A Simple Lower Bound for the Depth of Monotone Circuits Computing Clique Using a Communication Game

Home , Circuit complexity

A Simple Lower Bound for Monotone Clique Using a

Communication Game

Mikael Goldmann

Johan Hastad

Royal Institute of Technology

Sto ckholm, SWEDEN

Abstract squares used (space). For a Bo olean circuit we

would b e interested in the numb er of gates (its

We give a simple pro of that a monotone circuit

size) and the maximal distance from an input

for the k -clique problem in an n-vertex graph

to the output gate (its depth). These measures

p p

corresp ond to work and parallel time resp ec-

requires depth k ,whenk n=2 .

tively.

The pro of is based on an equivalence b etween

Since it has b een dicult to show non-trivial

the depth of a Bo olean circuit for a function

lower b ounds for general Bo olean circuits, one

and the numb er of rounds required to solvea

has chosen to study various restricted cir-

related communication problem. This equiv-

cuit mo dels. Anumber of lower b ounds have

alence was shown by Karchmer and Wigder-

b een shown for the size of Bo olean circuits of

son.

constant depth [Ajt83, FSS84,Has86, Raz87,

Warning: Essentially this pap er has

Smo87,Yao85].

b een published in Information Pro cess-

Another case studied is monotone circuits,

ing Letters and is hence sub ject to copy-

i.e. we only allow ^-gates and _-gates, but no

right restrictions. It is for p ersonal use

:-gates . Several interesting results for mono-

only.

tone circuits can b e found in [And85, Raz85,

AB87, KW88,RW89,RW90].

Key words. computational complexity, the-

In what follows we will b e lo oking at mono-

ory of computation, circuit complexity, for-

tone circuits where each gate has fanin at most

mula complexity, monotone circuits

2. In [KW88] Karchmer and Wigderson show

that a monotone circuit for st-connectivityin

an n-vertex graph has depth log n . As

1 Intro duction

part of their pro of they show that computing a

function f with a Bo olean circuit is connected

In complexity theory weareinterested in the

to the following communication game:

amount of resources that are required to com-

pute a certain function. For a Turing machine We have two players, player 1 and player 2,

the resources would typically be the number and they are each given an n-bit string, x

of transitions (time) and the number of tap e and y resp ectively, where x 2 f (1) and 1

y 2 f (0). The game pro ceeds in rounds. De nition 2 For an arbitrary set B and A

In each round player 1 can send player 2 a one B we de ne

jAj

bit message or vice versa. Their task is to nd

: (A)=

an index i so that x 6= y .

jB j

i i

There is also a monotone version of the game

Remark 1 This de nition is useful when we

where i should satisfy x = 1 and y =0. Note

i i

want to describe how much is known about

that for a monotone f there is always suchan

some element x. Suppose that, but we know

that x 2 A B: Suppose further that we

Karchmer and Wigderson [KW88] showed

know the structure of B, but that the structure

the following equivalence between circuit

of A is unknown, or very complicated. Then

depth and the numb er of rounds needed in the

the amount of information we have about x is

game:

given by the structure of B and (A). The

smal ler (A) is, the moreweknowabout x.

Theorem 1 (Karchmer and Wigderson)

For notational convenience weintro duce the

For a function f and an input length n, the

following shorthand:

number of rounds needed in the communica-

tion game equals the required depth of a circuit

De nition 3 For an arbitrary set B and an

computing f .

integer k

This is true b oth in the monotone and the gen-

= fA B jjAj = k g :

eral case.

Our main result is a simple pro of that

a circuit for the k -clique problem in an n-

3 Pro of outline

vertex graph requires depth k when k

We will showa lower b ound on the depth of 2

n=2 . We use the ab ove equivalence b e-

a circuit for CLIQUE(n, k). The idea b ehind

tween circuit depth and communication com-

the pro of is quite simple. By [KW88], what we

plexity.

needtodoistoshow that a proto col for the

related monotone communication game must

Raz and Wigderson have recently showed

use many rounds.

byamuch more complicated metho d that the

The communication version of the clique

clique problem requires depth (n)[RW90].

problem would b e as follows: Player 1 is given

a graph, G containing a k -clique and player

2 is given a graph, G , that do es not havea

2 Notation

k -clique. Their task is to nd an edge that is

presentin G but not in G .

1 2

As mentioned in the intro duction, we will b e

concerned with clique problem.

We will mo dify this by only lo oking at cer-

tain graphs. We then b ound the number of

rounds needed for this restricted set of inputs.

