex graph works in O(2n/3) time.
Besides the complexity in the worst case, a deeper estimation of the practical
performance of algorithms is provided by considering the average complexity of
algorithms, i.e., the mean values of the cost taken over certain sets of inputs with
0304-3975/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

220 Phan Dinh Dieu, Le Cong Thanh, Le Tuan Hoa
certain probability distributions on them. This investigation was made intensively
by Karp and others (see, e.g., [1, 6, 7]). An impressive result, due to Smale recently
[16], shows that Dantzig's simplex algorithm for the problem of linear programming
works in average polynomial time (with respect to the number of variables), although
it requires the exponential time in the worst case.
For examining the average complexity we have to randomize the input set, i.e.,
the set of instances of the considered problem. This can be done by giving probability
distributions on the sets In of inputs of size n (n = 1, 2,...). Then, the input itself
becomes a random variable, and the time complexity of a certain algorithm, as a
function of the input, becomes also a random variable, whose mean value is the
average time complexity T(n) of the given algorithm on the input set In.
In this paper we shall consider only the problems with discrete structure, for
which the set In is finite, and Ilnl-, oo as n ~ oo. Therefore, if the distributions given
on In are uniform, then any algorithm working in average polynomial time also has
a polynomial time complexity almost everywhere, i.e., for almost all of the inputs,
as n ~ oo. However, this does not hold when the distributions given on the sets I,
are not unifotJi~. Here, a property P is understood to hold almost everywhere, or
for almost all inputs as n ~ oo, if
lim (p(n)/lInl)= 1,
n---~OO
where p(n) is the number of inputs of size n for which P holds. However, the main
results (Theorems 3.5 and 4.4) are still valid if "almost everywhere" is understood
after Karp's definition in [6].
In Section 2 of this paper we shall consider the CLIQUE problem for several
nonuniform distributions on the input sets, and show that for certain values of the
parameter p in these distributions the CLIQUE problem can be solved in average
polynomial time. Of course, this does not imply the polynomial time complexity
almost everywhere.
Our attention will be concentrated on the case of uniform distributions on the
input sets. In this case the estimation of average complexity has an elementary
sense; and besides, from an estimation of average polynomial complexity immedi-
ately follows an estimation of polynomial complexity almost everywhere.
In earlier papers [10, 12, 13, 14] we have shown that for some NP-complete
problems--such as CLIQUEand other related problemsmthere are algorithms solving
them and working in average (and almost everywhere) time n°°°g n). Recently, such
an estimation of average time n°°°gn) (and almost certainly) was given also by
Lifsehitz and Pittel [8] for the algorithms solving a class of SET COVERINGproblems.
In Sections 3 and 4 we shall consider a class of CLIQUE problems, namely, the
problems CLIQUEN(n), which are the usual CLIQUE problem restricted to the set
of graphs of n vertices and <~N(n) edges. It will be proved in the Section 3 that
the problem CLIQUEN(n), when N(n)<~ n~ (0<e <2), can be solved in average
polynomial time; and in Section 4 it will be shown that when N(n) = [n ~] (0 < e < 2)

Average polynomial time complexity of some NP.complete problems 221
the problem CLIQUEN(n) still remains NP-complete. From these facts the main
result of the paper follows, which states that when N(n)=[n ~] (0<e<2), the
problem CLIQUEN(n)is simultaneously NP-complete and solvable in average, as
well as almost everywhere, polynomial time.
The CLIQUE problem is a typical one in the class of NP-complete problems. An
'efficient' algorithm can be derived from the Tarjan-Trojanowski algorithm finding
a maximum independent set [17]. However, in this paper we shall use a simple
algorithm, which first lists all maximal cliques, and then chooses among them one
of greatest size (or of a required size k). For listing all the maximal cliques of any
given graph we shall use the algorithm due to Tsukiyama, Ide, Ariyoshi and
Shirakawa in [18], which works in O(n.m.p.) time, where m = (~)- m', n, m', and
p being the numbers of vertices, edges, and maximal cliques of the given graph,
respectively. Hence we shall call the studied algorithm the TIAS-algorithm.
It is obvious that this TIAS-algorithm can find a maximum clique, or decide if a
given graph has a clique of k vertices, also in O(n'm'i~) time. Throughout this
paper by the name 'the CLIQUEproblem' we shall indifferently mean the problem
of finding a maximum clique of any given graph, or of deciding, for any given graph
G and number k, whether G has a clique of k vertices or not. Thus, the complexity
of the TIAS-algorithm essentially depends upon the number of all maximal cliques
of graphs, and our results in this paper will be deduced mainly from the estimations
of this number of all maximal cliques.
2. The CLIQUEproblem in different distributions on the data set
2.1. Average number of k-vertex cliques
The graphs considered in this paper are always assumed to be simple undirected
graphs without loops. Let G,~p be the generic random graph on the vertex set
V, = {1, 2,..., n} with
Prob(e~G,~p)=p (0~<p<~ 1),
for all edges e, which are drawn independently (see [3]).
Denote by ~n the set of all graphs with n vertices. For any graph G ~ (gn having
m edges (0~
< m ~<(~)) we have
Prob( p = G) = p m (1 --
These probabilities give us a distribution on the set ~n, which we denote by ~rp.
When p = ½we call ~rp the uniform distribution on the set (g,. In the general case,
the number of edges of G~.p,denoted a~.p, is a random variable having the binomial
distribution with parameters ((~), p):
Prob(a,.p = m) = pro(1 __p)(~)--m.

