Conference item
Fractional coverings, greedy coverings, and rectifier networks
- Abstract:
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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 techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area.
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)).
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 625.5KB, Terms of use)
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- Publisher copy:
- 10.4230/LIPIcs.STACS.2017.23
Authors
- Publisher:
- Schloss Dagstuhl
- Host title:
- LIPICS
- Journal:
- STACS 2017 More from this journal
- Volume:
- 66
- Pages:
- 23:1-23:14
- Publication date:
- 2017-02-24
- Acceptance date:
- 2016-11-12
- DOI:
- ISSN:
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1868-8969
- Keywords:
- Pubs id:
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pubs:681962
- UUID:
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uuid:d051af55-422b-4d8e-b1bd-58ea53df395b
- Local pid:
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pubs:681962
- Source identifiers:
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681962
- Deposit date:
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2017-02-27
Terms of use
- Copyright holder:
- Chistikov et al
- Copyright date:
- 2017
- Notes:
- © Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, Jeffrey Shallit. This article was presented at STACS 2017: 34th International Symposium on Theoretical Aspects of Computer Science (March 8-11, 2017: Hannover, Germany). Licensed under Creative Commons License CC-BY.
- Licence:
- CC Attribution (CC BY)
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