68,127 research outputs found
Boltzmann Sampling of Unlabelled Structures
International audienceBoltzmann models from statistical physics, combined with methods from analytic combinatorics, give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they do so by means of real-arithmetic computations. Here you'll find a collection of construction rules for such samplers, which applies to a wide variety of combinatorial classes, including integer partitions, necklaces, unlabelled functional graphs, dictionaries, series-parallel circuits, term trees and acyclic molecules obeying a variety of constraints
Approximation Algorithms for Facial Cycles in Planar Embeddings
Consider the following combinatorial problem: Given a planar graph G and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. This problem, called Max Facial C-Cycles, was first studied by Mutzel and Weiskircher [IPCO \u2799, http://dx.doi.org/10.1007/3-540-48777-8_27) and then proved NP-hard by Woeginger [Oper. Res. Lett., 2002, http://dx.doi.org/10.1016/S0167-6377(02)00119-0].
We establish a tight border of tractability for Max Facial C-Cycles in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that strengthening any of these conditions makes the problem polynomial-time solvable. Our main results are approximation algorithms for Max Facial C-Cycles. Namely, we give a 2-approximation for series-parallel graphs and a (4+epsilon)-approximation for biconnected planar graphs. Remarkably, this provides one of the first approximation algorithms for constrained embedding problems
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
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