Together with the random variable G~.p we shall consider also the following
random variables:
• ~:,..k.p(0~ k~ n): The number of all k-vertex cliques of G..n,
• ~,~p:the total number of cliques of G,~,,
• /~n,n: the total number of maximal cliques of G,..p.
For any random variable to, let Eto be the mean value of to, and Vto be the variance
of to.
We now recall some classical estimates in the theory of random graphs as follows.
Lemma 2.1. For any n > O, 0 < k <~n and 0 <~p <~1 we have
(a) E~:.,~p = (g)p@,
k
(b) V~.,kp=(£)p
2(g)Y~ (,~(~-_,~(p-@--l).
t----2
Proof. (a) Let Di (i = 1, 2,..., (~)) range over the k-vertex subsets of V,. Set
10 if D~ forms a clique of G,p,
Xi=
otherwise.
We have
~,,,k,p = ~) Xi.
i=I
Therefore,
E~:~.~, = Z E(X,) = Prob(X, = 1) = (~)p@.
i=1 i=1
(b) We now calculate V~.k,p 2 2
= E~,~p - (E~,~) . We have
2 _ ~) E(X, Xj)= ~) Prob[X,= IIXj= l]Prob(Xj-- 1).
E~,,,k.p--
/,j=l i,j= l
We note that if ]D~r~ Dj[ = t (0~
< t~ k), then
Prob[X, = 1 IXj = 1] = p(~)-(J).
For every t (0~
< t~
< k) there are (7)(~2~)(~Z~ pairs (D,, Dj) such that IDin Dj[ = t.
Hence, we have
k
2
E~:.,k.l,= Z Z Prob[ X, = 1[Xj = 1] Prob( Xj = 1)
t=O [Dtr~Dll=t
k
= Z (7)(r~-:)(~I~P@-(I)P (~)
t=O
(£)
= Z
t=2
k
+ Z
t=2

Average polynomial time complexity of some NP-complete problems 223
k
---(E~,~kp)2+ ~ (r~)(~(~-~)p2q)(p -(9- 1)
t=2
from which the value of V~:~,k,vfollows. []
2.2. Average number of cliques
Assume that 0<p~ I/log n. Let t~ = 1/p; we have a ~>log n (the probability p,
and therefore a, can depend on n).
Lemma 2.2. Let fl be a constant, ½< fl < 1. As n is large enough we have
n<fl-1/2)i°gn/l°g''-l/2< max E~,.k.~,< n(1-fl2/2)l°gn/l°g°t+fl/2.
l~k~n
Proof. For simplicity of notation we shall write Ek for E~:n,k,p. By elementary
computations we have
Ek+l n -- k
Ek -- k+ 1pk'
and hence,
log(Ek+~/Ek) = log(n -- k) - k log t~ -log(k+ 1).
Therefore, if k I>log n/log a, then
log( Ek+,/ Ek) < O.
Furthermore, for n large enough and with k ~</3 log n/log ot we have
log(Ek+~/Ek) > (1 -/3)log n -log log n -O(1) > 0.
Thus, the expression Ek, as a function of k (0 <~k <~n), attains its maximum at some
value kM in the interval
13log n < kM < log n
log t~ log a"
By some rough estimates we have
n ) nlogn/iogot"
n kM
kM
p(k2M)~ p~21og2 n/21og2 ot--fl log n/21og ot~ n-fl21°gn/21Og a n fl/2.
Therefore,
max gk < n (1-f12/2)lOg n/Ioga+fl/2.
k
On the other hand, we have
> >
kM kM i~ggn ] > niSl°g"/l°g<'-'
p(~) > piOg2,,12 log2,~-Iog ,,12log ~<I> n -log ,,12 log <~
+ 1/2.

