Remarks on Graph Complexity
(Extended Abstract) Satyanarayana V. Lokam 1
[email protected] Department of Mathematical and Computer Sciences Loyola University Chicago Chicago, IL 60626
Abstract. We revisit the notion of graph complexity introduced by
Pudlak, Rodl, and Savicky [PRS]. Using their framework, we show that suciently strong superlinear monotone lower bounds for the very special class of 2-slice functions would imply superpolynomial lower bounds for some other functions. Given an n-vertex graph G, the corresponding 2-slice function fG on n variables evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's, fG evaluates to 1 exactly when the pair of variables set to 1 corresponds to an edge in G. Combining our observations with those from [PRS], we can show, for instance, that a lower bound of n1+ (1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 (l) lower bound on the formula size of another explicit function g on l variables, where l = (log n). We consider lower bound questions for depth-3 bipartite graph complexity. We prove some weak lower bounds on this measure using algebraic methods. For instance, our results give a lower bound of ((log n= log log n)2 ) for bipartite graphs arising from Hadamard matrices, such as the Paleytype bipartite graphs. A lower bound of n (1) on the depth-3 complexity of an explicit bipartite graph would give superlinear size lower bounds on log-depth boolean circuits for an explicit function. Similarly, a lower bound of 2(log n) (1) would give an explicit language outside the class 2cc of the two-party communication complexity.
1 Introduction and Overview Proving superlinear (superpolynomial) lower bounds on the circuit size (formula size) of an explicit boolean function is a major challenge in theoretical computer science. On the other hand, remarkable results have been proved in restricted models such as constant depth circuits [Ha,Ra2,Sm], monotone circuits [AB,Ra3], and monotone formulas [RW,KaWi,RM]. In general, techniques from these results on restricted models are considered unlikely to be useful to attack the general question. In fact, there are functions with exponential lower bounds on the monotone complexity, but with polynomial upper bounds on their complexity over a complete basis [Ta]. However, for the special class of slice functions, monotone complexity and general complexity di er only by a 1
Part of the work done while the author was a postdoctoral fellow at University of Toronto.
small polynomial amount [Be]. An n-variable boolean function f is called a kslice function if f (x) = 1 whenever jxj > k and f (x) = 0 whenever jxj < k, where jxj denotes the number of 1's in the input assignment x. When jxj = k, f (x) may be nontrivially de ned. A function is called a slice function if it is a k-slice function for some 0 k n. There are NP-complete languages whose characteristic functions are slice functions [Du]. Hence, assuming P 6= NP, it is conceivable that superpolynomial size lower bounds over a complete basis may be provable by proving superpolynomial size monotone lower bounds for such slice functions. For the approach via slice functions described above to yield superpolynomial lower bounds, we must consider a nonconstant k, because a k-slice function has complexity at most O(nk ). But we will show in this paper that for constant k, in fact for k = 2, suciently strong superlinear lower bounds on k-slice functions would already imply superpolynomial lower bounds for some other functions derived from the 2-slice functions. More speci cally, a lower bound of n1+ (1) on (even monotone) formula size of an n-variable 2-slice function f implies a lower bound of 2 (l) on the formula size of an l-variable function g. We show this using the framework of graph complexity introduced by Pudlak, Rodl, and Savicky [PRS]. Note that the nontrivial part (for jxj = 2) of a 2-slice function is essentially a (labeled) graph. Conversely, with every n-vertex graph G we can associate an n-variable 2-slice function fG (see De nition 4). The model of graph complexity [PRS] is a common generalization of the models of boolean circuits and two-party communication complexity. Measures of graph complexity such as ane dimension and projective dimension of graphs have been proposed and studied in [PR2,Ra1,PR1] as criteria for lower bounds on formula size and branching program size of boolean functions. Separation questions about classes of two-party communication complexity [Yao,BFS] can be reformulated as lower bound questions about bipartite graph complexity as studied in [PRS]. In graph complexity, an n-vertex graph G is constructed or computed as follows. We are given a set of atomic n-vertex graphs (these are analogous to the input variables in a circuit or a formula), called generators. In