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Tuning as convex optimisation: a polynomial tuner for multi-parametric combinatorial samplers

Part of: Combinatorics Theory of computing Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  15 December 2021

Maciej Bendkowski ,
Olivier Bodini  and
Sergey Dovgal
Maciej Bendkowski
Affiliation:
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
Olivier Bodini
Affiliation:
Institut Galilée, Université Paris 13, 99 Avenue Jean Baptiste Clément, 93430 Villetaneuse, France.
Sergey Dovgal*
Affiliation:
Institut Galilée, Université Paris 13, 99 Avenue Jean Baptiste Clément, 93430 Villetaneuse, France.
*
*Corresponding author. E-mail: [email protected]
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Abstract

Combinatorial samplers are algorithmic schemes devised for the approximate- and exact-size generation of large random combinatorial structures, such as context-free words, various tree-like data structures, maps, tilings, RNA molecules. They can be adapted to combinatorial specifications with additional parameters, allowing for a more flexible control over the output profile of parametrised combinatorial patterns. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain sub-patterns in generated strings. However, such a flexible control requires an additional and nontrivial tuning procedure. Using techniques of convex optimisation, we present an efficient tuning algorithm for multi-parametric combinatorial specifications. Our algorithm works in polynomial time in the system description length, the number of tuning parameters, the number of combinatorial classes in the specification, and the logarithm of the total target size. We demonstrate the effectiveness of our method on a series of practical examples, including rational, algebraic, and so-called Pólya specifications. We show how our method can be adapted to a broad range of less typical combinatorial constructions, including symmetric polynomials, labelled sets and cycles with cardinality lower bounds, simple increasing trees or substitutions. Finally, we discuss some practical aspects of our prototype tuner implementation and provide its benchmark results.

Keywords

boltzmann samplersmultiparametric tuningconvex optimisationcombinatorial specificationsanalytic combinatoricscontext-free grammarsrandom samplingself-concordant barriers

MSC classification

Primary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Secondary: 05A99: None of the above, but in this section 68R05: Combinatorics
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 5 , September 2022 , pp. 765 - 811
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Maciej Bendkowski was partially supported by the Polish National Science Center grant 2016/21/N/ST6/01032 and a French Government Scholarship awarded by the French Embassy in Poland. Olivier Bodini and Sergey Dovgal were supported by the French project ANR project MetACOnc, ANR-15-CE40-0014. The current paper is an extended version of doi:10.1137/1.9781611975062.9 presented at ANALCO’18.

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