Therefore.
max Ek > n(13-1/2)l°gn/l°got-1/2.
k
[]
From Lemma 2.2 we obtain the following corollary.
Corollary 2.3. Let fl be a constant, ½< ~ < 1. As n is large enough we have
n (fl-l/2)lOgn/lOga-l/2 ( E~n,p< n (1-fl2/2)lOgn/l°g°t+3/2.
2.3. The case p = I/n ~
We now consider the case p= 1/n ~, where e is an arbitrarily small positive
constant. In this case we have the following result.
Lemma 2.4. Let p = 1/n ~. Then, as n is large enough, we have
(a) E~p < n 1/2e+3/2,
1
(b) Prob(/.~,,,p< n 1/2"+z)> 1-~nn"
Proof. (a) follows from the last inequality in Corollary 2.3, and (b) follows from
(a) and the following obvious inequality:
1
Prob(/z.., < .¢r-ffE/z~.p)
> I - ~ff. []
From this lemma we immediately obtain the following theorem.
Theorem 2.5. Assume thatp = 1/n ~, where e is a positive constant. With the distributions
¢rp given on the sets c~, (n > 0), the problem CLIQUE can be solved (by the TIAS-
algorithm) in average polynomial time.
Remark 2.6. Under the assumption of Theorem 2.5, the TIAS-algorithm also solves
the CLIQUE problem in polynomial time almost certainly in the following sense:
there exists a polynomial g(n) such that
Prob(T(n)<~g(n))-->l as n-->eo,
where T(n) is the random variable measuring the time required for the TIAS-
algorithm when working on the random graph G~,p.
Note that in this ease the distributions 7rp are not uniform, therefore, from the
property "in polynomial time almost certainly", the property "in polynomial time
almost everywhere" does not follow.

Remark 2.7. Theorem 2.5 remains valid when p< 1/n ~, where e is any positive
constant. Thus, we see that when the considered graphs are rather sparse, the CLIQUE
problem can be solved in average polynomial time.
However, as considered in what follows, this property does no longer hold if the
examined graphs are not sparse enough.
2.4. The case p =//log/,1
We now consider the case p = 1/log n (i.e., ot = log n), and we shall show that
with the corresponding distributions ~rp on the sets ~d, the TIAS-algorithm does no
longer work in average polynomial time.
Lemma 2.8. Assume that p = 1/log n. Then, for k = [log n/log a], we have
Prob(g~. >/,1~°~n/~°g~°g--~) > 1 - e~(/n)
when n is large enough, where e~(n)--> 0 as n-->0o.
Proof. Similarly as in the proof of Lemma 2.2, we obtain
E~n,k,p> ],/log n/2 log log n-l~2
We now estimate the value of V~,,,k,p. By Lemma 2.1 we have
k
v&~. = (7,)p21~) E a,,
t=2
where
a,=(,k)(~--,k)(p--(~)--l) (t=2, ..., k).
It is easy to see that, for t = 2,..., k- 1,
a,+x <~ (k- t)2 (log n)t < (k- t)2n < kZ"
a, 2(n-2k) 2(n-2k)
hence,
at+l < k2at.
V~,,,k,n
Therefore,
k 2
I' n~ .,2(2) ~.,
< (DpZ(gazE kz('-2) <~ ~ n
~kJF k 4 t'2.
t~2
Since
(1)
/,12
~a z ~ (~)/log n,
K

we finally have
1 E 2
Vsr'k'P < log n (~..k,l,) •
By putting t = E~,.k.p/~ in Tchebyshev's inequality:
Pr°b(l~'t;P - E¢"t;Pl ~>t) < ~< loCi-d-~'
we obtain
( (1) )
Prob sr~.k.p> 1 ~ E~:,.k., >1
lo,/iW
From (1) and (2) Lemma 2.8 follows. []
Lemma 2.9. Assume that p = 1/log n. When n is large enough we have
(a) erob(~,,a, > nlog n/21oglogn-2) > 1 -- e2(n),
(b) E/~.,p > n l°g n/21oglog n-2,
where e2(n)-->0 as n -->~.
ProoL Let k = [log n/log a] and h = [2 log n/log a]. With p = 1/log n, we obtain
Esr ,h,p= (~)pl~) < n21Ogn/loglogn(log n)-21°g2n/]oglogn+logn/loglog n
n -l°g2n/loglogn
Let E3(n ) : n l°g2n/l°gl°gn. It is obvious that
Prob(sC.,h.p < esl(n)E~.,ha,) > 1 -- es(n),
hence,
Prob(sr..h.p = 0) = Prob(~h,p < ) > 1 - e3(n ).
By virtue of Lemma 2.8, from (3) it follows that
Prob(~,~p> n l°gn/21°gl°gn-1 & ~n,h,p = 0) > 1 -- e2(n),
(2)
(3)
(4)
where e2(n) = et(n) + es(n) - el(n)ts(n) -->0 as n ->oo.
Suppose that G ~ ~,, and 0 < k ~ h. if G has no h-vertex cliques, i.e., every clique
of G does not contain h or more vertices, and if G has m k-vertex cliques, then
the number of maximal cliques of G must be greater than m/(~,), because every
maximal clique (with less than h vertices) cannot contain more than (~) k-vertex
cliques. Therefore, from (4) we obtain
Prob(/z,~p > nl°g"/21°gl°g"-'l(~:)) > 1 - e2(n). (5)

