## Superpolynomial lower bounds for monotone span programs (1996)

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@TECHREPORT{Babai96superpolynomiallower,

author = {László Babai and Anna Gál and Avi Wigderson},

title = {Superpolynomial lower bounds for monotone span programs},

institution = {},

year = {1996}

}

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### Abstract

In this paper we obtain the first superpolynomial lower bounds for monotone span programs computing explicit functions. The best previous lower bound was Ω(n 5/2) by Beimel, Gál, Paterson [BGP]; 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 n Ω(log n / log log n) 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.

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Citation Context ... adjacent if (x + y) (q−1)/k = 1 (in GF (q)). These bipartite graphs are regular of degree (q − 1)/k. (For this and other elementary facts about finite fields we refer the reader to Lidl–Niederreiter =-=[LN]-=-.) In order to construct the functions for which we are able to prove superpolynomial lower bounds, we need a set system for which one can control the sizes of t-wise intersections for t = t(n) → ∞. (... |

270 | Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms
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Citation Context ...Weil’s character sum estimates in the spirit of the classical paper by Graham and Spencer [GS]. Constructions of k-wise nearly independent random variables have been analysed in a similar spirit (cf. =-=[AGHP]-=-, [AMN]). In this paper we only describe the second approach. For the details of the first solution we refer to [BGKRSW]. 2 The power of span programs 2.1 Monotone span programs vs. monotone circuits ... |

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Citation Context ...and Wigderson [KW1]. Define the disjointness function on a pair x, y of u-bit vectors by DISJ (x, y) = 1 if and only if the sets represented by these vectors are disjoint. Theorem 2.4 ([KS, Ra4], cf. =-=[BFS]-=-) Any 1/3 error probabilistic communication protocol for DISJ requires Ω(u) communication bits. We prove the following. Theorem 2.5 Any monotone formula computing ODDFACTORn has size exp(Ω( √ n)). Pro... |

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Citation Context ... for monotone formula size. Lower bound techniques for monotone circuits and formulae are well known 2(e.g. Razborov [Ra1, Ra2, Ra3], Haken [Ha] for circuits, Karchmer-Wigderson [KW1], Raz-Wigderson =-=[RW]-=- for formulae). These techniques, however, do not appear to be adaptable to the study of monotone span programs. Beimel, Gál and Paterson [BGP] showed that monotone span programs can be strictly stron... |

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Citation Context ...re the entries of AG are aij ·xij, and the Boolean variable xij takes the values 1 or 0 depending on whether or not the edge (i, j) is present in G. ✷ It follows from the arguments of Buntrock et al. =-=[BDHM]-=- that testing singularity of a variable matrix can be performed by polynomial size span programs. [BDHM] have not considered the model of span programs, but their model is equivalent to span programs ... |

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Citation Context ...grams. [BDHM] have not considered the model of span programs, but their model is equivalent to span programs up to polynomial increase in the size of the computation [KW]. Allender, Beals and Ogihara =-=[ABO]-=- showed explicitly how to construct polynomial size span programs for testing feasibility of systems of linear equations (Theorem 2.12 [ABO]). This gives the following. Theorem 2.7 (implicit in [BDHM,... |

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Citation Context ... general class of Paley-type graphs in the same way as we used the “norm graphs” but the analysis is done via Weil’s character sum estimates in the spirit of the classical paper by Graham and Spencer =-=[GS]-=-. Constructions of k-wise nearly independent random variables have been analysed in a similar spirit (cf. [AGHP], [AMN]). In this paper we only describe the second approach. For the details of the fir... |

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Citation Context ...haracter sum estimates in the spirit of the classical paper by Graham and Spencer [GS]. Constructions of k-wise nearly independent random variables have been analysed in a similar spirit (cf. [AGHP], =-=[AMN]-=-). In this paper we only describe the second approach. For the details of the first solution we refer to [BGKRSW]. 2 The power of span programs 2.1 Monotone span programs vs. monotone circuits and for... |

13 |
Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs
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Citation Context ...e nearly independent random variables have been analysed in a similar spirit (cf. [AGHP], [AMN]). In this paper we only describe the second approach. For the details of the first solution we refer to =-=[BGKRSW]-=-. 2 The power of span programs 2.1 Monotone span programs vs. monotone circuits and formulae - Proof of Theorem 1.1 Here we give the proof of Theorem 1.1, a result that may be interpreted as an indica... |

13 | On proving lower bounds for circuit size - Karchmer - 1993 |

9 | The dealer’s random bits in perfect secret sharing schemes
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Citation Context ...g schemes. For details we refer to the survey by Stinson [St] and to the extensive literature listed in [BGP]. The best known lower bound for general secret sharing schemes is Ω(n 2 / log n) (Csirmaz =-=[Cs]-=-). This immediately implies the same lower bound for monotone span programs for explicit functions. This by-product of [Cs] was improved by Beimel, Gál, and Paterson [BGP] to an Ω(n 5/2 ) lower bound ... |

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Citation Context ...-00106, and a grant from the Israel Research Foundation, founded by the Israel Academy of Sciences and Humanities. Email: avi@cs.huji.ac.il . 11 Introduction 1.1 Span programs Karchmer and Wigderson =-=[KW]-=- introduced span programs as a linear algebraic model for computing Boolean functions. Let us consider a linear space W over some field K; let w = 0 be a specified vector called the root. A span prog... |

2 |
Counting bottlenecks to show monotone P=NP
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Citation Context ... lower bounds for monotone symmetric branching programs and for monotone formula size. Lower bound techniques for monotone circuits and formulae are well known 2(e.g. Razborov [Ra1, Ra2, Ra3], Haken =-=[Ha]-=- for circuits, Karchmer-Wigderson [KW1], Raz-Wigderson [RW] for formulae). These techniques, however, do not appear to be adaptable to the study of monotone span programs. Beimel, Gál and Paterson [BG... |

2 |
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Citation Context ... will go in the exponent of the lower bound.) The first idea we used for constructing such set systems was based on the recent progress made by Kollár, Rónyai, and Szabó 4on the Zarankiewicz problem =-=[KRS]-=-. The set system was obtained from the Paley-type graphs with special parameters called “norm graphs” by [KRS] and analysed by them using the elements of commutative algebra. This approach resulted in... |

2 | Oleĭnik: Estimates of the Betti numbers of real algebraic hypersurfaces, Mat - O - 1951 |

2 |
Razborov: A lower bound on the monotone network complexity of the logical permanent
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Citation Context ...quently monotone circuits of depth Ω( √ n)). For Boolean circuits we know that non-monotone circuits are more powerful than monotone circuits. Razborov’s lower bound for the perfect matching function =-=[Ra2]-=- gives a superpolynomial separation between monotone and nonmonotone circuits, and a result by É. Tardos [T] shows an exponential gap. No separation is known between monotone and non-monotone span pro... |

1 | Razborov: On the method of approximation - A - 1989 |