Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs

Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn Part F129452, Issue , 1996, Pages 603-611

  Babai, László ^a   Gál, Anna ^b   Kollár, János ^c   Rónyai, Lajos ^d   Szabó, Tibor ^e   Wigderson, Avi ^b,f  

^a THE UNIVERSITY OF CHICAGO   (United States)

^b INSTITUTE FOR ADVANCED STUDY   (United States)

^c UNIVERSITY OF UTAH   (United States)

^d INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

^e OHIO STATE UNIVERSITY   (United States)

^f HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; COMPUTATION THEORY; BOOLEAN FUNCTIONS; COMBINATORIAL MATHEMATICS; COMPUTATIONAL GEOMETRY; LOGIC CIRCUITS; MATHEMATICAL MODELS; POLYNOMIALS; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

ALGEBRAIC GEOMETRY; BIPARTITE GRAPHS; BIPARTITE SUBGRAPHS; COMMUTATIVE ALGEBRA; EXPLICIT CONSTRUCTIONS; EXTREMAL GRAPH THEORY; MONOTONE CIRCUIT; MONOTONE SPAN PROGRAMS;

GRAPH THEORY;

EXTREMAL BIPARTITE GRAPHS; MONOTONE SPAN PROGRAMS; SUPERPOLYNOMIAL LOWER BOUND;

EID: 0029711425     PISSN: 07378017     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/237814.238010     Document Type: Conference Paper

References (48)