Average polynomial time complexity of some NP-completeproblems 227
Since
(h) < 2h< 221ogn/Iogiogn<~n2/loglogn< n 1/2,
part (a) of Lemma 2.9 immediately follows from (5).
Part (b) is deduced from (a) in an obvious way. []
From Lemma 2.9 we obtain the following theorem.
Theorem 2.10. Assume that p = 1/log n. With the distribution 75, given on the sets ~
(n > 0), the TIAS-algorithm cannot solve the CLIQUEproblem in average polynomial
time. It cannot solve it in polynomial time almost certainly.
Remark 2.11. Theorem 2.10 remains valid when p = 1/logr n for any constant r> 0.
When p is constant, then the estimations in Lemma 2.9 will be of order n °(i°gn),
and the conclusion of Theorem 2.10 also holds in this case.
3. Average polynomial time complexity for CLIQUEN(n)problems
3.1. The problem CLIQUEN(n)
Let N(n) be a function of natural numbers, such that
0<- N(n)<~(D.
For any natural n > 0 we denote by ~n,N the class of all graphs on n vertices
having <~N(n) edges, and by ~.N the class of all graphs on n vertices having
exactly N(n) edges.
The problem CUQUE, restricted to the graphs in the class
~= U ~n,N ~ = ~-'-~,",,
will be denoted by CLIQUEN(n) (CLIQuEN(n)e, respectively).
We notice that
2q
~a.,~ = U ~.j.
j=l
In this section we shall consider the average time complexity of algorithms solving
the problems CLIQUEN(n) and CLIQUEN(n)e under the assumption of a uniform
distribution on the sets ~,~N and ~9~n,Nrespectively.
We denote by G~,N the random graph with n vertices having N(n) edges, and
by Gn,N the random graph with n vertices having <-N(n) edges.
In our case we have, for any G e ~n,N,
Prob(G~,N G) = / N]'

228 Phan Dinh Dieu, 12 Cong Thanh, Le Tuan Hoa
and, for any G E @..N,
Prob(G,,N=G)=I/~ ((~)~.
• j=oJ /
Some relations between the properties of random graph G~,~and G~,t¢ have been
examined, e.g., in [1]. By using these relations one can translate some properties
of G~,p into those of G.,N and vice versa. However, in this paper we shall estimate
some characteristics of G~,N and Gn•N by direct computations.
Let Ck(G), c(G), and ~(G) denote, respectively, the numbers of all k-vertex
cliques, of all cliques, and of all maximal cliques of the graph G. We shall consider
the following random variables:
=
= ), = c( Gn,,, ), = G..,, ).
Lemma 3.1. For all n > O, 0 <~k <~n, and 0 <~N <~(~), we have
-(~//N]"
Proof. Let
p= and q= k '
for short. Let D~ (i = 1, 2,..., q) denote the k-vertex subsets of V, = 1,..., n.
Similarly as in the proof of the Lemma 2.1, we set
~1 if D, forms a clique of G..N,
x,
t0 otherwise.
Then we have
q
=E X,,
5=1
and therefore,
q q
E~,~k.N= ~, EX,= • Prob(Xi= 1).
5=1 5~1
In this case we have
Prob(X5 = 1) = (('~) -(~ /[ (~)~
from which the lemma follows. []
(6)
3.2. The case N(n) <~n2-~
Let us now consider the case 0 ~<N(n) <~n2-~, where e is some (possibly arbitrarily
small) positive constant (0 < e ~< 2).

