The Linear-Array Conjecture in Communication Complexity Is False

linear array network consists of k+1 processors with links only between and (0≤i<k). It is required to compute some boolean function f(x,y) in this network, where initially x is stored at and y is stored at . Let be the (total) number of bits that must be exchanged to compute f in worst case. Clearly, , where D(f) is the standard two-party communication complexity of f. Tiwari proved that for almost all functions and conjectured that this is true for all functions. In this paper we disprove Tiwari's conjecture, by exhibiting an infinite family of functions for which is essentially at most . Our construction also leads to progress on another major problem in this area: It is easy to bound the two-party communication complexity of any function, given the least number of monochromatic rectangles in any partition of the input space. How tight are such bounds? We exhibit certain functions, for which the (two-party) communication complexity is twice as large as the best lower bound obtainable this way. Received: March 1, 1996     

On the Knowledge Complexity of

We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class . Thus, if the polynomial time hierarchy does not collapse, then -complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The first asserts that statistical zero knowledge is contained in [11, 2], while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in [19]. Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability is less than (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specifically, it is in ). In the standard setting of negligible error probability, there exist PSPACE-complete languages which have sub-linear knowledge complexity. However, if we insist, for example, that the error probability is less than , then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless . In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions , the verifier runs in time polynomial in and and fails with probability at most to detect an additive error in the entropy. We believe that this protocol is of independent interest. Subsequent to our work Goldreich and Vadhan [16] established that the problem of comparing the entropies of two samplable distributions if they are noticeably different is a natural complete promise problem for the class of statistical zero knowledge (). Received January 6, 2000 RID=" " ID=" " This research was performed while the authors were visiting the Computer Science Department at the University of Toronto, preliminary version of this paper appeared in [27] RID="*" ID="*" Partially supported by Technion V. P. R. Found––N. Haar and R. Zinn Research Fund. RID="†" ID="†" Partially supported by the grants OTKA T-029255, T-030059, FKFP 0607/1999, and AKP 2000-78 2.1.     

Separation of the Monotone NC Hierarchy

, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC ≠ monotone-P. 2. For every i≥1, monotone-≠ monotone-. 3. More generally: For any integer function D(n), up to (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const). Only a separation of monotone- from monotone- was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds: 1.  For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result. 2.  For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known. Received: December 19, 1997     

Superpolynomial Lower Bounds for Monotone Span Programs

monotone span programs computing explicit functions. The best previous lower bound was by Beimel, Gál, Paterson [7]; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weil's character sum estimates. We prove an lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields. Received: August 1, 1996     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Improved Pseudorandom Generators for Combinatorial Rectangles

We construct a pseudorandom generator which uses bits and approximates the volume of any combinatorial rectangle in to within error. This improves on the previous construction using bits by Armoni, Saks, Wigderson, and Zhou [4]. For a subclass of rectangles with at most nontrivial dimensions and each dimension being an interval, we also give a pseudorandom generator using bits. This again improves the previous upper bound by Chari, Rohatgi, and Srinivasan [5]. Received July 29, 1998     

Testing Monotonicity

at arguments of its choice, the test always accepts a monotone f, and rejects f with high probability if it is ε-far from being monotone (i.e., every monotone function differs from f on more than an ε fraction of the domain). The complexity of the test is O(n/ε). The analysis of our algorithm relates two natural combinatorial quantities that can be measured with respect to a Boolean function; one being global to the function and the other being local to it. A key ingredient is the use of a switching (or sorting) operator on functions. Received March 29, 1999     

Almost Optimal Dispersers

A disperser is a bipartite graph with the property that every subset A of of cardinality at least K, has at least fraction of the vertices of as neighbors. Such graphs have many applications in derandomization. Saks, Srinivasan and Zhou presented an explicit construction of a disperser with an almost optimal degree , for every . We extend their result for every parameter . Received November 12, 1998/Revised June 20, 2000 RID="*" ID="*" This work was partially done while the author was at the Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. This work was supported by a Phil Zacharia postdoctoral fellowship.     

Some Remarks on Obligatory Subsytems of Uncountably Chromatic Triple Systems

Dedicated to the memory of Paul Erdős An obligatory triple system is one that embeds into every triple system of uncountable chromatic number. It is proved that a triple system is obligatory iff every 2-connected component of it is. Obligatory triple systems are tripartite but not vice versa. Received January 13, 2000 RID="*" ID="*" Research partially supported by Hungarian National Research Grant T 19367.     

