The BIN_COUNTS Constraint: Filtering and Applications

We introduce the BIN_COUNTS constraint, which deals with the problem of counting the number of decision variables in a set which are assigned values that lie in given bins. We illustrate a decomposition and a filtering algorithm that achieves generalised arc consistency. We contrast the filtering power of these two approaches and we discuss a number of applications. We show that BIN_COUNTS can be employed to develop a decomposition for the $\chi^2$ test constraint, a new statistical constraint that we introduce in this work. We also show how this new constraint can be employed in the context of the Balanced Academic Curriculum Problem and of the Balanced Nursing Workload Problem. For both these problems we carry out numerical studies involving our reformulations. Finally, we present a further application of the $\chi^2$ test constraint in the context of confidence interval analysis.

[1]  Jean-Charles Régin,et al.  A Filtering Algorithm for Constraints of Difference in CSPs , 1994, AAAI.

[2]  I. Heller,et al.  14 . An Extension of a Theorem of Dantzig’s , 1957 .

[3]  Krzysztof Kuchcinski,et al.  JaCoP Library User's Guide , 2010 .

[4]  Gilles Pesant,et al.  SPREAD: A Balancing Constraint Based on Statistics , 2005, CP.

[5]  Pierre Schaus,et al.  Bound-consistent spread constraint , 2014, EURO J. Comput. Optim..

[6]  D. C. Hurst,et al.  Large Sample Simultaneous Confidence Intervals for Multinomial Proportions , 1964 .

[7]  J. Glaz,et al.  Simultaneous confidence intervals for multinomial proportions , 1999 .

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

[9]  Peter J. Stuckey,et al.  Finite Domain Bounds Consistency Revisited , 2004, Australian Conference on Artificial Intelligence.

[10]  P. Schaus.  Solving balancing and bin-packing problems with constraint programming , 2009 .

[11]  Alan K. Mackworth.  Consistency in Networks of Relations , 1977, Artif. Intell..

[12]  Mats Carlsson,et al.  Global Constraint Catalogue: Past, Present and Future , 2007, Constraints.

[13]  Stephane Rivaud.  Sampling Markov Models under Binary Equality Constraints Is Hard , 2016 .

[14]  Robert E. Tarjan,et al.  Improved Algorithms for Bipartite Network Flow , 1994, SIAM J. Comput..

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

[16]  Nicolas Beldiceanu,et al.  Extending CHIP in order to solve complex scheduling and placement problems , 1993, JFPL.

[17]  Jean-Guillaume Fages,et al.  Combining finite and continuous solvers , 2014, ArXiv.

[18]  Gilles Pesant,et al.  Achieving Domain Consistency and Counting Solutions for Dispersion Constraints , 2015, INFORMS J. Comput..

[19]  Yves Deville,et al.  The Deviation Constraint , 2007, CPAIOR.

[20]  Klaus Truemper,et al.  Matroid decomposition , 1992 .

[21]  Roberto Rossi,et al.  Statistical Constraints , 2014, ECAI.

[22]  Hsiuying Wang,et al.  Exact confidence coefficients of simultaneous confidence intervals for multinomial proportions , 2008 .

[23]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[24]  K. Pearson.  On the Criterion that a Given System of Deviations from the Probable in the Case of a Correlated System of Variables is Such that it Can be Reasonably Supposed to have Arisen from Random Sampling , 1900 .

[25]  Mark Lawley,et al.  Assigning patients to nurses in neonatal intensive care , 2002, J. Oper. Res. Soc..

[26]  Carlos Castro,et al.  Variable and Value Ordering When Solving Balanced Academic Curriculum Problems , 2001, ArXiv.

[27]  Yves Deville,et al.  A CP Approach to the Balanced Academic Curriculum Problem , 2007 .

[28]  François Pachet,et al.  Generating 1/f Noise Sequences as Constraint Satisfaction: The Voss Constraint , 2015, IJCAI.

[29]  Jean-Charles Régin,et al.  Generalized Arc Consistency for Global Cardinality Constraint , 1996, AAAI/IAAI, Vol. 1.

[30]  Thorsten Dickhaus,et al.  Simultaneous Statistical Inference , 2014, Springer Berlin Heidelberg.

[31]  Pascal Van Hentenryck,et al.  Scalable Load Balancing in Nurse to Patient Assignment Problems , 2009, CPAIOR.

[32]  Irit Katriel,et al.  Fast Bound Consistency for the Global Cardinality Constraint , 2003, CP.

[33]  Gilles Pesant.  Balancing Nursing Workload by Constraint Programming , 2016, CPAIOR.

[34]  Jean-Charles Régin,et al.  Global Constraints and Filtering Algorithms , 2004 .

[35]  Roberto Rossi,et al.  Randomness as a Constraint , 2015, CP.

[36]  Toby Walsh,et al.  CSPLIB: A Benchmark Library for Constraints , 1999, CP.

[37]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[38]  Nicolas Beldiceanu,et al.  Introducing global constraints in CHIP , 1994 .

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