Lemma 3.2. Under this assumption we have, for any k, 1 <~k <~n,
E~,kpc = ek( n, N) <~n9/4~
when n is large enough.
Proof. In view of Lemma 3.1 we have
~.k(n, N)<~n k N(N-1)'"(N-(2k) +1) k(N) ~
(~)((~)- 11 • • • ((~)- (2k)+ 1) ~<n (-~
/ n2-~ (~
<~nk~-~) ~ n (4+r)k/4-~.k214
since (~) > n 2-e/2 when n is large enough. By putting f(x) = (4+ e)x/4- ex2/4, we
have
f~4+ e~ (4+e) 2
maxfx)=fTfV/-
Therefore, we obtain, for any k, 1 ~<k ~<n,
~k( n, N) <-n(4+e)2/16e~ n 9/4e
as was to be proved. []
Corollary 3.3. Assume that 0<~ N(n) <-n 2-~, O< e <-2. Then we have, when n is large
enough,
(1) E~'~N<~ n9/4~+1, e ~<n9/4~+1,
E pbn,N ".~.
(2) Esr~,N~ n 9/4s+1, Ep..,pc ~ n 9/4~+1.
Proof. Part (1) immediately follows from Lemma 3.2 and the following obvious
relations:
E¢Xpc=E E# <<-
We put pj = Prob(G,,N ~ f~,j) (0~<j~
< N), We have
N PC
pj = 1 and E~n,pc--" ~, pjE~,,.j.
j=O j=O
From these equalities and (1), part (2) follows. []
Corollary3.4. Assume that 0 <~N (n ) <~n
all graphs G in ~,pc we have
2-~ (0 < e ~<2). Then, when n -->oofor almost
~(G)<n s/2.+~. (7)

Proof. From Corollary 3.3 and the following inequality:
Prob(/~,,,N < nl/4~El~) > 1- n-1/4~,
we obtain
Prob(/z,,N < n5/2~+1)> 1- o(1).
In our case of the uniform distribution on (g~,N,this means also that the fraction
of graphs G for which (7) holds tends to 1 as n-->oo; and thus, Corollary 3.4 has
been proved. []
By similar arguments Corollary 3.4 also holds for ~,N.
From Corollaries 3.3 and 3.4 we obtain the following theorem.
Theorem 3.5. When O<~N(n)<~n 2-~ (0<e<~2) the problems CLIQUEN(n) and
CUQUEN(n) e can be solved (by the TIAS-algorithm) in average, and also almost
everywhere, polynomial time.
Remark 3.6. By taking, for example, N(n)= [n3/4], i.e., e =I, we obtain a problem
CLIQUEN(n), which is solved by the TIAS-algorithm in average, and also almost
everywhere, in time O(nr).
The problem CLIQUEN(n), when N(n) = n, is solved by this algorithm in average,
and almost everywhere, in time O(nT).
Remark 3.7. Theorem 3.5 is still valid if "almost everywhere" is understood after
Karp's definition in [6]. In fact, from Corollary 3.3 we can obtain
qn = Prob(/z~,N ~: n5/2~+3)< n -2
def
and with these values of q, we obviously have ~--1 q, < oo.
3.3. The hard cases of CLIQuEN(n)
Now we consider the case when the function N(n) satisfies the following condition
<-N(n)<~ (or > 0) (8)
(log n) ~ (log n)a'
and we shall prove that in this case, the TIAS-algorithm cannot solve the problems
CLIQUEN(n) and CLIQUEN(n)c in average polynomial time.
Lemma 3.8. Let N satisfy condition (8), and k = [log n/ot log log n]. Then we have
Prob(~k,N > (1 k2
-~) E~k.N) > l-e,(n) (9)
where el (n) ->0 as n -->oo.

Proof. Put k = [log n/or log log n]. we estimate the value of V~k,N. Let Di and X~
(i = 1,..., q) be defined as in the proof of Lemma 3.1. Similarly as in the proof of
Lemma 2.1 we have again
q q
e 2
E(¢N,~N) = E E(X, Xs)= E
/,j = 1 /,j = 1
Prob(X~ = 1IXs = 1) Prob(Xs = 1).
In this case, Prob(Xj = 1) is defined by (6) and if ID, c~DjI= t (0~<t<~k), then
Prob(X, = llXs-- 1)= N-2(~+(~)]/ N-(~ 1"
Therefore,
where
E 2 q k
(~.,~N) =- E a,,
p t~0
q= k ' P= , and a, t]k-t]N-2(~+(~)]"
It is not hard to verify that when k = [log n/a log log n] and N satisfies condition
(8) we have
a 1 ---- max a,.
l~t~k
Therefore, by some simple computations, we obtain
hence,
k 3
e 2~
E (~¢,,,k.N) ~ ao +--al <~(E~,k.N) 2+--(E~,k,N) 2,
q p n
k 3
V£,,.k.N "~---( E~.,~N ) •
n
Now, putting 13= (k2/4-d)E£~.,~N in Tchebyshev's inequality:
VF'.~k,N
Prob(l~,~ - E~:~,~,I I>t3) < t~2
we immediately obtain (9) with el(n) = ~ log log n/log n. []
I-emma 3.9. Let N satisfy condition (8). Then we have
Pr0b(ft~,N :> nl°g'q3~'°gl°g')> 1- e2(n),
where e2(n) -> 0 as n -~ ~.
(10)

