Combinatorica

, Volume 19, Issue 3, pp 301–319

Superpolynomial Lower Bounds for Monotone Span Programs

  • László Babai
  • Anna Gál
  • Avi Wigderson
Original Paper

DOI: 10.1007/s004930050058

Cite this article as:
Babai, L., Gál, A. & Wigderson, A. Combinatorica (1999) 19: 301. doi:10.1007/s004930050058

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.

AMS Subject Classification (1991) Classes:  68Q05, 68R05, 05D05

Copyright information

© János Bolyai Mathematical Society, 1999

Authors and Affiliations

  • László Babai
    • 1
  • Anna Gál
    • 2
  • Avi Wigderson
    • 3
  1. 1.Department of Computer Science, University of Chicago; Chicago, IL 60637-1504, USA; E-mail: laci@cs.uchicago.eduUS
  2. 2.Dept. of Computer Sciences, The University of Texas at Austin; Austin, TX 78712, USA; E-mail: panni@cs.utexas.eduUS
  3. 3.Institute of Computer Science, Hebrew University; Jerusalem, Israel; E-mail: avi@cs.huji.ac.ilIL