De nition 1 We cal l the set of graphs on n

In particular, player 1, the clique player, re-

vertices containing a k -clique CLIQUE(n, k).

ceives a set q of k vertices, which corresp onds

to the graph that has a k -clique on the ver- We need to express subset size in the following

tices in q , and no edges other than those in way: 2

the clique. Player 2, the color player, receives rounds.

a k 1 coloring, c, of the vertices, corresp ond-

In section 4 we formally describ e the adver-

ing to a complete k 1-partite graph. The task

sary strategy that, given a proto col, nds a

of nding the \faulty" edge in the clique now

clique-coloring pair, (q; c), that requires many

translates into nding two vertices u; v 2 q

rounds. In section 5 we will prove that the

that have the same color. We call fu; v g a

pair (q; c) do es indeed require many rounds.

mono chromatic edge.

In a round the players are allowed to send

one bit each rather than only one of them

4 An adversary strategy

sending a bit. Since we are interested in the

for a proto col

number of rounds rather than the number of

bits transferred, this can only make life eas-

In the next section we givea lower b ound for

ier for them. Each bit that the clique/color

the depth of a monotone circuit for recognizing

player sends decreases the set of p ossible

CLIQUE(n, k). In this section we show how

cliques/colorings. The adversary strategy that

the pair of inputs that require the players to

we will use is to makes sure that an edge that

communicate for \many" rounds is chosen.

app ears in some remaining clique is bichro-

In our communication game the clique

matic in most remaining colorings, and the re-

player is given a set q of k vertices which cor-

maining colorings are 1 1 on the vertices that

resp ond to a clique. The color player is given

app ear in all remaining cliques.

a k 1-partition of the graph in the shap e of a

When a vertex app ears in many of the re-

k 1 coloring c of the graph. Their task is to

maining cliques we \ x" it i.e. we restrict the

agree on a mono chromatic edge , i.e. an edge

set of remaining cliques to those that contain

fu; v g q such that c(u)=c(v ).

this vertex. We then restrict the remaining

As the proto col pro ceeds we will lo ok at the

colorings to those that are 1 1 on the \ xed"

following sets:

vertices.

If an edge fu; v g is mono chromatic in many

V = f1; 2;:::ng is the set of vertices in a

of the remaining colorings, we restrict the set

graph,

of remaining colorings to those colorings c that

have c(u)=c(v ). Since fu; v g was mono chro-

Q is the set of cliques that remain after round

matic, u and v cannot both be xed vertices

since all remaining colorings are 1 1 on the

xed vertices. Assume that u is not xed. We

C is the set of colorings remaining after round

now restrict the cliques to those that do not

contain u, so an edge that app ears in some re-

maining clique is not mono chromatic in many

M V is a set of vertices that o ccur in every

of the remaining colorings.

clique after round t,

We continue this pro cess for k=4 rounds.

From the remaining cliques and colorings we

m = jM j,

t t

cho ose q and c. Since any edge, fu; v g, in q

is not mono chromatic for some of the p ossible

L V is a set of vertices that o ccur in no

choices for c, the clique player cannot knowof

clique after round t,

an edge in q that must be mono chromatic in

l = jL j . c. Thus a proto col requires more than k=4

t t 3

M M [fv g At the b eginning wehave

Q fq 2 Q j v 2 q g

Q = ;

Remark 2 Thus, (Q) increases by

at least a factor two since Q is now

C = fc : V ! [k 1]g ;

a subset of a smal ler set.

M = ;

We rep eat this step until no suchvertex

L = :

v can b e found.

We now describ e howto handle the proto-

3. Q Q

col. Recalling Remark 1, we consider Q and

C to b e subsets of the following sets:

C fc 2 C j c is 1 1onM g

M M .

Q q 2 j M q ;

4. The color player sends bit b .

C fc : V n L ! [k 1]g :

0 0

C fc 2 C jb =0g

Thus m together with (Q ) tells us how

t t

much the color player knows ab out the clique

1 0

C fc 2 C jb =1g

players k -set after round t.

0 1

Similarly, l and (C ) measure what the

C the larger of C and C

t t

clique player knows ab out the color players

L L

coloring.

The proto col is handled in the following

5. Find u; v 2 V n L where u 6= v

fashion. In each round we allow b oth play-

ers to send one bit each instead of just one of

such that (fc 2 C j c(u)=c(v )g)

them sending a bit.

2(C )=(k 1)

At round t the following happ ens:

i.e. at least twice as often as average.

1. The clique player sends bit b .

If such u and v are found

Q fq 2 Q jb =0g

t1 1

Since all c 2 C are 1 1onM we

can without loss of generality assume

Q fq 2 Q jb =1g

t1 1

u=2 M .

0 1

Q the larger of Q and Q

L L [fug

M M

C fc 2 C j c(u)=c(v )g

Remark 3 When a c is restrictedin

2. Find v 2 V n M so that

this way on L it can be seen as a

(fq 2 Q j v 2 q g)

function c : V n L ! [k 1]. Thus,

2(k m)(Q)=(n m)

(C ) increases by at least a factor

two.

i.e. at least twice as often as the av-

erage vertex.