Proof. Set k = [log n/o~ log log n], and h = [2(log n)~+l]. First we estimate E~n,h,N.
By virtue of Lemma 3.1 we have
:n~N(N-1)...(N-(~)-I-I)
E~n,h,N= h / (~)((~) - 1) ((:~)_ (h) + 1) <~ L(~')J
Since N satisfies condition (8) we have
E~:..h.N <~ 1 (n -- 1)(log n) ~']
1
(h)(1
( n~9_(h_l)log n _< n ~ 1 1
h ]:" ~ 2-'h--n (.°g n)* <--'n
From this it is easy to obtain
Prob(~h,N < log n) 1
>1
n log n
This also means that •
Prob(~h,N ----0) > 1 -
Now, from (9) and (11) we obtain
log n"
(11)
Prob(~N> 3/& ~h.N = 0) > 1 -- e2(n), (12)
where T=(1-k2/x/n) ESe~k,N, and e2(n)-->0 as n->oo.
By the same reasoning as in the proof of Lemma 2.9, it follows from (12) that
Prob (/~,.N > 7/(kh)) > 1 -- e2(n ). (13)
Taking into account condition (8) we get
(T)_(1-k2/vrn)(~) N(N-1) . . . (N-(~+ I)
(~) (~)((~)- 1) - • • ((~)- (~ + 1)
I> 2h 2(log n)~/2 ] >Ink/(2h)k(log n)~(~)
It is easy to see that
(2h) k <~ n(a+l)/a+2/alosiogn,
therefore, we obtain
y/(h) nlog./3 lo og.,
(log n) a(~)<~ n (k-1)/2,
from which, together with (13), Lemma 3.9 follows. []

Averagepolynomialtimecomplexityof some NP-completeproblems 233
Remark 3.10. From (10) we can deduce that
ElZ~N > n l°gn/4"l°gl°gn. (14)
This inequality means that in the case when N(n) satisfies (8), the average number
of maximal cliques of graphs in the classes ~,,N cannot be bounded by any
polynomial of n. And, since the distributions considered on the sets ~n,N are uniform,
inequality (10) also means that as n -*oo, for almost all graphs G in the class c~n,N,
the number of maximal cliques of G cannot be bounded by any given polynomial
of n.
Thus we obtain the following result.
Theorem 3.11. Assume that
<~N(n)<~ (a >0).
(log n) ~ (log n) ~
Then, the TIAS-algorithm cannot solve the problem CLIQUEN(n)e in average poly-
nomial time. It cannot solve this problem in polynomial time almost everywhere either.
Remark 3.12. We notice that EI~..N=~,j~opjEp~j, where pj=Prob(G,,,,,~ q3~j).
Obviously, we have
PJ----j // j=oJ /
This expression pj attains its maximum at some value J in the interval
(log n) ~ <~J<~ -(log n) ~
and therefore, estimation (14) is still valid for J instead of N. Besides, we have
N < n2. Therefore, we obtain
EP,n,N > niOgn/4ot log log n--2
Thus, the TIAS-algorithm cannot solve the problem CLIQUEN(n) in average poly-
nomial time.
4. NP-completeness of some CLIQUEN(n) problems
4.1. The cases of NP-completeness
In this section we shall find a class of functions N(n) for which the problems
CLIQUEN(n) and CLIQUEN(n) e are NP-complete.
For the sake of convenience, the problem CLIQuEN(n) (CLIQuEN(n) e) will be
understood as follows:

