Minimum Rank Research Articles

An upper bound on the minimum rank of a symmetric Toeplitz matrix completion problem

We consider a symmetric Toeplitz matrix completion problem, of which the matrix possesses special row and column structures. It has wide applications in diverse areas and is well-known to be computationally NP-hard. This note presents an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on conclusions from the trigonometric moment problem and the semi-infinite problem. We prove that the upper bound is less than twice the number of the active constraints of the associated semi-infinite problem. Moreover, it is less than twice the number of linear constraints of the problem.

Self-handicapping strategies in educational context: construction and validation of the Brazilian Self-Handicapping Strategies Scale (EEAPREJ)

Self-handicapping strategies refer to the set of choices and attitudes adopted to minimize blame for failure and increase the value of success in achievement situations. This paper aims to describe the stages of construction and the psychometric analysis of a scale to measure the self-handicapping strategies of university students. In study 1, the major steps for the construction of the scales and initial results are reported. The internal consistency indices were acceptable and the principal component analysis revealed factors with little explanation of data variance. In study 2, data from a sample of 834 students from several undergraduate courses of different Brazilian universities were subjected to exploratory factor analysis using the minimum rank factor analysis (MRFA) method and the matrix of polychoric correlations. The parallel analysis criterion for factor retention indicated the one-factor solution as the best fit to data. The importance of having a valid and reliable measure to assess self-handicapping strategies in educational contexts and the promising use of the scale in actions to improve learning in higher education are discussed.

Robustness of rank minimization heuristics for form-finding of tensegrity structures

The form-finding process for tensegrity structures with a given topology seeks a distribution of force densities that satisfies stability and equilibrium conditions. A rank minimization problem regarding the force density matrix can yield feasible distributions of force densities. A direct computation of the matrix rank is difficult and is generally avoided in practice. In a recent contribution by Wang [Engineering Structures 2021;227:111419], a use the trace heuristic for form-finding is proposed. The trace heuristic does not guarantee recovery of the minimal rank. Insufficient rank deficiency leads to a degenerated geometry of the tensegrity. In this contribution, the approach is extended to log-det heuristic and a comparison between both methods is carried out focusing on robustness of the form-finding procedure for tensegrities. Robustness of the form-finding procedure is understood here as the ability to recover at least one feasible tensegrity design. Several examples show that the robustness is mainly influenced by the topology of the structure, constraints on force densities and the admissible set for force densities. Furthermore, the proposed extension with log-det heuristic outperforms the original method in terms of robustness, however it increases the computational cost.

Subgraph Complementation and Minimum Rank

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2 (G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leqslant c_2 (G)\leqslant \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following are equivalent for any graph $G$ with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$ is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$ over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as described in which every vertex appears an even number of times; and iv. for every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is twice the minimum number of tricliques whose symmetric difference of edge sets is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of the order, size, and vertex cover number of $G$. Finally, we show that the class of graphs with $c_2(G)\leqslant k$ is hereditary and finitely defined. For odd $k$, the sets of minimal forbidden induced subgraphs are the same as those for the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set for $c_2(G) \leqslant 2$.

On the minimum semidefinite rank of signed graphs

The (real) minimum semidefinite rank of a signed graph is the minimum rank among all real symmetric positive semidefinite matrices associated to the graph and having the given sign pattern. We give a new lower bound for the minimum semidefinite rank of a signed multigraph and show it equals a new upper bound for signed complete multigraphs. This allows a complete characterization of signed multigraphs with minimum semidefinite rank two. We also determine the minimum semidefinite rank of all signed wheel graphs.