each step of the computation, we can perform a set-operation on the sets of edges of graphs we have constructed so far. Starting with the generators and performing the allowed operations on intermediate graphs, we would like to obtain G as the result of the computation. Complexity of G is the minimum cost of such a computation to construct G. By stipulating the structure of the computation (such as circuits), the set of generators (such as complete bipartite graphs), and the allowed setoperations (such as union and intersection), we get various measures of the cost of the computation and a corresponding de nition of the complexity of G. With a boolean function f on 2l variables, we can naturally associate a bipartite graph Gf with color classes f0; 1gl such that (x; y) is an edge of Gf i f (x; y) = 1. Let n = 2l , so that G is an nn bipartite graph. In [PRS], it is shown that a lower bound of (log n) on the circuit size (formula size) complexity of
Gf , with complete bipartite graphs as generators and union and intersection as operators, would imply a lower bound of (l) on the boolean circuit size (formula size) of f . Hence superlogarithmic (superpolylogarithmic) lower bounds on
the complexity of some explicit bipartite graphs would yield superlinear (superpolynomial) lower bounds on the circuit size (formula size) of explicit boolean functions, resolving long-standing open questions in computational complexity. In this paper, we make the simple observation that lower bounds on graph complexity are implied by lower bounds on 2-slice functions. We prove that a lower bound of n log n (n), for any (n) = !(1), on the monotone formula size of a 2-slice function implies a lower bound of (n) on the formula complexity of the corresponding graph. Similar results hold for circuit size. Combining this relation with the results from [PRS] we prove that suciently strong lower bounds on the monotone complexity of the very special class of 2-slice functions imply lower bounds on the complexity of general boolean functions. Next, we consider lower bounds on graph complexity. As mentioned above, the model of graph complexity is more general than the models of boolean circuits and two-party communication complexity. Thus, proving lower bounds on graph complexity is even harder. However, studying the graph-theoretic structure of boolean functions may provide insights into their complexity. Understanding the properties of graphs that imply high graph complexity may suggest candidate functions with high boolean complexity. Lower bound arguments for such boolean functions may in turn exploit tools from the well-studied area of graph theory. Pudlak, Rodl, and Savicky [PRS] prove some nontrivial formula size lower bounds in graph complexity. Their lower bounds are on star-formula complexity of graphs. A star-formula for a graph is a formula with union and intersection operators and \stars" as the generators, where a star is a complete bipartite graph with a single vertex on one side and all other vertices on the other side. In this paper, we focus on the bipartite-formula complexity of graphs. Here the generators are complete bipartite graphs, and the operators are union and intersection. Results in this more general model translate more readily into the frameworks of boolean function complexity and two-party communication complexity. In [PRS], a lower bound of (n log n) is proved for the star-formula complexity of some bipartite graphs. This immediately implies a lower bound of (log n) on the bipartite-formula complexity. To get any new lower bounds on boolean formula complexity, this needs to be improved to at least (log3 n). The methods used in [PRS] cannot give bounds beyond (log n). Furthermore, [PRS] also show that certain Ramsey type properties of graphs (absence of large cliques and independent sets) do not imply strong enough lower bounds on graph complexity to give new results in boolean function complexity. We prove lower bounds on depth-3 bipartite-formula complexity of some bipartite graphs. Our results give a lower bound of ((log n= log log n)2 ) for bipartite graphs arising from Hadamard matrices such as the Paley-type bipartite graphs. In fact, our lower bounds are expressed in terms of the spectrum of the 1 incidence matrix associated with the bipartite graph. Our methods are based