234 PhanDinh Dieu,Le CongThanh,Le TuanHoa
INSTANCE: A graph G= ( V, E) with IEI ~<N(I VI) (or IEI = N(I VI), respectively);
and a positive integer k < lVl.
QUESTION: Does the graph G contain a clique of size k or more?
Theorem 4.1. Let e be a rational number, 0 < e < 2.
(1) The problem CLIQUEN(n), where N(n) >i n ~, is NP-complete.
(2) The problem CLIQUEN(n)e, where N(n)= [n~], is NP-complete.
Proof. Since CLIQUEN(n) and CLIQUEN(n)e are subproblems of the usual problem
CLIQUE, it is obvious that they belong to the class NP. In order to prove the
NP-completeness of these problems we only need to show that the usual problem
CLIQUE can be reduced to them by a function computable in polynomial time.
We notice that since e is a rational number, the expressions [n 2/~] and [(2n) 2/~]
are computable in polynomial time of n.
(1) Consider the problem CLIQUEN(n), with N(n)>t n ~. The reduction can be
simply realized as follows: Let (G = ( V, E), k) be a given instance of the problem
CLIQUE, with V={1,..., n}. We take n'=[n2/~]+l and put V'={1,..., n'}. Let
G'= (V', E') be the graph obtained from G by adding the isolated vertices n +
1,..., n'. Put k'= k: We have
IE'I=IEI n 2 n <~N(n').
Thus, (G', k') is an instance of the problem CLIQUEN(n) and it is clear that G' has
a clique of k' vertices iff G has a clique of k vertices. This reduction can be realized
by a function computable in polynomial time.
(2) Now we consider the problem CLIQUEN(n) e. Let (G--- (V, E), k) be a given
instance of the problem CLIQUE. We shall construct an instance (G'= (V', E'), k')
of the problem CLIQUEN(n)~ with N(n)= [n ~] as follows: Suppose that n = Ivl,
and V={1,2,..., n}. The graph G will have n'=[(2n) 2/~] vertices, and V'=
{1, 2,..., n'}. Since e < 2, for n large enough we have n'> 4n, therefore, for construct-
ing the procedure of reduction we can consider only n's such that n'> 4n. The
vertex set W is partitioned into five subsets V1, V2,..., Vs, where
V~={(i-1)n+ l,..., in} (i = 1, 2, 3,4),
4
vs= v' U v,.
i~l
The set of edges E' of the graph G' is given by:
E'= ( ~-J Ei) k'J( V1X V2U V2X V3U V1X
where V~x V
s denotes the set of all edges joining a vertex in V~with a vertex in Vj,
and E~ (i = 1,..., 5) are defined as follows.
(a) E~ = E, the set of edges of G; and E5 = 0. Thus, the number of edges in the set
E, u Es u V, x V2 u VEX V3u V,x V3
is equal to 3n2+ [EI.

(b) In order to define the sets E:, Ea, and E4 we first introduce the following
ordering on the set U 4
,=2 V~x vii: let (k, l)e V~x V~ and (k', l')e Vjx Vj be two
elements of this set; we define
(k, l) < (k', l') iff either i <j, or i =j and k < k',
or i =j, k = k', and 1< l'.
Now let m = N(n')- (3n2+ IE[), and let W be the set of the first m edges in the
set U 4
i=2 vi x vi ordered as above. Since N(n)= [n ~] and n'= [(2n)2/'], it is easy
to verify that
(2)<m < 3(2), (15,
and thus, W is well defined. The sets Ei (i = 2, 3, 4) are defined by
E,= Wn(V, x V,),
here V~x V~ denotes the set of all edges joining a vertex of V~with another vertex
of E.
Thus, the graph G'= (V', E') with n' vertices and N(n') edges is completely
defined. It remains to define the number k'. For this purpose we set kl = k and
k: = n. From (15), it follows that either (~) < m < 2(~), or 2(~) <~m < 3(~). In the last
case we set k3 = n, and in the first case we set k3 to be the unique integer satisfying
the following inequalities:
½12n-( k3-1 )](k3 - 2) < m -(~) <~½(2n- k3)(k3-1 ).
Then we define k' = kl + k2+ k3.
It is easy to verify that the above described procedure of constructing (G', k')
from (G, k) can be realized by an algorithm working in polynomial time.
Note that every clique of G' of size ~>k' contains only vertices in V1u V2u V3
and, by the above definition, we can see that G' contains a clique of size 1>k' if
and only if G contains a clique of size ~>/c
Thus, the problem CLIQUEN(n)¢ is NP-complete. []
Remark 4.2. We can prove that the problem CLIQUEN(n) is also NP-complete when
N(n) is given by N(n) = [(~) - n~], where e is a rational number, 0< e <2.
In order to prove the NP-completeness of CLIQUEN(n)e in this case, we take
n'=[n2/~]. We also have n'>4n when n is large enough. The set W= V~u- • -w V5
is defined as above, and the set E' is given by
5 )
E'= E, u V'x v'U (v,x v,) ,
i=l i=1
where E1 = E, E5 = V5x Vs, and Ei (i = 2, 3, 4) are defined as follows.
Put m = N(n')-[(~2')-4(~)+[EI]; we can see that 0<~ m<~3(~). We again define
4
W to be the set of m first edges in U~=2 v~x v~, and E~= Wc~ ( V~x V~) (i = 2, 3, 4).

