On minimal π-circuits of closing contacts for symmetric functions with threshold 2

Lozhkin, S. A.

doi:10.1515/156939205776368887

Cited by 2 publications

(1 citation statement)

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“…1 Apart from purely combinatorial considerations, the interest in this problem is motivated by its applications in formula and switching-circuit complexity of the Boolean threshold-2 function (which takes on the value 1 if and only if at least two of its inputs are set to 1). For more context, see treatments by Radhakrishnan [33] and Lozhkin [26]. Our lower bound is obtained in a slightly more restrictive setting, because of explicit asymmetry: for OR 2 (T n ), one needs to cover entries (i, j) with i < j in the matrix; in biclique coverings of undirected graphs, it suffices to cover either of (i, j) and (j, i).…”

Section: Discussion and Related Workmentioning

confidence: 99%

Fractional coverings, greedy coverings, and rectifier networks

Chistikov,

Iván,

Lubiw

et al. 2015

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A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i, j) iff there is a path from the jth source to the ith sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century.We explore two well-known techniques that have hitherto found little to no applications in this theory. Both of them build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. We obtain new results by using fractional and greedy coverings (defined in the standard way).First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpiński (disjointness) matrix. The previous upper bound is O(n 1.28 ), and we improve it to O(n 1.17 ), while the best known lower bound is Ω(n 1.16 ). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)).

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Section: Discussion and Related Workmentioning

confidence: 99%