1. N. Alon and R. Boppana: The monotone circuit complexity of Boolean functions. Combinatorica 7(1987), 1-22.
2. N. Alon, O. Goldreich, J. Hastad, R. Peralta: Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3(1992), 289-304.
3. A. E. Andreev: On a family of Boolean matrices. Vestnik Mask. Univ. Ser. 1 (mat.-mech.) 41(1986), 97-100 (in Russian), English translation: Moscow Univ. Math. Bull. 41(1986), 79-82.
4. M. F. Atiyah, I. G. Macdonald: Introduction to Commutative Algebra. Addison-Wesley, 1969.
5. Y. Azar, R. Motwani, J. Naor: Approximating probability distributions using small sample spaces. Combmatorica, to appear.
6. L. Babai, P. Frankl, J. Simon: Complexity classes in communication complexity theory. In: Proc. 27th IEEE FOCS, 1986, pp. 337-347.
7. L. Babai, N. Nisan, M. Szegedy: Multiparty protocols, pseudorandom generators for Logspace, and time-space tradeoffs. J. Computer and Sys. Sci. 45(1992), 204-232.
8. L. Babai, A. Gal, A. Wigderson: Superpolynomial lower bounds for monotone span programs, in preparation.
9. A. Beimel, A. Gal and M. Paterson: Lower bounds for monotone span programs. In Proc. 36th IEEE FOCS, Milwaukee WI 1995, pp. 674-681.
10. S. J. Berkowitz: On computing the determinant in small parallel time using a small number of processors. Inform. Process. Lett. 18(1984), 147-150.
11. B. Bollobas: Extremal Graph Theory. Academic Press, 1978.
12. N. Bourbaki: Algibre Commutative. Hermann, Paris, 1961-1965.
13. W. G. Brown: On graphs that do not contain a Thomsen graph. Canad. Math. Bull. 9(1966), 281-289.
14. G. Buntrock, C. Damm, H. Hertrampf, and C. Meinel: Structure and importance of the logspace-mod class. Math. Systems Theory 25(1992), 223-237.
15. L. Csirmaz: The size of a share must be large. In A. De Santis, ed., Advances in Cryptology - Eurocrypt'94, preproceedings, 1994.
16. L. Csirmaz: The dealer's random bits in perfect secret sharing schemes. Preprint, Mathematical Inst. Hungarian Acad. Sci., 1994.
17. P. Erdõs: Some remarks on the theory of graphs. Bull. A. M. S. bf 53(1947), 292-294.
18. P. Erdõs: On sequences of integers no one of which divides the product of two others and on some related problems. Isvestija Nautshno-Issl. Inst. Mat. i Meh. Tomsk 2 (1938) 74-82, (Mitteilungen des Forschungsinstitutes fur Math. und Mechanik Univ. Tomsk).
19. P. Erdõs, A. Rényi, V. T. Sós: On a problem of graph theory. Studia Sci. Math. Hungar. 1(1966), 215-235.
20. P. Erdos, J. Spencer: Probabilistic Methods in Combinatorics. Academic Press, London - New York, Akademiai Kiadó, Budapest, 1974.
21. Z. Füredi: New asymptotics for bipartite Turán numbers. J. Comb. Theory-A, to appear.
22. R. L. Graham, J. H. Spencer: A constructive solution to a tournament problem. Canad. Math. Bull. 14(1971), 45-48.
23. Armin Haken: Counting bottlenecks to show monotone P≠NP. In Proc. 36th IEEE FOCS, Milwaukee WI 1995, pp. 36-40.
24. B. Kalyanasundaram and G. Schnitger: The probabilistic communication complexity of set intersection. In Proc. of Structure in Complexity Theory, 1987, pp. 41-49.
25. M. Karchmer: On proving lower bounds for circuit size. In Proc. 8th Ann. Symp. Structure in Complexity Theory, IEEE 1993, pp. 112-118.
26. M. Karchmer and A. Wigderson: On span programs. In Proc. 8th Ann. Symp. Structure in Complexity Theory, IEEE 1993, pp. 102-111.
27. M. Karchmer and A. Wigderson: Monotone Circuits for Connectivity require Super-Logarithmic Depth. In SIAM Journal on Discrete Mathematics, Vol 3, No. 2, 1990, pp. 255-265.
28. J. Kollár, L. Ronyai, T. Szabó: Norm-graphs and bipartite Turán numbers. Combinatorica, to appear.
29. T. Kovári, V. T. Sós, P. Turán: On a problem of K. Zarankiewicz. Colloquium Math. 3(1954), 50-57.
30. R. Lidl, H. Niederreiter: Introduction to Finite Fields and their Applications, Cambridge University Press, 1986.
31. H. Matsumura: Commutative Ring Theory, Cambridge University Press, 1989.
32. K. Mehlhorn: Some remarks on Boolean sums. Acta Inform. 12(1979), 371-375.
33. K. Mulmuley: A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7(1987), 101-104.
34. E. I. Nečiporuk: On a Boolean matrix. Problemy Kibernet. 21(1969), 237-240.
35. English translation in Systems Theory Res., 21 (1971) 236-239.
36. N. Pippenger: On another Boolean matrix. Theoretical Computer Science 11(1980), 49-56.
37. D. Quillen: Projective modules over a polynomial ring, Inventiones Mathematicae 36(1976), 167-171.
38. R. Raz and A. Wigderson: Monotone Circuits for Matching require Linear Depth. Journal of the ACM, Vol 39, 1992, pp. 736-744.
39. A. A. Razborov: Lower bounds for the monotone complexity of some Boolean functions. Soviet Math. Doklady, 31 (1985) 354-357.
40. A. A. Razborov: A lower bound on the monotone network complexity of the logical permanent. Mat. Zametki 37(1985), 887-900.
41. A. A. Razborov: On the method of approximation. In Proc. 21st ACM STOC, 1989, pp. 167-176.
42. A. A. Razborov: On the distributional complexity of disjointness. Theoretical Computer Science, 106, 1992, No. 2, pp. 385-390.
43. W. M. Schmidt: Equations over Finite Fields: An Elementary Approach. Lect. Notes in Math. vol. 536, Springer Verlag, 1976.
44. I. R. Shafarevich: Basic Algebraic Geometry, 2nd revised and expanded ed., Springer Verlag, Berlin, 1994.
45. D. R. Stinson: An explicitation of secret sharing schemes. Design, Codes and Cryptography 2(1992), 357-390.
46. A. A. Suslin, Projective modules over a polynomial ring are free, Soviet Math. Dokl. 17(1976), 1160-1164.
47. I. Wegener: The Complexity of Boolean Functions. Wiley-Teubner 1987.
48. A. Wigderson: The fusion method for lower bounds in circuit complexity. In: Combinatorics, Paul Erdos is Eighty, (Volume 1), D. Miklos, V. T. S6s, T. Szonyi, eds., Bolyai Society Mathematical Studies 1, Budapest 1993, pp. 453-467.