![](bg8.png)
uation functions to have specific mathematical prop-
erties. This is the case of MOVCSP classes MOVC-
SPs with Directional Substitutable Valuation Func-
tions only (L(MODS)). As usefulness of L(MODS)
MOVCSP class even when the studied problem is not
limited to these functions, we have proposed a Direc-
tional Substitutable decomposition algorithm. As a
natural extension of this work, in order to validate the
practical use of L (MODS), we will compare a prob-
lem decomposition scheme for MOVCSPs that uses
Pareto-based Soft Arc Consistency only with a de-
composition scheme witch, in addition, takes advan-
tage of L(MODS).
REFERENCES
Ali, L. B., Helaoui, M., and Naanaa, W. (2019). Pareto-
based soft arc consistency for multi-objective val-
ued csps. In Rocha, A. P., Steels, L., and van den
Herik, H. J., editors, Proceedings of the 11th In-
ternational Conference on Agents and Artificial In-
telligence, ICAART 2019, Volume 2, Prague, Czech
Republic, February 19-21, 2019, pages 294–305.
SciTePress.
Carbonnel, C. and Cooper, M. C. (2016). Tractability in
constraint satisfaction problems: a survey. Constraints
An Int. J., 21(2):115–144.
Cohen, D. A., Cooper, M. C., Creed, P., Marx, D., and Sala-
mon, A. Z. (2012). The tractability of CSP classes
defined by forbidden patterns. J. Artif. Intell. Res.,
45:47–78.
Cohen, D. A., Cooper, M. C., and Jeavons, P. (2008a). Gen-
eralising submodularity and horn clauses: Tractable
optimization problems defined by tournament pair
multimorphisms. Theor. Comput. Sci., 401(1-3):36–
51.
Cohen, D. A., Jeavons, P., and Gyssens, M. (2008b). A uni-
fied theory of structural tractability for constraint sat-
isfaction problems. J. Comput. Syst. Sci., 74(5):721–
743.
Cooper, M. C., Duchein, A., Mouelhi, A. E., Escamocher,
G., Terrioux, C., and Zanuttini, B. (2016). Broken
triangles: From value merging to a tractable class of
general-arity constraint satisfaction problems. Artif.
Intell., 234:196–218.
Cooper, M. C. and Zivn
´
y, S. (2011). Hybrid tractability
of valued constraint problems. Artif. Intell., 175(9-
10):1555–1569.
Cooper, M. C. and Zivn
´
y, S. (2012). Tractable triangles and
cross-free convexity in discrete optimisation. J. Artif.
Intell. Res., 44:455–490.
De, P., Dunne, E. J., Ghosh, J. B., and Wells, C. E. (1997).
Complexity of the discrete time-cost tradeoff problem
for project networks. Oper. Res., 45(2):302–306.
Debels, D. and Vanhoucke, M. (2007). A decomposition-
based genetic algorithm for the resource-constrained
project-scheduling problem. Oper. Res., 55(3):457–
469.
Freuder, E. C. (1991). Eliminating interchangeable values
in constraint satisfaction problems. In Dean, T. L. and
McKeown, K. R., editors, Proceedings of the 9th Na-
tional Conference on Artificial Intelligence, Anaheim,
CA, USA, July 14-19, 1991, Volume 1, pages 227–233.
AAAI Press / The MIT Press.
Greco, G. and Scarcello, F. (2011). Structural tractabil-
ity of constraint optimization. In Lee, J. H., editor,
Principles and Practice of Constraint Programming -
CP 2011 - 17th International Conference, CP 2011,
Perugia, Italy, September 12-16, 2011. Proceedings,
volume 6876 of Lecture Notes in Computer Science,
pages 340–355. Springer.
Greco, G. and Scarcello, F. (2013). Constraint satisfac-
tion and fair multi-objective optimization problems:
Foundations, complexity, and islands of tractability.
In Rossi, F., editor, IJCAI 2013, Proceedings of the
23rd International Joint Conference on Artificial In-
telligence, Beijing, China, August 3-9, 2013, pages
545–551. IJCAI/AAAI.
Helaoui, M. and Naanaa, W. (2013). Modularity-based de-
compositions for valued CSP. Ann. Math. Artif. Intell.,
67(2):165–187.
Helaoui, M., Naanaa, W., and Ayeb, B. (2013).
Submodularity-based decomposing for valued CSP.
Int. J. Artif. Intell. Tools, 22(2).
Lecoutre, C., Roussel, O., and Dehani, D. E. (2012). WCSP
integration of soft neighborhood substitutability. In
Milano, M., editor, Principles and Practice of Con-
straint Programming - 18th International Conference,
CP 2012, Qu
´
ebec City, QC, Canada, October 8-12,
2012. Proceedings, volume 7514 of Lecture Notes in
Computer Science, pages 406–421. Springer.
Naanaa, W. (2008). A domain decomposition algorithm for
constraint satisfaction. ACM J. Exp. Algorithmics, 13.
Naanaa, W., Helaoui, M., and Ayeb, B. (2009). A do-
main decomposition algorithm for constraint satisfac-
tion and optimization. In JFPC 2009 - Actes des Cin-
quiemes Journees Francophones de Programmation
par Contraintes.
Schiex, T., Fargier, H., and Verfaillie, G. (1995). Valued
constraint satisfaction problems: hard and easy prob-
lems. In In Proceedings of the 14
th
IJCAI, 631–637,
Montr
´
eal, Canada.
Schlesinger, D. (2007). Exact solution of permuted sub-
modular minsum problems. In Yuille, A. L., Zhu,
S. C., Cremers, D., and Wang, Y., editors, Energy Min-
imization Methods in Computer Vision and Pattern
Recognition, 6th International Conference, EMM-
CVPR 2007, Ezhou, China, August 27-29, 2007, Pro-
ceedings, volume 4679 of Lecture Notes in Computer
Science, pages 28–38. Springer.
Tavana, M., Abtahi, A., and Damghani, K. K. (2014).
A new multi-objective multi-mode model for solv-
ing preemptive time-cost-quality trade-off project
scheduling problems. Expert Syst. Appl., 41(4):1830–
1846.
van Beek, P. and Manchak, D. W. (1996). The design and
experimental analysis of algorithms for temporal rea-
soning. J. Artif. Intell. Res., 4:1–18.
Vanhoucke, M. (2005). New computational results for the
discrete time/cost trade-off problem with time-switch
constraints. Eur. J. Oper. Res., 165(2):359–374.
Soft Directional Substitutable based Decompositions for MOVCSP
225