Efficient Algorithms for Maximum Induced Matching Problem in Permutation and Trapezoid Graphs

We first design an 𝒪(n2) solution for finding a maximum induced matching in permutation graphs given their permutation models, based on a dynamic programming algorithm with the aid of the sweep line technique. With the support of the disjoint-set data structure, we improve the complexity to 𝒪(m+n). Consequently, we extend this result to give an 𝒪(m+n) algorithm for the same problem in trapezoid graphs. By combining our algorithms with the current best graph identification algorithms, we can solve the MIM problem in permutation and trapezoid graphs in linear and 𝒪(n2) time, respectively. Our results are far better than the best known 𝒪(mn) algorithm for the maximum induced matching problem in both graph classes, which was proposed by Habib et al.

Several results on compact metrizable spaces in $$\mathbf {ZF}$$

In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

FACTORIALS OF INFINITE CARDINALS IN ZF PART II: CONSISTENCY RESULTS

For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:(1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$, where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition.(2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$.(3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.

Dominance structure of assemblages is regulated over a period of rapid environmental change

Ecological assemblages are inherently uneven, with numerically dominant species contributing disproportionately to ecosystem services. Marked biodiversity change due to growing pressures on the world's ecosystems is now well documented. However, the hypothesis that dominant species are becoming relatively more abundant has not been tested. We examined the prediction that the dominance structure of contemporary communities is shifting, using a meta-analysis of 110 assemblage timeseries. Changes in relative and absolute dominance were evaluated with mixed and cyclic-shift permutation models. Our analysis uncovered no evidence of a systematic change in either form of dominance, but established that relative dominance is preserved even when assemblage size (total N ) changes. This suggests that dominance structure is regulated alongside richness and assemblage size, and highlights the importance of investigating multiple components of assemblage diversity when evaluating ecosystem responses to environmental drivers.

8. Permutation Models for Relational Data

A common problem in sociology, psychology, biology, geography, and management science is the comparison of dyadic relational structures (i.e., graphs). Where these structures are formed on a common set of elements, a natural question that arises is whether there is a tendency for elements that are strongly connected in one set of structures to be more—or less—strongly connected within another set. We may ask, for instance, whether there is a correspondence between golf games and business deals, trade and warfare, or spatial proximity and genetic similarity. In each case, the data for such comparisons may be continuous or discrete, and multiple relations may be involved simultaneously (e.g., when comparing multiple measures of international trade volume with multiple types of political interactions). We propose here an exponential family of permutation models that is suitable for inferring the direction and strength of association among dyadic relational structures. A linear-time algorithm is shown for MCMC simulation of model draws, as is the use of simulated draws for maximum likelihood estimation (MCMC-MLE) and/or estimation of Monte Carlo standard errors. We also provide an easily performed maximum pseudo-likelihood estimation procedure for the permutation model family, which provides a reasonable means of generating seed models for the MCMC-MLE procedure. Use of the modeling framework is demonstrated via an application involving relationships among managers in a high-tech firm.