on approximating polynomials for boolean functions [NS] and rank lower bounds on real matrices under sign-preserving perturbations [KW]. Our lower bounds are still too weak to imply any new results in boolean circuit or communication complexity. We will note that depth-3 bipartite-formula complexity is related to depth-3 boolean formulas and the class 2cc of the twoparty communication complexity model [BFS]. Lower bounds on depth-3 boolean circuits have been pursued in much of recent research [HJP,PPZ,PSZ]. One motivation for \strongly" exponential depth-3 lower bounds comes from Valiant's [Va] result that depth-3 lower bounds of 2!(l= log log l) for an l-variable boolean function would imply superlinear size lower bounds on log-depth circuits computing thatp function. Currently, the best known lower bound on depth-3 circuits is (l1=4 2 l ) for the parity function [PPZ]. One approach to develop tools for depth-3 lower bounds in boolean complexity is to understand the corresponding (more general) question in graph complexity. Such an approach might lead to graph-theoretic criteria for depth-3 lower bounds in boolean complexity (just as it lead to the notions of ane and projective dimensions as criteria for unrestricted boolean complexity). A lower bound of n (1) on the depth-3 bipartite-formula complexity of an explicit bipartite graph would give the \strongly" exponential lower bounds mentioned above and hence would imply superlinear size lower bounds on logdepth circuits for an explicit function. But the current lower bounds pl on depth-3 bipartite-formulas are too weak even to give (the current best) 2 lowerp bound on depth-3 boolean formulas. In fact, generalizing the best known 2 l lower bound (or even the weaker bound of 2l ) on depth-3 boolean complexity to the framework of graph complexity would resolve a long-standing open question log n
in communication complexity: a lower bound of 2 on depth-3 bipartiteformula complexity of an explicit bipartite graph gives an explicit language outside the class 2cc of the two-party communication complexity. Such strong bounds on graph complexity, however, remain as interesting open questions in this area. (1)
(1)
2 Models of Graph Complexity The complexity of a graph measures the diculty of constructing a target graph using a given collection of primitive graphs, called generators, and a given basis of operations on sets of edges. All the graphs involved are assumed to have the same set of vertices, typically V = f1; : : : ; ng. A set operation on graphs refers to the operation on the corresponding edge sets. For instance, the result of G1 [ G2 on graphs G1 = (V; E1 ) and G2 = (V; E2 ) is the graph G = (V; E1 [ E2 ). Models of graph complexity are de ned analogous to the standard models of circuits and formulas where the generator graphs play the role of input variables and the set operations play the role of gates. We now give some formal de nitions, most of which are based on [PRS]. Fix a set of generator graphs G with vertex set V and a basis O of set operations. A graph circuit with generators G and basis O is a sequence of equations
or gates g1 ; : : : ; gs such that each gi , for i = 1; : : : ; s, is of the form
Gi = H for some H 2 G , or
Gi = Gj Gk where 2 O and j; k < i.
(Here we are assuming for simplicity that is binary; analogous equations can be written for operations of other arities.) The output graph Gs of the last gate is the graph computed by the circuit. The circuit complexity of a graph G, with respect to generators G and basis O, is the smallest s for which there exists a circuit that computes G. As usual, we can imagine a circuit to be a directed acyclic graph (DAG) with the input nodes (of in-degree 0) labeled by the generator graphs and the internal nodes (gates) labeled by set operations from the basis. The target graph appears at the root (of out-degree 0) of this DAG. The length of the longest path in the DAG is the depth of the circuit. Number of nodes is its size. A graph formula is a graph circuit in which the out-degree of each gate is at most one. Thus a graph formula can be represented as a tree with the leaves labeled by generator graphs and the internal nodes labeled by operations from the basis. The size of a formula is the number of leaves in its tree. The formula complexity of a graph is the smallest size of a formula that computes the graph (with respect to a xed set of generators and a basis). We can also de ne the natural restricted models such as constant depth graph circuits and formulas. In these models, we allow unbounded fanin and assume that the operations from the basis are naturally extendable to an unbounded number of operands (for example, union, intersection, and symmetric di erence of sets have this property). In what follows, we concentrate on formula complexity. Similar de ntions and results can be stated for circuit complexity. In this paper, we will consider graph complexity with the set operations of union and intersection only. We will naturally want the sets of generators to be complete in the sense that every graph should be constructible from these generators and using \ and [ operators in a circuit or a formula. De nitions 1 and 2 below give two such sets of generators.
