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Probabilistic Communication Complexity of Boolean Relations
 Proc. of the 30th FOCS
, 1989
"... In [KW] it was prooved that for every boolean function f there exist a communication complexity game R f such that the minimal circuitdepth of f exactly equals to the communication complexity of R f . If f is monotone then there also exists a game R m f with communication complexity exactly equal ..."
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Cited by 23 (7 self)
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In [KW] it was prooved that for every boolean function f there exist a communication complexity game R f such that the minimal circuitdepth of f exactly equals to the communication complexity of R f . If f is monotone then there also exists a game R m f with communication complexity exactly equals to the monotone depth of f . It was also prooved in [KW] that the communication complexity of R m st\Gammaconnectivity is\Omega\Gamma/34 2 n), or eqivalently that the monotone depth of the st connectivity function is\Omega\Gamma/38 2 n). In this paper we consider the games R f and R m f in a probabilistic model of communication complexity, and prove that the communication complexity of R m st\Gammaconnectivity is \Omega\Gamma/20 2 n) even in the probabilistic case. We also prove that in every NC1 circuit for st connectivity at least a constant fraction of all input variables must be negated. 1 Introduction In standard communication complexity, two players are trying to compute...
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
The Fusion Method for Lower Bounds in Circuit Complexity
 Keszthely (Hungary
, 1993
"... This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both t ..."
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Cited by 10 (0 self)
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This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both the "approximation method" of Razborov [13] and the "topological approach" of Sipser [15] for proving circuit lower bounds. In Karchmer's setting the lower bound prover shows that a given circuit C is too small for computing a given function f by contradiction, in the following way. She tries to combine (or 'fuse', as we propose calling it) correct accepting computations of inputs in f \Gamma1 (1) by C into an incorrect accepting computation of an input in f \Gamma1 (0). It turns out that this "Fusion Method" reduces the dynamic computation of f by C into a static combinatorial cover problem, which provides the lower bound. Moreover, different restrictions on how we can fuse computations ...
Lower Bounds for Perfect Matching in Restricted Boolean Circuits
, 1993
"... We consider three restrictions on Boolean circuits: bijectivity, consistency and multilinearity. Our main result is that Boolean circuits require exponential size to compute the bipartite perfect matching function when restricted to be (i) bijective or (ii) consistent and multilinear. As a consequen ..."
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We consider three restrictions on Boolean circuits: bijectivity, consistency and multilinearity. Our main result is that Boolean circuits require exponential size to compute the bipartite perfect matching function when restricted to be (i) bijective or (ii) consistent and multilinear. As a consequence of the lower bound on bijective circuits, we prove an exponential size lower bound for monotone arithmetic circuits that compute the 01 permanent function. We also define a notion of homogeneity for Boolean circuits and show that consistent homogeneous circuits require exponential size to compute the bipartite perfect matching function. Motivated by consistent multilinear circuits, we consider certain restricted (\Phi; ) circuits and obtain an exponential lower bound for computing bipartite perfect matching using such circuits. Finally, we show that the lower bound arguments for the bipartite perfect matching function on all these restricted models can be adapted to prove exponential low...