Proposed RPL routing protocol in the IoT applications

SummaryRouting protocol for low‐power and lossy networks (RPL) is a proactive routing protocol that automates the path development process using objective functions (OFs). The IETF defines two types of RPL objective functions, minimum rank with hysteresis objective function (MRHOF) and OF0. OF0 is based on the minimum hop count metric while the MRHOF is based on the expected transmission count (ETX) metric. Because of the single metric, both MRHOF and OF0 will suffer from long hops when selecting the routs to the sink, and select inefficient routes maybe paths contain nodes with small residual energy, which may cause consume the nod's energy faster than other nodes. The other problem of RPL is unbalanced chosen by parents that make bottleneck nodes that cause more network delay and high packet loss ratio because of the high congestion nodes, especially the nodes that locate near the sink node. In this article, we proposed an improvement RPL objective function that considers three metrics (load, residual energy, and ETX metrics) instead of one metric, to solve the MRHOF problems. The performance of the proposed protocol shows the enhancement in the RPL performance in terms of PDR, total energy consumption, and many other parameters.

Positive semidefinite solution to matrix completion problem and matrix approximation problem

In this paper, firstly, we discuss the following matrix completion problem in the spectral norm: ?(A B B* X)?2 < 1 subject to (A B B* X) ? 0. The feasible condition for the above problem is established, in this case, the general positive semidefinite solution and its minimum rank are presented. Secondly, applying the result of the above problem, we also study the matrix approximation problem: ?A ? BXB*?2 < 1 subject to A ? BXB* ? 0, where A ? Cm?m?, B ? Cm?n, and X ? Cn?n?.

Inertia Sets of Semicliqued Graphs

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.

Nonmonotonic reasoning from conditional knowledge bases with system W

For nonmonotonic reasoning in the context of a knowledge base mathcal {R} containing conditionals of the form If A then usually B, system P provides generally accepted axioms. Inference solely based on system P, however, is inherently skeptical because it coincides with reasoning that takes all ranking models of mathcal {R} into account. System Z uses only the unique minimal ranking model of mathcal {R}, and c-inference, realized via a complex constraint satisfaction problem, takes all c-representations of mathcal {R} into account. C-representations constitute the subset of all ranking models of mathcal {R} that are obtained by assigning non-negative integer impacts to each conditional in mathcal {R} and summing up, for every world, the impacts of all conditionals falsified by that world. While system Z and c-inference license in general different sets of desirable entailments, the first major objective of this article is to present system W. System W fully captures and strictly extends both system Z and c-inference. Moreover, system W can be represented by a single strict partial order on the worlds over the signature of mathcal {R}. We show that system W exhibits further inference properties worthwhile for nonmonotonic reasoning, like satisfying the axioms of system P, respecting conditional indifference, and avoiding the drowning problem. The other main goal of this article is to provide results on our investigations, underlying the development of system W, of upper and lower bounds that can be used to restrict the set of c-representations that have to be taken into account for realizing c-inference. We show that the upper bound of n − 1 is sufficient for capturing c-inference with respect to mathcal {R} having n conditionals if there is at least one world verifying all conditionals in mathcal {R}. In contrast to the previous conjecture that the number of conditionals in mathcal {R} is always sufficient, we prove that there are knowledge bases requiring an upper bound of 2n− 1, implying that there is no polynomial upper bound of the impacts assigned to the conditionals in mathcal {R} for fully capturing c-inference.

The treewidth of 2-section of hypergraphs

Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|f\cap g|\le 1$ for any $f,g\in F$ with $f\not=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,v\in V([H]_2)$, $uv\in E([H]_2)$ if and only if there is $ f\in F$ such that $u,v\in f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2\cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.

On reduction of moduli schemes of abelian varieties with definite quaternion multiplications

Dans cet article, nous faisons une étude initiale sur les espaces de modules de type D en caractéristique positive p≠2, où le nombre premier p peut être ramifié dans la donnée définissant l’espace de modules. Nous classifions explicitement les classes d’isogénie des groupes p-divisibles avec structures supplémentaires en question. Nous étudions également la réduction des espaces de modules de type D de rang minimal.