De nition 1?VFix the vertex set V where jV j = n. The set of stars is de ned to be S = fG :G = K ;n? g. 2
1
1
For a graph G on V , the star-formula complexity of G, denoted LS (G), is the smallest size of a graph formula computing G using the stars S as the set of generators and [ and \ as the basis.
A standard counting argument shows that LS (G) = (n2 = log n) for most graphs G. It is also known [Bu,PRS] that for all graphs G, LS (G) = O(n2 = log n). Pudlak et al. [PRS] show that the star-formula complexity of the complement of a graph can be substantially larger than the complexity of the graph itself.
We are especially interested in the complexity of bipartite graphs because of their direct relevance to lower bounds on boolean circuits and commmunication complexity. De nition 2 Fix the color classes U and V . Let B denote the following set of complete bipartite graphs: B = fA V : A U g [ fU B : B V g: For a bipartite graph G U V , the bipartite-formula complexity of G is the smallest size of a graph formula computing G using B as the set of generators and [ and \ as the basis. Bipartite-formula complexity of G is denoted by LB (G). The following relation holds between bipartite complexity and star complexity. Proposition 1 (PRS) Let G U V be a bipartite graph, jU j = jV j = n. Then LS (G) LB (G) n + 2n, where the stars are on the vertex set U [ V . Another observation of [PRS] is useful to translate a lower bound on starcomplexity of general, i.e. not necessarily bipartite, graphs to a lower bound on bipartite graphs: Proposition 2 (PRS) Let G be a graph on W , jW j = 2n. Then there exists a partition W = U [ V , jU j = jV j = n, such that the bipartite graph G \ U V satis es LS (G) : LS (G \ U V ) dlog 2ne 2
De nition 3 Let f be a boolean function on 2l variables, written as f : f0; 1gl f0; 1gl ?! f0; 1g. Let n := 2l . The n n bipartite graph Gf f0; 1gl f0; 1gl is de ned by including the edge (x; y) in Gf i f (x; y) = 1, where x; y 2 f0; 1gl.
Note that the and and or operations on boolean functions correspond to and intersection operations on the edge sets of their corresponding graphs. In other words, Gf ^f = Gf \ Gf and Gf _f = Gf [ Gf . This suggests a syntactic transformation of a boolean formula (assuming all negations are pushed to the leaves) into a graph formula. But what about the input literals of the boolean formula? The literals are simply the projection funtions and the graphs corresponding to projection functions are complete bipartite graphs isomorphic to Kn=2;n and Kn;n=2. For instance, Gxi is the complete bipartite graph fx 2 f0; 1gl : xi = 1g f0; 1gl. Thus each literal can be translated into a generator in B. With this transformation of a boolean formula for f into a bipartite-formula for Gf , it follows that LB (Gf ) L(f ); (1) where L(f ) is the minimum size formula (with tight negations) computing f . Given an n n bipartite graph G, where n is a power of 2, we can clearly de ne a function f such that Gf = G. Thus we get the following criterion for lower bounds on boolean formula size:
union
1
2
1
2
1
2
1
2
Proposition 3 (PRS) An explicit n n bipartite graph G, where n = 2l, with lower bound of LB (G) (log n) would give an explicit function f on l varaibles with formula size lower bound L(f ) (l). Since the proof of this proposition is essentially syntactic, similar relations hold for several restricted models of formulas and circuits as well. Note, however, that graph complexity of Gf could be much smaller than the boolean complexity of f . This is because in a bipartite-formula we have access to an exponential (in n) number of generators B, whereas the transformation above uses only the 2 log n \canonical" generators corresponding to the projection functions. In fact, the generators in B, in case of graphs G f0; 1gl f0; 1gl, can be viewed as de ning (arbitrary) boolean functions of either the rst l or the last l variables. This interpretaion captures the connection between twoparty communication complexity and graph complexity. In particular, bounded depth bipartite formulas of size 2(log log n)c de ne the complexity classes of the \polynomial hierarchy" of the two-party communication model [BFS].
