Abstract
In constraint satisfaction, pairwise consistency (PWC) is a well-known local consistency improving generalized arc consistency in theory but not often in practice. A popular approach to enforcing PWC enforces arc consistency on the dual encoding of the problem, allowing to reuse existing AC algorithms. In this paper, we explore the benefit of this simple approach in the optimization context of cost function networks and soft local consistencies. Using a dual encoding, we obtain an equivalent binary cost function network where enforcing virtual arc consistency achieves virtual PWC on the original problem. We experimentally observed that adding extra non-binary cost functions before the dual encoding results in even stronger bounds. Such supplementary cost functions may be produced by bounded variable elimination or by adding ternary zero-cost functions. Experiments on (probabilistic) graphical models, from the UAI 2022 competition benchmark, show a clear improvement when using our approach inside a branch-and-bound solver compared to the state-of-the-art.
This research was funded by the grants ANR-18-EURE-0021 and ANR-19-P3IA-0004. It receives support from the Genotoul (Toulouse) Bioinformatic platform.
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Notes
- 1.
This corresponds to full PWC because unary constraints may appear in these pairs.
- 2.
This is unsurprising because the strongest bound that can be derived using EPTs is obtained using a linear program which includes pairwise consistency constraints [34].
- 3.
https://uaicompetition.github.io/uci-2022, see MPE and MMAP entries.
- 4.
- 5.
Options -A -P=1000 -T=1000 -vacint -vacthr -rasps -raspsini in toulbar2-vacint.
- 6.
Called double encoding in [26], it allows more flexibility to enforce various levels of consistency from GAC to PWC depending on the selected channeling constraints.
- 7.
- 8.
It is done only if the median degree in the original problem is less than 8, eliminating variables with a current degree less than or equal to the original median degree.
- 9.
With additional options -i -pils -p = -8 -O = −3 -t = 1. A triangle is defined by three variables involved in three binary cost functions. The score of a triangle is given by the average cost in the three functions. Triangles with the largest score are selected first. This approach allows to simulate soft path inverse consistency [22].
- 10.
See toulbar2-ipr results on the UAI 2022 Tuning Leader Board. Multiple runs of VNS with increasing floating-point precision were done with a total amount of time of \(\frac{1}{2}\)h. The remaining time is allocated to HBFS-VPWC. Each search procedure gives its best solution found to the next search procedure. On UAI 2022 tuning instances, this approach found 119 best solutions, ranking first among our 7 tested methods.
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Montalbano, P., Allouche, D., de Givry, S., Katsirelos, G., Werner, T. (2023). Virtual Pairwise Consistency in Cost Function Networks. In: Cire, A.A. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2023. Lecture Notes in Computer Science, vol 13884. Springer, Cham. https://doi.org/10.1007/978-3-031-33271-5_27
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