Abstract
Concepts of consistency have long played a key role in constraint programming but never developed in integer programming (IP). Consistency nonetheless plays a role in IP as well. For example, cutting planes can reduce backtracking by achieving various forms of consistency as well as by tightening the linear programming (LP) relaxation. We introduce a type of consistency that is particularly suited for 0–1 programming and develop the associated theory. We define a 0–1 constraint set as LP-consistent when any partial assignment that is consistent with its linear programming relaxation is consistent with the original 0–1 constraint set. We prove basic properties of LP-consistency, including its relationship with Chvátal-Gomory cuts and the integer hull. We show that a weak form of LP-consistency can reduce or eliminate backtracking in a way analogous to k-consistency. This work suggests a new approach to the reduction of backtracking in IP that focuses on cutting off infeasible partial assignments rather than fractional solutions.
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References
Apt, K.R.: Principles of Constraint Programming. Cambridge University Press, Cambridge (2003)
Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)
Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973)
Davis, E.: Constraint propagation with intervals labels. Artif. Intell. 32, 281–331 (1987)
Freuder, E.C.: Synthesizing constraint expressions. Commun. ACM 21, 958–966 (1978)
Freuder, E.C.: A sufficient condition for backtrack-free search. Commun. ACM 29, 24–32 (1982)
Hooker, J.N.: Input proofs and rank one cutting planes. ORSA J. Comput. 1, 137–145 (1989)
Hooker, J.N.: Integrated Methods for Optimization, 2nd edn. Springer, Heidelberg (2012). https://doi.org/10.1007/978-1-4614-1900-6
Hooker, J.N.: Projection, consistency, and George Boole. Constraints 21, 59–76 (2016)
Mackworth, A.: Consistency in networks of relations. Artif. Intell. 8, 99–118 (1977)
Montanari, U.: Networks of constraints: fundamental properties and applications to picture processing. Inf. Sci. 7, 95–132 (1974)
Quine, W.V.: The problem of simplifying truth functions. Am. Math. Mon. 59, 521–531 (1952)
Quine, W.V.: A way to simplify truth functions. Am. Math. Mon. 62, 627–631 (1955)
Régin, J.C.: Global constraints: a survey. In: Milano, M., Van Hentenryck, P. (eds.) Hybrid Optimization: The Ten Years of CPAIOR, pp. 63–134. Springer, New York (2010). https://doi.org/10.1007/978-1-4419-1644-0_3
Tsang, E.: Foundations of Constraint Satisfaction. Academic Press, London (1983)
Van Hentenryck, P.: Constraint Satisfaction in Logic Programming. MIT Press, Cambridge (1989)
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Davarnia, D., Hooker, J.N. (2019). Consistency for 0–1 Programming. In: Rousseau, LM., Stergiou, K. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019. Lecture Notes in Computer Science(), vol 11494. Springer, Cham. https://doi.org/10.1007/978-3-030-19212-9_15
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DOI: https://doi.org/10.1007/978-3-030-19212-9_15
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