3 2-Slice Functions In this section, we relate graph complexity to the boolean complexity of 2-slice functions.
De nition 4 A boolean function f : f0; 1gn ?! f0; 1g is a 2-slice function if f (x) = 0 for all x with jxj < 2 and f (x) = 1 for all x with jxj > 2. On inputs x with jxj = 2, f may be nontrivially de ned. The inputs x such that jxj = 2 and f (x) = 1 can be identi ed with the edges of a graph G with vertex set f1; : : : ; ng in an obvious way. Conversely, every graph G on n vertices gives rise to a 2-slice function fG on n variables: Given a graph G = ([n]; E ), we de ne the function fG by 8 1 > >
2, 0 2, fG (x) = > 1 ifif jjxxjj < = 2 and X 2 E (G), > : 0 if jxj = 2 and X 62 E (G). Here X denotes the set with characteristic vector x, i.e., X = fi : xi = 1; 1 i ng for x 2 f0; 1gn.
Recall that for 1 i n, the star Si is de ned as the graph with vertex set V (Si ) = f1; : : :; ng and edge set E (Si ) = ffi; j g : j 6= ig.
Lemma 1. Let Tg be a star-formula computing the graph G = ([n]; E ). Let Tb be the boolean formula obtained from Tg by replacing ['s by _'s and \'s by ^'s and the star Si by the variable xi . Let f : f0; 1gn ?! f0; 1g be the function computed by Tb . Then,8x with jxj = 2; f (x) = 1 i X 2 E:
Proof : By induction on the number of operators in Tg that computes G.
In the base case we have zero operators in Tg and this computes a single star Si for some i; 1 i n. The corresponding Tb is xi . Clearly the inputs X of size 2 such that f (x) = 1 are exactly the edges of Si . For the inductive step, rst consider the case when the top gate of Tg is [. Let Tg = Tg [ Tg , and let Tgi compute Gi for i = 1; 2, so that G = G1 [ G2 . Correspondingly, we get Tb = Tb _ Tb and f = f1 _ f2, where Tbi computes fi . If X 2 G, then X 2 Gi for i = 1 or i = 2. By induction hypothesis fi (x) = 1 and therefore f (x) = 1. Conversely, suppose f (x) = 1 and jxj = 2. Then for at least one i 2 f1; 2g, we have fi (x) = 1. By induction hypothesis fi (x) = 1 i X 2 Gi when jxj = 2. Thus X 2 Gi and therefore X 2 G. The proof when the top gate of Tg is \ is similar. 1
2
1
2
Theorem 1 Let G be an n-vertex graph and fG be the associated 2-slice function on n variables. Let Lmon(fG ) be the f g (monotone) formula complexity and, or
of fG and let LS (G) be the star-formula complexity of the graph G. Then,
Lmon(fG ) LS (G) + O(n log n):
Proof : Let f be the function from Lemma 1 obtained from an optimal starformula for G. Note that fG f ^ Thn _ Thn, where Thnk denotes the k-th 2
3
threshold function on n variabes. Now we use the fact that for constant k there are monotone formulas of size O(n log n) to compute Thnk [Fr].
Theorem 2 Let f : f0; 1gn ?! f0; 1g be a 2-slice function where n = 2l. Suppose that Lmon(f ) n log n (log n), for some (log n) = !(1). Then there exists a function g : f0; 1gm ?! f0; 1g such that L(g) = ( (m)). Proof : Let f be a 2-slice function on n variables where n = 2l, and let G be the associated graph. From Theorem 1, LS (G) Lmon (f ) ? O(n log n): Using Proposition 2, there exists an n=2 n=2 bipartite subgraph H of G such that LS (H ) LS (G)=O(log n): From Proposition 1, LB (H ) LS (H )=n ? 2: Combining the three inequalities, we have LB (H ) (Lmon(f )=n log n).
