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

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A Simple Lower Bound for the Depth of Monotone Circuits Computing Clique Using a Communication Game 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 2 p p 3 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 2 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 1 of transitions (time) and the number of tap e and y resp ectively, where x 2 f (1) and 1 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 i. 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 B This is true b oth in the monotone and the gen- = fA B jjAj = k g : eral case. k Our main result is a simple pro of that a circuit for the k -clique problem in an n- 3 Pro of outline p vertex graph requires depth k when k We will showa lower b ound on the depth of 2 p 3 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 1 2 is given a graph, G , that do es not havea 2 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 t matic, u and v cannot both be xed vertices t, 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 t now restrict the cliques to those that do not t, 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 t of the remaining colorings. clique after round t, p 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 t choices for c, the clique player cannot knowof clique after round t, an edge in q that must be mono chromatic in p 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 V Q = ; 0 Remark 2 Thus, (Q) increases by k at least a factor two since Q is now C = fc : V ! [k 1]g ; 0 a subset of a smal ler set.
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