On the Constant-Depth Complexity of k-Clique ¤ Benjamin Rossman Massachusetts Institute of Technology 32 Vassar St., Cambridge, MA 02139 [email protected] ABSTRACT General Terms We prove a lower bound of !(nk=4) on the size of constant- Theory depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound 2 Keywords of !(nk=89d ) due to Beame where d is the circuit depth. k-clique, constant-depth circuits, circuit complexity, AC0, Our lower bound has the advantage that it does not depend ¯rst-order logic, bounded variable hierarchy on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeo®. Our k-clique lower bound derives from a stronger result of 1. INTRODUCTION (n) Constant-depth circuits have been one of the most fruit- independent interest. Suppose fn : f0; 1g 2 ¡! f0; 1g is a sequence of functions computed by constant-depth circuits ful settings for complexity lower bounds. (Here we refer of size O(nt). Let G be an Erd}os-R¶enyi random graph with to logical circuits comprised of :, ^ and _ gates with un- vertex set f1; : : : ; ng and independent edge probabilities n¡® bounded fan-in.) Early results of Ajtai [1] and Furst, Saxe 1 and Sipser [12] proved that the parity problem (given an n- where ® · 2t¡1 . Let A be a uniform random k-element subset of f1; : : : ; ng (where k is any constant independent of bit string, are there an even number of 1's?) does not have polynomial-size constant-depth circuits. This result, often n) and let KA denote the clique supported on A. We prove stated as PARITY 2= AC0, was subsequently sharpened by that fn(G) = fn(G [ KA) asymptotically almost surely. These results resolve a long-standing open question in Yao [28] and Hºastad [15] among others, who eventually es- ¯nite model theory (going back at least to Immerman in tablished that exponential-size constant-depth circuits are 1982). The m-variable fragment of ¯rst-order logic, denoted required. by FOm, consists of the ¯rst-order sentences which involve In this paper we consider the k-clique problem for con- stant values of k. The k-clique problem asks, given a graph at most m variables. Our results imply that the bounded n 1 2 m G on n vertices (speci¯ed by a bitstring of length ), does variable hierarchy FO ½ FO ½ ¢ ¢ ¢ ½ FO ½ ¢ ¢ ¢ is strict 2 in terms of expressive power on ¯nite ordered graphs. It G contain a k-clique? Unlike the parity problem, the k- was previously unknown that FO3 is less expressive than clique problem has polynomial-size constant-depth circuits (for every constant k). In particular, there is an obvious full ¯rst-order logic on ¯nite ordered graphs. n depth-2 circuit consisting of k ^-gates at the bottom level (deciding each potential k-clique) feeding into a single _-gate on top. A natural question is whether there exist signi¯- Categories and Subject Descriptors cantly smaller constant-depth circuits for the k-clique prob- o(k) F.2.2 [Theory of Computation]: Nonnumerical Algo- lem (say, of size n ). 0 rithms and Problems|Computations on discrete structures; Lower bounds on the complexity of problems within AC F.4.1 [Mathematical Logic and Formal Languages]: traditionally involve size-depth tradeo®s. Lynch [23] in 1986 Mathematical Logic|Model theory established the ¯rst size-depth tradeo® for the k-clique prob- p 1:5 lem by proving a lower bound of n­( k=d ) on the size of depth-d circuits solving the k-clique problem on n-vertex ¤ 2 Supported by a National Defense Science and Engineering graphs. This was improved to !(nk=89d ) by Beame [7] in Graduate Fellowship. 1990 (in the context of CRCW PRAM's). These results were based on developments in the switching lemma technology originally introduced by Hºastad [15] for studying the par- ity problem. For his lower bound, Beame proved powerful switching lemmas in the setting of random graphs (see [8]). Permission to make digital or hard copies of all or part of this work for Size-depth tradeo®s of this sort, however, are unsatisfac- personal or classroom use is granted without fee provided that copies are tory inasmuch as they degrade in the exponent of n as d is not made or distributed for profit or commercial advantage and that copies taken to be a larger constant. (This seems to be an unavoid- bear this notice and the full citation on the first page. To copy otherwise, to able consequence of using switching lemmas in the usual republish, to post on servers or to redistribute to lists, requires prior specific p way.) For depth d = k, the lower bounds of Lynch and permission and/or a fee. n Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$5.00. Beame no longer beat the trivial lower bound of 2 (the 1 ¡® number of input bits). These lower bounds thus do not rule 0 < ® · 2t¡1 , let G = ER(n; n ) be an Erd}os-R¶enyi ran- out the possibility that the 100-clique problem has depth-10 dom graph and let A be a uniform random set of k vertices of 2 circuits of size O(n ). G. Then fn(G) = fn(G [ KA) asymptotically almost surely. In this paper we create a more satisfactory state of a®airs by proving a lower bound of !