Quick Approximation to Matrices and Applications

m ×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, R?dl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. From our matrix approximation, the above graph algorithms and the Regularity Lemma and several other results follow in a simple way. We generalize our approximations to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems. Received: July 25, 1997     

On a Degree Property of Cross-Intersecting Families

Motivated by some applications in computational complexity, Razborov and Vereshchagin proved a degree bound for cross-intersecting families in [1]. We sharpen this result and show that our bound is best possible by constructing appropriate families. We also consider the case of cross-t-intersecting families. Received October 28, 1999     

Ryser's Conjecture for Tripartite 3-Graphs

We prove that in a tripartite 3-graph . Received March 17, 1999     

Bounds on Pairs of Families with Restricted Intersections

such that for any , . We are interested in the maximum product , given r and L. We give asymptotically optimal bounds for L containing only elements of s<q residue classes modulo q, where q is arbitrary (even non-prime) and s is a constant. As a consequence, we obtain a version of the Frankl–R?dl result about forbidden intersections for the case of two forbidden intersections. We also give tight bounds for L={0,...,k}. Received: August 5, 1998     

Existence of Polyhedral Embeddings of Graphs

It is proved that the decision problem about the existence of an embedding of face-width 3 of a given graph is NP-complete. A similar result is proved for some related decision problems. This solves a problem raised by Neil Robertson. Received July 6, 1998 RID="*" ID="*" Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1–0502–0101–98.     

The Size of 3-Cross-Free Families

We give a short and simple proof for the theorem that the size of a 3-cross-free family is linear in the size of the groundset. A family is 3-cross-free if it has no 3 pairwise crossing members. Received July 1, 1998 RID="*" ID="*" Research supported by the Netherlands Organization for Scientific Research (NWO) and the OTKA T 029772 project.     

Mixed and Isoperimetric Estimates on the Log-Sobolev Constants of Graphs and Markov Chains

Two types of lower bounds are obtained on the log-Sobolev constants of graphs and Markov chains. The first is a mixture of spectral gap and logarithmic isoperimetric constant, the second involves the Gaussian isoperimetric constant. The sharpness of both types of bounds is tested on some examples. Product generalizations of some of these results are also briefly given. Received March 1, 1999/Revised July 17, 2000 RID="*" ID="*" Research supported in part by the NSF Grant No. DMS–9803239. The author greatly enjoyed the hospitality of CIMAT, Gto, Mexico, where part of this work was done.     

Combinatorics of Monotone Computations

, and (ii) arbitrary real-valued non-decreasing functions on variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial lower bounds for circuits computing explicit functions even when . The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion also implies corresponding lower bounds for the length of cutting planes proof in the propositional calculus. Received: July 2, 1996/Revised: Revised June 7, 1998     

Consistently There Is No Non Trivial CCC Forcing Notion with the Sacks or Laver Property

Dedicated to the memory of Paul Erdős Received March 2, 2000 RID=" " ID=" " The research supported by the Israel Science Foundation (founded by the Israel Academy of Sciences). Publication number 723. We would like to thank Martin Goldstern for many improvements in this paper.     

Superpolynomial Size Set-systems with Restricted Intersections mod 6 and Explicit Ramsey Graphs

Dedicated to the memory of Paul Erdős We construct a system of subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all non-prime-power moduli m in place of m=6. This result is in sharp contrast with results of Frankl and Wilson (1981) for prime power moduli and gives strong negative answers to questions by Frankl and Wilson (1981) and Babai and Frankl (1992). We use our set-system to give an explicit Ramsey-graph construction, reproducing the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981). Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994). Received January 15, 1996/Revised August 2, 1999     

Equivalence in Finite-Variable Logics is Complete for Polynomial Time

of first-order logic whose formulas contain at most k variables (for some ). We show that for each , equivalence in the logic is complete for polynomial time. Moreover, we show that the same completeness result holds for the powerful extension of with counting quantifiers (for every ). The k-dimensional Weisfeiler–Lehman algorithm is a combinatorial approach to graph isomorphism that generalizes the naive color-refinement method (for ). Cai, Fürer and Immerman [6] proved that two finite graphs are equivalent in the logic if, and only if, they can be distinguished by the k-dimensional Weisfeiler-Lehman algorithm. Thus a corollary of our main result is that the question of whether two finite graphs can be distinguished by the k-dimensional Weisfeiler–Lehman algorithm is P-complete for each . Received: March 23, 1998