Since H is a bipartite graph with n=2 = 2l?1 vertices on each side, we can identify its color classes with f0; 1gl?1 and de ne a function g on m = 2l ? 2 variables such that the Gg = H (as de ned in Section 2) and from Inequality 1 in Section 2, we get L(g) LB (H ) (Lmon(f )=n log n) : It follows that a lower bound of n log n (log n) on Lmon(f ) would give a lower bound of ( (m)) on L(g), since m = O(log n).
Corollary 1 An explicit n-variable 2-slice function f with Lmon(f ) n , 1+
for a constnst > 0, would give an explicit m-variable function g such that L(g) = 2 (m) .
4 Depth-3 Lower Bounds In this section, we consider lower bounds on depth-3 bipartite formulas computing bipartite graphs G U V , jU j = jV j = n. Recall that the leaves of the formula are graphs from B = fA V : A U g [ fU B : B V g: Let us rst observe that the bottom gates of a bipartite formula need not have fan-in more than 2: if the bottom gate is an \, then it actually computes a complete bipartite graph A B , where A U and B V , and this can be written as the intersection of at most two graphs from B; if the bottom gate is a [, then it is easy to see that it computes the complement of a complete bipartite graph A B , and again the complement of a complete bipartite graph can be written as a union of at most two graphs from B. Without loss of generality, we consider [\[ formulas. By the remark above, we can write such a formula as G = [i \j Gij , where Gij is the complement of a complete bipartite graph, i.e., Aij Bij for some Aij U and Bij V . Our lower bound proof is based on approximating polynomials for the or function [NS] and variation ranks under sign-preserving changes [KW]. Nisan and Szegedy [NS] give the following construction of -approximating polynomials for the or function. They assume a constant . The re ned analysis to bring out the dependence on is due to Hayes and Kutin [HK]. Proof is omitted from this extended abstract. Lemma 2 (NS,HK). The or-function ofp n boolean variables can be -approximated for every by a real polynomial of degree at most O( n log(2=)). More precisely, p 0 < < 1=2, there is a real polynomial p of degree at most O( n log(2=)) such that for every x 2 f0; 1gn, jor(x) ? p(x)j . In the following, we will use the notation exp(x) to denote cx for some constant c > 1 (c may depend on the context). For a biprtite graph G U V , we will let G(x; y) = 1 if (x; y) 2 G and G(x; y) = 0 if (x; y) 62 G. Lemma 3. Suppose the n n bipartite graph H U V is written as a union of d complete bipartite graphs:
H=
d [ i=1
(Ai Bi ); where Ai U; Bi V:
Then, for every , where 0 < < 1=2, there is a real matrix MH such that For all (x; y) 2 Up V , jMH (x; y) ? H (x; y)j , rank(MH ) exp( d log(2=) log d).
Proof : Let R be the incidence matrix of H , and similarly let Ri be the incidence matrices of the complete bipartite graphs Ai Bi , 1 i d, covering H . Note
that R is simply the entry-wise or of the Ri . Furthermore, each Ri is of rank one as a real matrix. We obtain MH from R using the approximating polynomials for the or-function given by Lemma 2.
Suppose p p(z1 ; : : : ; zd) is an -approximating polynomial of degree k := c d log(2=) for the or-function of d boolean variables. Syntactically substitute the matrix Ri for zi in this polynomial, but interpret the product as entry-wise product of matrices, i.e., a monomial zi zj is replaced by Ri Rj , where for matrices A and B , (A B )(x; y) := A(x; y)B (x; y). Note that if A and B are rank-1 matrices, then A B is also a rank-1 matrix. Thus, a monomial zi zit is replaced by the rank-1 0-1 matrix Ri Rit . The matrix obtained by computing the polynomial p(R1 ; : : : ; Rd ) in this way gives us the desired matrix MH . It is clear that MH (x; y) = p(R1 (x; y); : : : ; Rd (x; y)). From the properties of p, it is easy to see that for all x; y, jMH (x; y) ? H (x; y)j . Since MH is a 1
1
linear combination of rank-1 matrices, one for each monomial, it follows that rank of M is at most the number of monomials in p which is bounded by Pk ?d H j =0 j exp(k log d). Lemma 4. Let G be an n n bipartite graph G U V . If G is realized by a depth-3 bipartite formula L i.e.,
G=
di t \ [ i=1 j =1
(Aij Bij ); where Aij U; Bij V;
then there exists a matrix M such that i) If G(x; y) = 0, then jM (x; y)j 1=3, ii) If G(x; y) = 1, then =3, p 2=3 p M (x; y) t + 1P iii) rank(M ) texp( L log L), where L = ti=1 di denotes the length of the formula L.
