Original Paper

Combinatorica

, Volume 19, Issue 3, pp 301-319

First online:

Superpolynomial Lower Bounds for Monotone Span Programs

  • László BabaiAffiliated withDepartment of Computer Science, University of Chicago; Chicago, IL 60637-1504, USA; E-mail: laci@cs.uchicago.edu
  • , Anna GálAffiliated withDept. of Computer Sciences, The University of Texas at Austin; Austin, TX 78712, USA; E-mail: panni@cs.utexas.edu
  • , Avi WigdersonAffiliated withInstitute of Computer Science, Hebrew University; Jerusalem, Israel; E-mail: avi@cs.huji.ac.il

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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