Hidden Tractable Classes: From Theory to Practice

Tractable classes constitute an important issue in CP, at least from a theoretical viewpoint. But they are not actually used in practice. Either their treatment is too costly for time complexity or, even if there exist efficient algorithms to manage them, they do not appear in the real problems. We propose here to address this issue thanks to the notion of hidden tractable classes. Such classes are based on a known tractable class C, and a transformation t, and are defined by sets of instances P such that their transformation using t is in C, that is t (P) in C. We propose a general framework to study such notions. After, we focus our study on the tractable class BTP, and several transformations which are the filterings classically used in CP. We show then that the use of filterings allows sometimes to highlight the occurrence of BTP in the benchmarks generally considered for solver comparisons, i.e. That BTP is sometimes "hidden" in the benchmarks. Thus, this approach allows to extend the set of known tractable classes.

[1]  Roland H. C. Yap,et al.  An optimal coarse-grained arc consistency algorithm , 2005, Artif. Intell..

[2]  Christian Bessiere,et al.  From Restricted Path Consistency to Max-Restricted Path Consistency , 1997, CP.

[3]  Philippe Jégou,et al.  A hybrid tractable class for non-binary CSPs , 2013, 2013 IEEE 25th International Conference on Tools with Artificial Intelligence.

[4]  Eugene C. Freuder.  A Sufficient Condition for Backtrack-Free Search , 1982, JACM.

[5]  Eugene C. Freuder,et al.  Neighborhood Inverse Consistency Preprocessing , 1996, AAAI/IAAI, Vol. 1.

[6]  Robert M. Haralick,et al.  Increasing Tree Search Efficiency for Constraint Satisfaction Problems , 1979, Artif. Intell..

[7]  Eugene C. Freuder,et al.  Contradicting Conventional Wisdom in Constraint Satisfaction , 1994, ECAI.

[8]  Christian Bessiere,et al.  Domain Filtering Consistencies , 2011, J. Artif. Intell. Res..

[9]  Ugo Montanari,et al.  Networks of constraints: Fundamental properties and applications to picture processing , 1974, Inf. Sci..

[10]  Bernard A. Nadel,et al.  Tree search and ARC consistency in constraint satisfaction algorithms , 1988 .

[11]  Christophe Lecoutre,et al.  Constraint Networks: Techniques and Algorithms , 2009 .

[12]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[13]  Bart Selman,et al.  Backdoors To Typical Case Complexity , 2003, IJCAI.

[14]  Georg Gottlob,et al.  A Comparison of Structural CSP Decomposition Methods , 1999, IJCAI.

[15]  Philippe Jégou.  Decomposition of Domains Based on the Micro-Structure of Finite Constraint-Satisfaction Problems , 1993, AAAI.

[16]  Martin C. Cooper,et al.  Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination , 2010, Artif. Intell..

[17]  Christian Bessiere,et al.  Optimal and Suboptimal Singleton Arc Consistency Algorithms , 2005, IJCAI.

[18]  Toby Walsh,et al.  Handbook of Constraint Programming , 2006, Handbook of Constraint Programming.

[19]  Martin C. Cooper,et al.  Characterising Tractable Constraints , 1994, Artif. Intell..