Proof : Let G ; : : : ; Gt be the input graphs to the top gate, i.e., G = [ti Gi . 1
=1
Since each Gi , i = 1; : : : ; t, is an intersection of complements of complete bipartite graphs, its complement, Gi is computed by a union of complete bipartite graphs. Thus we can apply Lemma 3 to these complements Gi . Let Mi0 be the real matrix given by Lemma 3 that i -approximates Gi , where i := di =3L. Let Mi := J ? Mi0 , where J is the n n all-ones matrix. It is obvious that Mi i -approximates Gi . Furthermore, p p p rank(Mi ) 1 + rank(Mi0 ) exp( di log(L=di ) log di ) exp( L log L). Let M := M1 + + Mt . We want to see the relation between M and G: If G(x; y) = 0,Pthen 8i; Gi (x; y) = 0, and hence 8i; jMi (x; y)j i . It follows that jM (x; y)j ti=1 i = 1=3. If G(x; y) = 1, then 9i;PGi (x; y) = 1 and for this P i, 1 ? i Mi (x; y ) 1+ i . Hence, we have 1 ? i ? j6=i j M (x; y) tj=1 (1 + j ). So, in this case, 2=3 M (x; y) t + 1=3. P p p Moreover, rank(M ) ti=1 rank(Mi ) texp( L log L). We now show that for some "interesting" graphs G any matrix satisfying (i) and (ii) of Lemma 4 must have a large rank and hence conclude a lower bound on the depth-3 complexity of G using (iii).
Lemma 5 (KW). Let A be an n n 1 matrix and let B be a real matrix such that 1 jbij j and sign(aij ) = sign(bij ) for all i; j . Then rank(B ) n =(kAk ), where kAk is the largest eigenvalue of the matrix AA and A is 2
2
2
the conjugate transpose of A.
Theorem 3 Let G be an n n bipartite graph and let AG be its 1 incidence matrix, i.e., AG (x; y) = ?1 if (x; y) is an edge of G and AG (x; y) = +1 if (x; y) is not an edge of G. Then any depth-3 bipartite formula for G must have size at least a constant times log2 (n=kAGk) : log log2 (n=kAGk) Proof : Given a depth-3 formula for G of size L, let M be the matrix given by Lemma 4. De ne B := 3J ? 6M , where J is the n n all-ones matrix. Note that if G(x; y) = 0, then B (x; y) 1, and if G(x; y) = 1, then B (x; y) ?1 and that jB (x; y)j is always at most 6t. Hence B is a sign-preserving variation of AG and we can apply Lemma 5: rank(B ) = (n =(tkAGk )). On the other hand, rank(Bp) rank p (M ) + 1. So, from Lemma 4, part iii), we get that rank(B ) t exp( L log L). Combining the two estimates on rank(B ) and observing that t L, we get p p n exp( L log L) = ( kAG k ): 2
2
2
2
Solving for L proves the theorem. An n n Hadamard matrix is a 1 matrix such that HH > = nI . It is obvious p that kH k = n. Corollary 2 For any graph G such that AG is an Hadamard matrix, the depth3 bipartite formula complexity of G is at least ((log n= log log n)2 ). An example of such a graph is the Paley-type bipartite graph.
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