A note on the complexity of the causal ordering problem

In this note we provide a concise report on the complexity of the causal ordering problem, originally introduced by Simon to reason about causal dependencies implicit in systems of mathematical equations. We show that Simon's classical algorithm to infer causal ordering is NP-Hard-an intractability previously guessed but never proven. We present then a detailed account based on Nayak's suggested algorithmic solution (the best available), which is dominated by computing transitive closure-bounded in time by O ( | V | ź | S | ) , where S ( E , V ) is the input system structure composed of a set E of equations over a set V of variables with number of variable appearances (density) | S | . We also comment on the potential of causal ordering for emerging applications in large-scale hypothesis management and analytics.

[1]  Nicholas T. Carnevale,et al.  ModelDB: A Database to Support Computational Neuroscience , 2004, Journal of Computational Neuroscience.

[2]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[3]  David A. Bader,et al.  Graph Algorithms , 2011, Encyclopedia of Parallel Computing.

[4]  David Serrano,et al.  Constraint Management in Conceptual Design , 1989 .

[5]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[6]  Fábio Porto,et al.  Managing Scientific Hypotheses as Data with Support for Predictive Analytics , 2014, Computing in Science & Engineering.

[7]  Fábio Porto,et al.  Υ-DB: Managing scientific hypotheses as uncertain data , 2014, Proc. VLDB Endow..

[8]  Ronald L. Rivest,et al.  Introduction to Algorithms, third edition , 2009 .

[9]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[10]  Takeaki Uno,et al.  An Efficient Algorithm for Solving Pseudo Clique Enumeration Problem , 2008, Algorithmica.

[11]  Milind Dawande,et al.  On Bipartite and Multipartite Clique Problems , 2001, J. Algorithms.

[12]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[13]  Arlindo L. Oliveira,et al.  Biclustering algorithms for biological data analysis: a survey , 2004, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[14]  Herbert A. Simon,et al.  Causality and Model Abstraction , 1994, Artif. Intell..

[15]  David S. Johnson,et al.  The NP-Completeness Column: An Ongoing Guide , 1982, J. Algorithms.

[16]  Nicolas Le Novère,et al.  BioModels Database: a repository of mathematical models of biological processes. , 2013, Methods in molecular biology.

[17]  P. Pandurang Nayak,et al.  Causal Approximations , 1992, Artif. Intell..

[18]  H. Simon,et al.  Causal Ordering and Identifiability , 1977 .

[19]  Paul P. Maglio,et al.  Data is dead... without what-if models , 2011, Proc. VLDB Endow..

[20]  Marek J. Druzdzel,et al.  A note on the correctness of the causal ordering algorithm , 2008, Artif. Intell..

[21]  Peter Hunter,et al.  Integration from proteins to organs: the IUPS Physiome Project , 2005, Mechanisms of Ageing and Development.

[22]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[23]  P. Pandurang Nayak.  Automated Modelling of Physical Systems , 1995 .

[24]  Bernardo Gonçalves,et al.  Managing large-scale scientific hypotheses as uncertain and probabilistic data , 2014, ArXiv.

[25]  P. Hunter,et al.  Integration from proteins to organs: the Physiome Project , 2003, Nature Reviews Molecular Cell Biology.

[26]  Richard M. Karp,et al.  A n^5/2 Algorithm for Maximum Matchings in Bipartite Graphs , 1971, SWAT.

[27]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .