Combinatorial Computation Research Articles

Calibrating the Discrete Boundary Conditions of a Dynamic Simulation: A Combinatorial Approximate Bayesian Computation Sequential Monte Carlo (ABC-SMC) Approach.

This paper presents a novel adaptation of the conventional approximate Bayesian computation sequential Monte Carlo (ABC-SMC) sampling algorithm for parameter estimation in the presence of uncertainties, coined combinatorial ABC-SMC. Inference of this type is used in situations where there does not exist a closed form of the associated likelihood function, which is replaced by a simulating model capable of producing artificial data. In the literature, conventional ABC-SMC is utilised to perform inference on continuous parameters. The novel scheme presented here has been developed to perform inference on parameters that are high-dimensional binary, rather than continuous. By altering the form of the proposal distribution from which to sample candidates in subsequent iterations (referred to as waves), high-dimensional binary variables may be targeted and inferred by the scheme. The efficacy of the proposed scheme is demonstrated through application to vibration data obtained in a structural dynamics experiment on a fibre-optic sensor simulated as a finite plate with uncertain boundary conditions at its edges. Results indicate that the method provides sound inference on the plate boundary conditions, which is validated through subsequent application of the method to multiple vibration datasets. Comparisons between appropriate forms of the metric function used in the scheme are also developed to highlight the effect of this element in the schemes convergence.

On Fractional ID-\((g,f)\)-factor-critical Covered Graphs

A graph \(G\) is called a fractional ID-\((g,f)\)-factor-critical covered graph if for any independent set \(I\) of \(G\) and for every edge \(e \in E(G-I)\), \(G-I\) has a fractional \((g,f)\)-factor \(h\) such that \(h(e) = 1\). We give a sufficient condition using degree condition for a graph to be a fractional ID-\((g,f)\)-factor-critical covered graph. Our main result is an extension of Zhou, Bian, and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-\(k\)-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013) 229–236] and Yashima’s previous result [T. Yashima, A degree condition for graphs to be fractional ID-\([a,b]\)-factor-critical, Australasian Journal of Combinatorics 65(2016) 191–199].

Note on the moment generating function of the multivariate normal distribution

We present a streamlined proof of a formula for the derivatives of the moment generating function of the multivariate normal distribution. We formulate it in terms of the summation of the contractions by pairings, which encodes a combinatorial computation procedure. We give two applications. First, we provide a simple proof of Isserlis’ theorem and derive a formula for the moments of the multivariate normal distribution. Second, we calculate the moments of the product of a finite number of correlated normally and lognormally distributed random variables.

The projector algorithm: A simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs

The Voronoi diagram is a certain geometric data structure which has found numerous applications in various scientific and technological fields. The theory of algorithms for computing 2D Euclidean Voronoi diagrams of point sites is rich and useful, with several different and important algorithms. However, this theory has been quite steady during the last few decades in the sense that no essentially new algorithms have entered the game. In addition, most of the known algorithms are serial in nature and hence cast inherent difficulties on the possibility to compute the diagram in parallel. In this paper we present the projector algorithm: a new and simple algorithm which enables the (combinatorial) computation of 2D Voronoi diagrams. The algorithm is significantly different from previous ones and some of the involved concepts in it are in the spirit of linear programming and optics. Parallel implementation is naturally supported since each Voronoi cell (actually, even just portions of one cell) can be computed independently of the other cells. A new combinatorial structure for representing the cells (and any convex polytope) is described along the way and the computation of the induced Delaunay graph is obtained almost automatically.

Sagbi combinatorics of maximal minors and a Sagbi algorithm

The maximal minors of a matrix of indeterminates are a universal Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the other hand it is known that they are not always a universal Sagbi basis. By an experimental approach we discuss their behavior under varying monomial orders and their extensions to Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm which is organized in a Singular script and falls back on Normaliz for the combinatorial computations. In comparison to packages in the current standard distributions of Macaulay 2, version 1.21, and Singular, version 4.2.1 and a package intended for CoCoA 5.4.2, it extends the range of computability by at least one order of magnitude.

Two Formulas for F-Polynomials

Abstract We discuss a product formula for $F$-polynomials in cluster algebras and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock–Goncharov decomposition of mutations. We conclude by expanding this product formula as a sum and illustrate applications. This expansion provides an explicit combinatorial computation of $F$-polynomials in a given seed that depends only on the $\textbf {c}$-vectors and $\textbf {g}$-vectors along a finite sequence of mutations from the initial seed to the given seed.

The Distance Spectral Radii of Hypercacti

Hypergraphs can model many concepts in computer science, engineering, psychology, etc. The spectral theory is an important research topic, which is widely used in combinatorial mathematics, computer science and social science. This paper mainly concerns the structure of extremal hypergraphs determined by the distance spectral of hypergraphs. A hypercactus is a connected k-uniform hypergraph with any two cycles sharing at most one common vertex. For hypercacti with a fixed number of vertices and cycles, we completely determine the hypercactus structure with minimum distance spectral radius.

Phonetic categorization relies on motor simulation, but combinatorial phonological computations are abstract

To identify a spoken word (e.g., dog), people must categorize the speech steam onto distinct units (e.g., contrast dog/fog,) and extract their combinatorial structure (e.g., distinguish dog/god). However, the mechanisms that support these two core functions are not fully understood. Here, we explore this question using transcranial magnetic stimulation (TMS). We show that speech categorization engages the motor system, as stimulating the lip motor area has opposite effects on labial (ba/pa)- and coronal (da/ta) sounds. In contrast, the combinatorial computation of syllable structure engages Broca’s area, as its stimulation disrupts sensitivity to syllable structure (compared to motor stimulation). We conclude that the two ingredients of language—categorization and combination—are distinct functions in human brains.

A direct method for constructing distribution-free tolerance regions

Distribution-free tolerance regions are hyper-rectangles in terms of the number of variables that include at least a pre-specified proportion of normal subjects with a confidence bounded from below by a prescribed probability. This paper has two main goals. The first is to propose an innovative method for constructing multidimensional tolerance regions that work well when the only assumption that can be made about the underlying distribution is that it is continuous. Although our proposal is, in essence, an extension of the Wilks-Wald method to higher dimensions, this research is far less immediate and straightforward than it has appeared to many authors, who, moreover, have not really used it in practical work. In particular, the order statistics dividing the regions are not fixed beforehand, but are determined by an optimization procedure. The second goal is to suggest a way of overcoming a problem of practical importance concerning the order in which the variables are included, which has remained unsolved since the introduction of the argument almost eighty years ago. This device facilitates the search for a good solution while avoiding being drawn away into the black hole of combinatorial computations. Simulation and applications to laboratory medicine data illustrate the advantages of using the method presented in this paper.

Sequential sampling of junction trees for decomposable graphs

The junction-tree representation provides an attractive structural property for organising a decomposable graph. In this study, we present two novel stochastic algorithms, referred to as the junction-tree expander and junction-tree collapser, for sequential sampling of junction trees for decomposable graphs. We show that recursive application of the junction-tree expander, which expands incrementally the underlying graph with one vertex at a time, has full support on the space of junction trees for any given number of underlying vertices. On the other hand, the junction-tree collapser provides a complementary operation for removing vertices in the underlying decomposable graph of a junction tree, while maintaining the junction tree property. A direct application of the proposed algorithms is demonstrated in the setting of sequential Monte Carlo methods, designed for sampling from distributions on spaces of decomposable graphs. Numerical studies illustrate the utility of the proposed algorithms for combinatorial computations on decomposable graphs and junction trees. All the methods proposed in the paper are implemented in the Python library trilearn.

The Erdős Matching Conjecture and concentration inequalities

More than 50 years ago, Erdős asked the following question: what is the maximum size of family F of k-element subsets of an n-element set if it has no s+1 pairwise disjoint sets? This question attracted a lot of attention recently, in particular due to its connection to various combinatorial, probabilistic and theoretical computer science problems. Improving the previous best bound due to the first author, we prove that |F|≤(nk)−(n−sk), provided n≥53sk−23s and s is sufficiently large. The bound on |F| is sharp since the family of all k-sets that intersect some fixed s-element set has such size and has no s+1 pairwise disjoint sets. We derive several corollaries concerning Dirac thresholds and deviations of sums of random variables. We also obtain several related results.

Normality of Closures of Orthogonal Nilpotent Symmetric Orbits

We study closures of conjugacy classes in the symmetric matrices of the orthogonal group and we determine which one are normal varieties. In contrast to the result for the symplectic group where all classes have normal closure, there is only a relatively small portion of classes with normal closure. We perform a combinatorial computation on top of the same methods used by Kraft-Procesi and Ohta.

An Integrated Optimization-Learning Framework for Online Combinatorial Computation Offloading in MEC Networks

Mobile edge computing (MEC) is a promising paradigm to accommodate the increasingly prosperous delay-sensitive and computation-intensive applications in 5G systems. To achieve optimum computation performance in a dynamic MEC environment, mobile devices often need to make online decisions on whether to offload the computation tasks to nearby edge terminals under the uncertainty of future system information (e.g., random wireless channel gain and task arrivals). The design of an efficient online offloading algorithm is challenging. On one hand, the fast-varying edge environment requires frequently solving a hard combinatorial optimization problem where the integer offloading decision and continuous resource allocation variables are strongly coupled. On the other hand, the uncertainty of future system parameters makes it hard for the online decisions to satisfy long-term system constraints. To address these challenges, this article overviews the existing methods and introduces a novel framework that efficiently integrates model-based optimization and model-free learning techniques. We suggest some promising future research directions for online computation offloading control in MEC networks.

On arithmetic intersection numbers on self-products of curves

We give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω ^ 2 \hat {\omega }^2 of the dualizing sheaf of a curve in terms of Zhang’s invariant φ \varphi . As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.

On Wiener Polarity Index and Wiener Index of Certain Triangular Networks

A topological index of graph G is a numerical quantity which describes its topology. If it is applied to the molecular structure of chemical compounds, it reflects the theoretical properties of the chemical compounds. A number of topological indices have been introduced so far by different researchers. The Wiener index is one of the oldest molecular topological indices defined by Wiener. The Wiener index number reflects the index boiling points of alkane molecules. Quantitative structure activity relationships (QSAR) showed that they also describe other quantities including the parameters of its critical point, density, surface tension, viscosity of its liquid phase, and the van der Waals surface area of the molecule. The Wiener polarity index has been introduced by Wiener and known to be related to the cluster coefficient of networks. In this paper, the Wiener polarity index W p G and Wiener index W G of certain triangular networks are computed by using graph-theoretic analysis, combinatorial computing, and vertex-dividing technology.

Molecular Descriptors of Some Chemicals that Prevent COVID-19.

Topological index is a numerical molecular descriptor that plays an important role in structure-property/structure-activity modeling. A large number of works on multiplicative degree based indices have been developed. However, no attention is paid to investigating their chemical significance. Investigation of the chemical importance of such indices is needed. The computation of topological indices for different chemical structures and networks is a current topic of interest in mathematical chemistry. The objective of the present work is to examine the usefulness of the multiplicative degree based indices in quantitative structure property/activity relationship modeling. In addition, we intend to compute the indices for some anti-COVID-19 chemicals. The regression analysis for octane data set is performed using MATLAB and Excel to check the predictability of the indices. The sensitivity test is conducted to examine the isomer discrimination ability. To study the indices for chemical structures preventing COVID-19, different combinatorial computation methods are utilized. The regression models governing the structural dependence of different properties and activities are derived. The supremacy of the indices as useful molecular descriptors compared to some well-known and most used descriptors is established. Explicit expressions of the indices for hydroxychloroquine, remdesivir (GS-5734) and theaflavin are obtained. As the indices are shown to have remarkable efficiency in quantitative structure property/activity relationship modeling and isomer discrimination, the outcomes can predict different properties and activities of the chemicals under consideration.

Supervised deep learning prediction of the formation enthalpy of complex phases using a DFT database: The [formula omitted]phase as an example

Machine learning (ML) methods are becoming the state-of-the-art in numerous domains, including material sciences. In this manuscript, we demonstrate how ML can be used to efficiently predict several properties in solid-state chemistry applications, in particular, to estimate the heat of formation of a given complex crystallographic phase (here, the σ−phase, tP30, D8b). Based on an independent and unprecedented large first principles dataset containing about 10,000 σ−compounds with n=14 different elements, we used a supervised learning approach to predict all the ∼500,000 possible configurations. From a random set of ∼1000 samples, predictions are given within a mean absolute error of 23meVat−1 (∼2kJ mol−1) on the heat of formation and ∼0.06Å on the tetragonal cell parameters. We show that deep neural network regression results in a significant improvement in the accuracy of the predicted output compared to traditional regression techniques. We also integrated descriptors having physical nature (atomic radius, number of valence electrons), and we observe that they improve the model precision. We conclude from our numerical experiments that the learning database composed of the binary-compositions only, plays a major role in predicting the higher degree system configurations. Our results open a broad avenue to efficient high-throughput investigations of the combinatorial binary computations for multicomponent complex intermetallic phase prediction.

Dynamicallyelongated associative toehold for tuning DNA circuit kinetics and thermodynamics.

Associative toehold is a powerful concept enabling efficient combinatorial computation in DNA circuit. A longer association length boosts circuit kinetics and equilibrium signal but results in higher leak rate. We reconcile this trade-off by using a hairpin lock design to dynamically elongate the effective associative toehold length in response to the input target. Design guidelines were established to achieve robust elongation without incurring additional leakages. Three hairpin initiators with different combinations of elongated associative toehold (4 → 6 nt, 5 → 8 nt and 6 → 9 nt) were shortlisted from the design framework for further discussion. The circuit performance improved in terms of reaction kinetics, equilibrium signal generated and limit of detection. Overall, the elongated associative toehold served as a built-in function to stabilize and favour the forward, desired reaction when triggered.

On vertex PI index of certain triangular tessellation networks

Abstract The Wiener index, due to its many applications is considered to be one of very important distance-based index. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The PI index like other distance related indices has great disseminating power. The index was firstly investigated by Khadikar et al. (2001), they have probed the chemical applications of the PI index. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. Recently, the vertex Padmarkar-Ivan (PIv) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v. In this paper, the vertex PIv index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology.

Computing intersections of closed geodesics on the modular curve

In a recent work of Duke, Imamoḡlu, and Tóth, the linking number of certain links on the space SL(2,Z)\\SL(2,R) is investigated. This linking number has an alternative interpretation as the intersection number of closed geodesics on the modular curve, which is the focus of this paper. By relating the intersection number to a combinatorial computation involving rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. A formula for the total intersection of a pair of positive coprime fundamental discriminants is also derived, which can be thought of as a real quadratic analogue of a classical result of Gross and Zagier. The paper ends with numerical computations and distribution questions relating to intersection numbers.