Now, we denote by 1the greatest integer such that m = (1-2)(~), and then take
k:=n for 2<~j<l, ~= 1 for 1<j<~4, and k~ to be the integer satisfying the
inequalities:
½12n- (k~- 1)](kl-2) < m-(l-2)(~)<~½(2n-k,)(kl - 1).
We define k5= IVs[ = n'-4n, and finally we take
k'= k + k2 + k3+ k4+ ks.
It is not very hard to verify that G' contains a clique of size k' or more if and only
if G contains a clique of size k or more. Thus CLIQUE can be reduced to
CLIQUEN(n) e, and CLIQUEN(n) e is NP-complete.
Remark 4.3. The problem CLIQUEN(n) can be solved in polynomial time, i.e.,
belongs to the class P, if either N(n)= O(log n), or N(n)= (~)-O(log n).
4.2. Main result
Combining the results formulated in Theorems 3.5 and 4.1 we obtain the main
result.
Theorem 4.4. The problems CLIQUEN(n) and CLIQUEN(n) e, when N(n)= [n~], is
a rational number such that 0 < e < 2, are NP-complete problems which can be solved
in average polynomial time, and even in polynomial time almost everywhere.
Remark 4.5. In view of the Remark 3.6 we see that the problem CLIQUEN(n), when
N(n) = [n3/4], is an example of an NP-complete problem which can be solved by
a deterministic algorithm in average, and also almost everywhere, in time O(n6).
The problem CLIQuEN(n) with N(n)= n is also NP-complete, and it is solved
in average (and also almost everywhere) time O(nT).
Acknowledgment
The authors would like to thank the (unknown) referee for his or her comments
and suggestions which helped them to improve the paper.
References
[1] D. Angluin and L.G. Valiant, Fast probabilistic algorithms for hamiltonian circuits and matchings,
9th Ann. ACM Symp. on Theory of Computing (1977) 30-41.
[2] B. Apolloni and S. DiGregorio, A probabilistie analysis of a new satisfiability algorithm, RAIRO
lnforr~ Thdor. 16 (3) (1982) 201-223.
[3] P. Erdrs and J. Spencer, Probabilistic Methods in Combinatorics (Academic Press, New York/Lon-
don, 1974).

[4] M.R. Garey and D.S. Johnson, Computers and Intractability (Freeman, San Francisco, 1979).
[5] G.R.Grimmet and C.J.H. McDiarmid, On colouring random graphs, Math. Proc. Cambridge Philos.
Soc. 77 (1975) 313-324.
[6] R. Karp, The probabilistic analysis of some combinatorial search algorithms, in: J.F. Traub, ed.,
Algorithms and Complexity: New Directions and Recent Results (Academic Press, New York/London,
1976), 1-19.
[7] R. Karp and M. Sipser, Maximum matchings in sparse random graphs, 22ndAnn. Symp. Foundations
of Computer Science (1981 ) 364-375.
[8] V. Lifschitz and B. Pittel, The worst and the most probable performance of a class of set-covering
algorithms, SIAM J. Comput. 12(2) (1983) 329-346.
[9] Le Cong Thanh, Estimations of some parameters of finite graphs and applications, Elektron.
lnformationsverarb. Kybernet. 13 (1977) 505-521 (in Russian).
[10] Le Cong Thanh and Phan Dinh Dieu, Asymptotical estimations of some parameters of finite graphs
and their applications, Acta Math. Vietnam. 3 (1978) 51-79 (in Russian).
[11] Phan Dinh Dieu, On the complexity ofthe problem of state minimization for incompletely specified
sequential automata, Elektron. lnformationsverarb. Kybernet. 11 (1975) 3-17.
[12] Phan Dinh Dieu, Asymptotical estimation of some characteristics of finite graphs, RAIRO Inform.
Thdor. 11 (1977) 157-174.
[13] Phan Dinh Dieu, Some investigations of finite graphs and their applications, Proc. FCT Conference,
Berlin (1979) 343-348.
[14] Phan Dinh Dieu, On the average complexity of some discrete optimization problems, Proc. IFIP
Working Conf. on System Modelling and Optimization, Hanoi (1983) 51-59.
[15] R. Shroeppel and A. Shamir, A T= 0(2 n/2) and S=0(2 n/4) algorithm for certain NP-complete
problems, SIAM J. Comput. 10(3) (1981)456-464.
[16] S. Smale, On the average number of steps of the simplex method of linear programming, Math.
Programming 27 (1983) 241-262.
[17] R.E. Tarjan and A.E. Trojanowski, Finding a maximum independent set, SlAM Z Comput. 6(3)
(1977) 537-546.
[18] S. Tsukiyama, M. Ide, H. Ariyoshi and L Shirakawa, A new algorithm for generating all the maximal
independent sets, SlAM J. Comput. 6 (1977) 505-517.

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