(nk=4) on the size of depth-d This result directly implies our k-clique lower bound. circuits for the k-clique problem for all constants k and d. (In fact, our lowerp bound holds even for slightly increasing Theorem 1.2. For every constant k, the k-clique problem d = d(n) = o( log n) and perhaps moderately increasing on n-vertex graphs requires constant-depth circuits of size k = k(n) as well.) By contrast, our lower bound implies !(nk=4). that 100-clique does not have size O(n25) circuits of any constant depth. Thus it may now be said that, in some In fact, Theorem 1.1 implies the even stronger assertion reasonable sense, no constant-depth circuits for the k-clique that for all ` > k, no constant-depth circuits of size O(nk=4) problem signi¯cantly beat the naive circuits of depth 2 and distinguish between the class of graphs which contain an size O(nk). `-clique and the class of graphs which contain no k-clique. Unlike traditional size-depth tradeo®s, our approach does not involve developing sharper switching lemmas for the 1.2 Corollaries in Logic problem at hand. In fact, we require nothing stronger than Our k-clique lower bound has some nice corollaries in logic Hºastad's original switching lemma. Rather, our technique (¯nite model theory [18, 22]), which answer long-standing breaks from past approaches in the innovative way that the open questions, via the well-known descriptive complexity switching lemma is used. A key technical notion we intro- characterization of ¯rst-order logic in terms of the circuit f;G 0 duce is the s-bounded clique-sensitive core Thsi (A) of a set class AC [6, 11, 21]. A of vertices in a graph G with respect to a graph-function f The m-variable fragment of ¯rst-order logic, denoted by f;G m (see x3.1). Thsi (A) is a subset of A with certain nice prop- FO , consists of the ¯rst-order sentences which involve at erties. In particular, if we add to G the clique supported most m variables. (Sentences of FOm may contain more f;G than m quanti¯ers, as variables may be reused.) For exam- on Thsi (A), then the value of f on the resulting graph is no longer sensitive to the further addition of any subclique ple, the following sentence (in the language of simple graphs of A up to size s. This new technical notion (which puts a with a symmetric binary relation » denoting adjacency) ex- twist on the familiar concept of sensitive inputs) allows for presses \diameter is at most 4" using only 3 variables: a novel inductive argument on circuits (Lemma 3.6). Ulti- 8x8y9z 9y (x » y) ^ (y » z) ^ 9x (z » x) ^ (x » y) : mately, our approach leads to an e®ective new way of using the switching lemma in conjunction with a union bound over More generally, 3 variables su±ce to express \diameter is at the nodes in a circuit. most d" for every constant d. Bounded variable fragments m m We remark that the complexity of the k-clique problem FO (and their in¯nitary counterparts L1;!) are important has been studied in various models of computation besides objects of study in both ¯nite and classical model theory constant-depth circuits. In the setting of monotone circuits, (see [14, 18]). The chain of m-variable fragments FO1 ½ it is known that the k-clique problem (for speci¯c increasing FO2 ½ ¢ ¢ ¢ ½ FOm ½ ¢ ¢ ¢ is known as the bounded (or ¯nite) k = k(n)) requires exponential size [3, 13, 26]. A super- variable hierarchy. On ¯nite graphs without a linear order, polynomial lower bound was even proved for circuits with a this hierarchy is strict in terms of expressive power for the bounded number of negation gates [5]. Lower bounds have simplest of reasons: the sentence \there exist at least m also been investigated in the context of branching programs vertices" is expressible in FOm but not in FOm¡1. and decision trees [27].