Polynomial tuning of multiparametric combinatorial samplers

TL;DR

This paper introduces a polynomial-time tuning algorithm for multiparametric Boltzmann samplers, enabling efficient control over multiple combinatorial parameters in large structures through convex optimization.

Contribution

It presents a novel polynomial-time tuning method based on convex optimization for multiparametric combinatorial samplers, improving flexibility and efficiency.

Findings

Efficient tuning algorithm demonstrated on polyomino tilings.

Applicable to rational, algebraic, and Pólya structures.

Achieves controlled sampling with multiple parameters.

Abstract

Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like structures. In their multiparametric variants, these samplers allow to control the profile of expected values corresponding to multiple combinatorial parameters. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain subpatterns in strings. However, such a flexible control requires an additional non-trivial tuning procedure. In this paper, we propose an efficient polynomial-time, with respect to the number of tuned parameters, tuning algorithm based on convex optimisation techniques. Finally, we illustrate the efficiency of our approach using several applications of rational, algebraic and P\'olya structures including polyomino tilings with prescribed tile frequencies, planar trees with a given specific node degree distribution, and weighted partitions.