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Inferring (Biological) Signal Transduction Networks via Transitive Reductions of Directed Graphs

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Abstract

In this paper we consider the p-ary transitive reduction (TR p ) problem where p>0 is an integer; for p=2 this problem arises in inferring a sparsest possible (biological) signal transduction network consistent with a set of experimental observations with a goal to minimize false positive inferences even if risking false negatives. Special cases of TR p have been investigated before in different contexts; the best previous results are as follows:

  1. (1)

    The minimum equivalent digraph problem, that correspond to a special case of TR1 with no critical edges, is known to be MAX-SNP-hard, admits a polynomial time algorithm with an approximation ratio of 1.617+ε for any constant ε>0 (Chiu and Liu in Sci. Sin. 4:1396–1400, 1965) and can be solved in linear time for directed acyclic graphs (Aho et al. in SIAM J. Comput. 1(2):131–137, 1972).

  2. (2)

    A 2-approximation algorithm exists for TR1 (Frederickson and JàJà in SIAM J. Comput. 10(2):270–283, 1981; Khuller et al. in 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 937–938, 1999).

In this paper, our contributions are as follows:

  1. •

    We observe that TR p , for any integer p>0, can be solved in linear time for directed acyclic graphs using the ideas in Aho et al. (SIAM J. Comput. 1(2):131–137, 1972).

  2. •

    We provide a 1.78-approximation for TR1 that improves the 2-approximation mentioned in (2) above.

  3. •

    We provide a 2+o(1)-approximation for TR p on general graphs for any fixed prime p>1.

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Authors and Affiliations

  1. Department of Physics, Pennsylvania State University, University Park, PA, 16802, USA

    Réka Albert

  2. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Bhaskar DasGupta

  3. Dipartimento di Scienze dei Linguaggi, della Comunicazione e degli Studi Culturali, Università degli Studi di Bergamo, Bergamo, 24129, Italy

    Riccardo Dondi

  4. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    Eduardo Sontag

Authors
  1. Réka Albert
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  2. Bhaskar DasGupta
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  3. Riccardo Dondi
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  4. Eduardo Sontag
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Corresponding author

Correspondence to Bhaskar DasGupta.

Additional information

R. Albert’s research was partly supported by a Sloan Research Fellowship in Science and Technology.

B. DasGupta’s research was partly supported by NSF grants DBI-0543365, IIS-0612044 and IIS-0346973.

E. Sontag’s research was partly supported by NSF grants EIA 0205116 and DMS-0504557.

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Albert, R., DasGupta, B., Dondi, R. et al. Inferring (Biological) Signal Transduction Networks via Transitive Reductions of Directed Graphs. Algorithmica 51, 129–159 (2008). https://doi.org/10.1007/s00453-007-9055-0

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