We rep eat this step until no suchvertices

u and v can b e found. If such v exists 4

6. Q fq 2 Q j L \ q = g Remark 4 This shows that a monotone cir-

cuit for CLIQUE(n, (n=2) ) must have depth

C C

t 3

n). Alon and Boppana [AB87 ] have (

proved a lower bound for the size of a mono-

L L

tone circuit for CLIQUE(n, k) that implies a

( n= log n) lower bound on the depth of such

5 The lower b ound

a circuit. Apart from the minor improvement,

we feel that our proof is simpler.

We are now ready to prove a lower b ound

We will prove Prop osition 2 by induction

for the depth of a monotone circuit for

over t, the numb er of rounds used. Before we

CLIQUE(n, k). If we handle the proto col as

do so we need relationships b etween (C ) and

describ ed the following is true.

(C ), and b etween (Q )and(Q ).

t1 t

Prop osition 2 Let t satisfy the fol lowing in-

Lemma 4 After step 3 we have

equalities:

(m +1)

(C ) 1 (C ):

t ; (1)

k 1

t : (2)

Pro of: Let us use the following shorthand:

(u;v )

C = fc 2 C j c(u)=c(v )g :

Then the fol lowing inequalities hold:

m 2t

We observe that we always have for t > 0,

(Q ) 2 ; (3)

l 2t

di erent u; v 2 V n L

(C ) 2 : (4)

2(C )

(u;v )

Assuming that this is correct, wegetalower

C < :

k 1

b ound on circuit depth by

This follows by our choice of C and from step

Theorem 3 For k recognizing n=2

5 in round t 1.

k -cliques in a graph with n vertices requires

k . depth

(C ) = (fc 2 C j c 1 1onM g)

t1 t

1 0

Pro of: For suchvalues of k (2) will always

[

C B

(u;v )

be satis ed as long as (1) is. Run the proto-

C B

C = (C )

A @

col T = k=4 rounds. We obtain Q , C ,

T T

u;v 2M

u6=v

M and L . Givetheplayers some input pair

T T

(q; c) 2 Q C . Wehave q \ L = .

(u;v )

T T T

(C ) C

If T rounds were sucient the clique player

u;v 2M

would now know an edge fx; y g q that is

u6=v

mono chromatic edge in all c 2 C . This im-

2 m

1 (C )

plies x 2 L _ y 2 L , i.e. q \ L 6= , and we

T T T

2 k 1

havea contradiction.

(m +1)

The theorem now follows by the equiva-

> 1 (C ):

k 1

lence between circuit depth and communica-

tion complexity stated in Theorem 1. 5

Lemma 5 After step 6 we have

(C )

l 2t+1

2kl

by induction 2 : (6)

(Q ) 1 (Q ):

Wenow go on to the second part of round

Pro of: After step 2 wehave for all v 2 V n M

t, where the color player sends one bit. The

0 0

b ounds established for (Q ) and (C ) allow

t t

2(k m )(Q )

us to nish the pro of. Wegetby Remark 3:

(fq 2 Q j v 2 q g) <

n m

l l 0

t t1

(C ) 2 (C )

Since L V n M we have the same b ound

t t

for v 2 L .

l l l 2t+1

t t1 t1

2 2 using (6)

[

l 2t

0 0

2 :

(Q ) = (Q ) fq 2 Q j v 2 q g

t t

v 2L

This shows that (4) holds. Since (C ) 1we

2(k m )l

t t

know that l 2t. We apply this to Lemma 5:

(Q ) 1

n m

2kl

(Q ) 1 (Q )

1 (Q ):

4kt

1 (Q )

Pro of of Prop osition 2: We wish to show

(Q using (2) )

that equations (3) and (4) hold for all t that

m 2t+1

satisfy (1) and (2).

2 using (5)

Since we have (Q ) = (Col ) = 1 and

0 0

m 2t

2 :

m = l = 0 the prop osition is true for t =0.

0 0

Now assume that the prop osition holds for

This completes the pro of of the prop osi-

the rst t 1 rounds. First we use the induc-

tion.

) tion hyp othesis to givelower b ounds for (Q

and (C ). By Remark 2 weget:

6 Acknowledgment

0 m m

t t1

(Q ) 2 (Q )

We are grateful to Mauricio Karchmer for his

m 2t+1

by induction 2 : (5)

comments and suggestions on a draft of this

article. We also thank Noga Alon for helpful

Since (Q ) 1wehave that m 2t 1.

discussions.

When we apply this to Lemma 4 we obtain:

(m +1)

(C ) ) 1 (C

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