Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs

This paper contains two main results. The first is an explicit construction of bipartite graphs which do not contain certain complete bipartite subgraphs and have maximal density, up to a constant factor, under this constraint. This construction represents the first significant progress in three decades on this old problem in extremal graph theory. The construction beats the previously known probabilistic lower bound on density. The proof uses the elements of commutative algebra and algebraic geometry (theory of ideals, integral extensions, valuation rings). The second result concerns monotone span programs. We obtain the first superpolynomial lower bounds for explicit functions in this model. The best previous lower bound was $\Omega(n^{5/2})$ by Beimel, Gal, Paterson (FOCS’95); our analysis exploits a general combinatorial lower bound criterion from that paper. We give two proofs of superpolynomial lower bounds; one based on an analysis of Paley-type bipartitie graphs via Weil’s character sum estimates. A third result demonstrates the 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.

[1]  László Csirmaz,et al.  The Size of a Share Must Be Large , 1994, Journal of Cryptology.

[2]  Zoltán Füredi,et al.  New Asymptotics for Bipartite Turán Numbers , 1996, J. Comb. Theory, Ser. A.

[3]  László Csirmaz The Size of a Share Must Be Large , 1994, EUROCRYPT.

[4]  Armin Haken,et al.  Counting bottlenecks to show monotone P/spl ne/NP , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[5]  Bala Kalyanasundaram,et al.  The Probabilistic Communication Complexity of Set Intersection , 1992, SIAM J. Discret. Math..

[6]  Lajos Rónyai,et al.  Norm-graphs and bipartite turán numbers , 1996, Comb..

[7]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[8]  Mauricio Karchmer,et al.  On proving lower bounds for circuit size , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[9]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[10]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[11]  A. Suslin Projective modules over a polynomial ring , 1974 .

[12]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[13]  I. Shafarevich,et al.  Basic algebraic geometry 1 (2nd, revised and expanded ed.) , 1994 .

[14]  Avi Wigderson,et al.  Monotone circuits for connectivity require super-logarithmic depth , 1990, STOC '88.

[15]  Douglas R. Stinson,et al.  An explication of secret sharing schemes , 1992, Des. Codes Cryptogr..

[16]  Alexander A. Razborov,et al.  On the method of approximations , 1989, STOC '89.

[17]  Nicholas Pippenger,et al.  On Another Boolean Matrix , 1980, Theor. Comput. Sci..

[18]  Stuart J. Berkowitz,et al.  On Computing the Determinant in Small Parallel Time Using a Small Number of Processors , 1984, Inf. Process. Lett..

[19]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[20]  Ran Raz,et al.  Monotone circuits for matching require linear depth , 1990, STOC '90.

[21]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[22]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[23]  W. Schmidt Equations over Finite Fields: An Elementary Approach , 1976 .

[24]  Noam Nisan,et al.  Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs , 1992, J. Comput. Syst. Sci..

[25]  R. Graham,et al.  A Constructive Solution to a Tournament Problem , 1971, Canadian Mathematical Bulletin.

[26]  E. Szemerédi,et al.  On a problem of graph theory , 1967 .

[27]  P. Erdös Some remarks on the theory of graphs , 1947 .

[28]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[29]  Yossi Azar,et al.  Approximating Probability Distributions Using Small Sample Spaces , 1998, Comb..

[30]  Miles Reid,et al.  Commutative Ring Theory , 1989 .

[31]  Alexander A. Razborov,et al.  On the Distributional Complexity of Disjointness , 1992, Theor. Comput. Sci..

[32]  Anna Gál,et al.  Lower bounds for monotone span programs , 1994, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[33]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[34]  Avi Wigderson,et al.  On span programs , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[35]  Peter Frankl,et al.  Complexity classes in communication complexity theory , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[36]  J. Spencer Probabilistic Methods in Combinatorics , 1974 .

[37]  K. Mulmuley A fast parallel algorithm to compute the rank of a matrix over an arbitrary field , 1987, Comb..

[38]  W. G. Brown On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.