Document Open Access Logo

Fractional Coverings, Greedy Coverings, and Rectifier Networks

Authors Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, Jeffrey Shallit

Thumbnail PDF


  • Filesize: 0.61 MB
  • 14 pages

Document Identifiers

Author Details

Dmitry Chistikov
Szabolcs Iván
Anna Lubiw
Jeffrey Shallit

Cite AsGet BibTex

Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, and Jeffrey Shallit. Fractional Coverings, Greedy Coverings, and Rectifier Networks. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 23:1-23:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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 j-th source to the i-th 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-Sierpinski (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 Omega(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)).
  • rectifier network
  • OR-circuit
  • biclique covering
  • fractional covering
  • greedy covering


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Joan Boyar and Magnus Gausdal Find. Cancellation-free circuits in unbounded and bounded depth. Theor. Comput. Sci., 590:17-26, 2015. URL:
  2. Parinya Chalermsook, Sandy Heydrich, Eugenia Holm, and Andreas Karrenbauer. Nearly tight approximability results for minimum biclique cover and partition. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 235-246, 2014. URL:
  3. V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979. Google Scholar
  4. Dameng Deng, P. C. Li, G. H. J. van Rees, and Yuan Zhang. The Stein-Lovasz theorem and its applications to some combinatorial arrays. Journal of Combinatorial Mathematics and Combinatorial Computing, 77:17-31, 2011. Google Scholar
  5. Magnus Find, Mika Göös, Matti Järvisalo, Petteri Kaski, Mikko Koivisto, and Janne H. Korhonen. Separating OR, SUM, and XOR circuits. Journal of Computer and System Sciences, 82(5):793-801, 2016. Google Scholar
  6. Hermann Gruber and Markus Holzer. Finding lower bounds for nondeterministic state complexity is hard. Electronic Colloquium on Computational Complexity (ECCC), 13(027), 2006. URL:
  7. G. Hansel. Nombre minimal de contacts de fermeture nécessaires pour réaliser une fonction booléenne symétrique de n variables. C. R. Acad. Sc. Paris, 258(25):6037-6040, 1964. In French. Google Scholar
  8. Szabolcs Iván, Ádám Dániel Lelkes, Judit Nagy-György, Balázs Szörényi, and György Turán. Biclique coverings, rectifier networks and the cost of ε-removal. In Descriptional Complexity of Formal Systems - 16th International Workshop, DCFS 2014, Turku, Finland, August 5-8, 2014. Proceedings, pages 174-185, 2014. URL:
  9. Stasys Jukna. Extremal Combinatorics. Springer-Verlag, 2011. 2nd edition. Google Scholar
  10. Stasys Jukna and Alexander S. Kulikov. On covering graphs by complete bipartite subgraphs. Discrete Mathematics, 309(10):3399-3403, 2009. URL:
  11. Stasys Jukna and Igor Sergeev. Complexity of linear Boolean operators. Foundations and Trends in Theoretical Computer Science, 9(1):1-123, 2013. Available at URL:
  12. Howard Karloff. Linear Programming. Birkhäuser, 2008. 2nd printing. Google Scholar
  13. Marek Karpinski and Alexander Zelikovsky. Approximating dense cases of covering problems. In Network design: connectivity and facilities location, volume 40 of DIMACS, pages 169-178. AMS, 1998. Google Scholar
  14. Gyula Katona and Endre Szemerédi. On a problem of graph theory. Studia Scientiarum Mathematicarum Hungarica, 2:23-28, 1967. Google Scholar
  15. R. E. Krichevskii. A minimal monotone contact scheme for a Boolean function of n variables. In Diskretnyj Analiz (Discrete Analysis), volume 5, pages 89-92. Institute for Mathematics in the Siberian Section of the Academy of Sciences, Novosibirsk, 1965. In Russian. Google Scholar
  16. Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997. Google Scholar
  17. László Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13(4):383-390, 1975. URL:
  18. László Lovász. A kombinatorika minimax tételeiről. Matematikai Lapok, 26:209-264, 1976. In Hungarian. Google Scholar
  19. László Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A, 25(3):319-324, 1978. URL:
  20. S. A. Lozhkin. On minimal π-circuits for monotone symmetric functions with threshold 2. Diskretnaya Matematika, 17(4):108-110, 2005. In Russian. English translation in Discrete Mathematics and Applications 15(5) (2005), 475-477. Google Scholar
  21. Kurt Mehlhorn. Some remarks on Boolean sums. Acta Inf., 12:371-375, 1979. URL:
  22. Robert Morris. Some theorems on sorting. SIAM J. Appl. Math., 17:1-6, 1969. Google Scholar
  23. E. I. Nechiporuk. Rectifier networks. Soviet Physics Doklady, 8:5-7, March 1963. Google Scholar
  24. E. I. Nechiporuk. On the topological principles of self-correction. Problemy Kibernetiki, 21:5-102, 1969. In Russian. English translation in: Systems Theory Res. 21 (1970), 1-99. Google Scholar
  25. Nicholas Pippenger. On another Boolean matrix. Theor. Comput. Sci., 11:49-56, 1980. URL:
  26. Jaikumar Radhakrishnan. Entropy and counting. In J. C. Misra, editor, Computational Mathematics, Modelling and Algorithms. Narosa Publishers, New Delhi, 2003. Available online at URL:
  27. Alexander Sapozhenko. On the complexity of disjunctive normal forms obtained with a gradient algorithm. In Diskretnyj Analiz (Discrete Analysis), volume 21, pages 62-71. Institute for Mathematics in the Siberian Section of the Academy of Sciences, Novosibirsk, 1972. In Russian. Google Scholar
  28. S. N. Selezneva. Lower bound on the complexity of finding polynomials of Boolean functions in the class of circuits with separated variables. Computational Mathematics and Modeling, 24(1):146-152, 2013. URL:
  29. Alexander A. Sherstov. Strong direct product theorems for quantum communication and query complexity. SIAM J. Comput., 41(5):1122-1165, 2012. URL:
  30. N. J. A. Sloane. On-line encyclopedia of integer sequences. Electronic resource at URL:
  31. S. K. Stein. Two combinatorial covering theorems. J. Comb. Theory, Ser. A, 16(3):391-397, 1974. URL:
  32. T. G. Tarján. Complexity of lattice-configurations. Studia Scientiarum Mathematicarum Hungarica, 10:203-211, 1975. Google Scholar
  33. Yu. L. Vasilyev and V. V. Glagolev. Metrical properties of disjunctive normal forms. In S. V. Yablonsky and O. B. Lupanov, editors, Discrete mathematics and mathematical questions of cybernetics, pages 99-148. Nauka, Moscow, 1974. In Russian. Google Scholar
  34. J. L. Wassiljew and W. W. Glagolew. Metrische Eigenschaften alternativer Normalformen. In S. W. Jablonski and O. B. Lupanow, editors, Diskrete Mathematik und Mathematische Fragen der Kybernetik, volume 71 of Mathematische Reihe, pages 100-144. Birkhäuser Basel, 1980. Translation of [33]. In German. URL:
  35. Valerie L. Watts. Fractional biclique covers and partitions of graphs. Electr. J. Comb., 13(1), 2006. URL:
  36. Ingo Wegener. A new lower bound on the monotone network complexity of Boolean sums. Acta Inf., 13:109-114, 1980. URL:
  37. David P. Williamson and David B. Shmoys. The design of approximation algorithms. Cambridge University Press, 2011. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail