Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBoltzmann Samplers, Polya Theory, and Cycle Pointing
130
0
0.0
(
0
)
Added by
Manuel Bodirsky
Publication date
2010
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an unbiased way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on Polya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.
rate research
Read More
Modeling binary and categorical data is one of the most commonly encountered tasks of applied statisticians and econometricians. While Bayesian methods in this context have been available for decades now, they often require a high level of familiarity with Bayesian statistics or suffer from issues such as low sampling efficiency. To contribute to the accessibility of Bayesian models for binary and categorical data, we introduce novel latent variable representations based on Polya Gamma random variables for a range of commonly encountered discrete choice models. From these latent variable representations, new Gibbs sampling algorithms for binary, binomial and multinomial logistic regression models are derived. All models allow for a conditionally Gaussian likelihood representation, rendering extensions to more complex modeling frameworks such as state space models straight-forward. However, sampling efficiency may still be an issue in these data augmentation based estimation frameworks. To counteract this, MCMC boosting strategies are developed and discussed in detail. The merits of our approach are illustrated through extensive simulations and a real data application.
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from a general result about relational systems, which also implies the similar fact about high girth and high arboricity used in the proofs. These facts concern graphs and digraphs of high girth and low degree; we contrast them by considering acyclic colorings of tournaments (which have low girth and high degree). We prove that even though acyclic two-colorability of tournaments is known to be NP-complete, random acyclically r-colorable tournaments allow recovering an acyclic r-coloring in deterministic linear time, with high probablity.
The second authors $omega$, $Delta$, $chi$ conjecture proposes that every graph satisties $chi leq lceil frac 12 (Delta+1+omega)rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $chi$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called skeletal graphs.
A power assignment is an assignment of transmission power to each of the wireless nodes of a wireless network, so that the induced graph satisfies some desired properties. The cost of a power assignment is the sum of the assigned powers. In this paper, we consider the dual power assignment problem, in which each wireless node is assigned a high- or low-power level, so that the induced graph is strongly connected and the cost of the assignment is minimized. We improve the best known approximation ratio from $frac{pi^2}{6}-frac{1}{36}+epsilonthickapprox 1.617$ to $frac{11}{7}thickapprox 1.571$. Moreover, we show that the algorithm of Khuller et al. for the strongly connected spanning subgraph problem, which achieves an approximation ratio of $1.61$, is $1.522$-approximation algorithm for symmetric directed graphs. The innovation of this paper is in achieving these results via utilizing interesting properties for the existence of a second Hamiltonian cycle.
We introduce a definition of admissibility for subintervals in interval exchange transformations. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We characterize the admissible intervals using a branching version of the Rauzy induction. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