Energy‐efficient fog computing in Internet of Things based on Routing Protocol for Low‐Power and Lossy Network with Contiki

SummaryThe traditional centralized cloud computing (CC) model faces a range of problems with the exponential growth of the Internet of Things (IoT) applications, like high latency, reduced bandwidth, and network instability. Fog computing (FC) takes the cloud closer to IoT computers to overcome these problems. Rather than moving them to the cloud, the FC provides local IoT data processing and storage on IoT computers. This paper focuses on the Routing Protocol for Low‐Power and Lossy Network (RPL), a universally routing protocol for a static environment that is used for reducing energy consumption, delay, and packet loss with data aggregation at the border router using a fog simulation model in Contiki Cooja. The objective function (OF) choice has an impact on network topology, as each node selects a set of potential parents to send to the destination. However, no systematic analysis of the effects of OF behavior in the RPL environment has been undertaken. Here, three different OFs, Objective Function Zero (OF0), Advanced Objective Function Zero (AOF0), and Minimum Rank with Hysteresis Objective Function (MRHOF) for RPL in the static environment for different node numbers, have been compared. The findings demonstrate that altering all three OFs has a significant impact on RPL. The energy consumption is reduced in the case of the AOF0 in the fog node by 50.86%, which is less than the case of the OF0 and MRHOF function. Extensive simulations show that AOF0 outperforms the existing OFs.

Development of a creativity orientation scale using EFA

In a pilot study, an exploratory factor analysis using a minimum rank factor extraction method and an oblique (Promin) rotation of 30 self-report items on a proposed Creativity Orientation Scale (COS) was conducted on a snowball sample (n = 237). The purpose was to gain more in-depth understanding of attitudes toward and perceptions of creativity and creative individuals within the framework of social identity theory. Using the optimal implementation of Parallel Analysis (PA) retention method, a three-factor solution provided the clearest extraction. Factor 1 (creative-averse orientation) accounted for 37.32% of the variance and had seven items. Factor 2 (creative-approach orientation) accounted for 18.76% of the variance and had eight items. Factor 3 accounted for 8.31% of the variance and had only two items and therefore was not considered salient. Following rotation, these three items accounted for 64.40% of the total variance.

Assessment of Participation within the International Classification of Functioning, Disability, and Health (ICF): The Turkish Validity and Reliability of the Participation Scale.

Objective The aim of the study was to investigate the reliability and validity of the Turkish version of the Participation Scale (P-Scale). Methods A total of 152 students, with a mean age of 20.27 ± 2.19, participated in the study. Sociodemographic information (age, gender, and family income) was recorded; all participants completed the P-Scale twice with a 15-day interval. Translation and cross-cultural adaptation were performed to analyze the validity and reliability of the P-Scale. Cronbach's alpha and McDonald's alpha were used for scale reliability statistics and explanatory (EFA) and confirmatory (CFA) factor analysis; Mardia's multivariate normality and assumption tests were used for the validity of the scale. The factor extraction methods were minimum rank factor analysis in EFA and weighted least squares mean and variance adjusted estimator polychoric correlation matrix in CFA. Results Internal consistency of the scale was found good with Cronbach's alpha (0.852) and excellent with McDonald's alpha (0.924). The EFA and CFA resulted in two-factored structure, with the explained variance found to be higher than 30%. Conclusions Analysis demonstrated that the P-Scale had a satisfactory level of reliability and validity in Turkish university students.

Uniform forcing and immune sets in graphs and hypergraphs

Zero forcing is an iterative coloring process on a graph that has been widely used in such different areas as the modeling of propagation phenomena in networks and the study of minimum rank problems in matrices and graphs. This paper deals with zero forcing on hypergraphs. (Representing a network by a hypergraph allows us to account for its community structure and for more general interactions between different subsets of nodes.)We consider two natural generalizations to hypergraphs of zero forcing on graphs (one of them already known) and, for each one of these generalizations we look into two clutters that play a significant role in the forcing process: the clutter of minimal forcing sets and the one of minimal immune sets. A formulation of immune sets in terms of neighborhoods (hence without making reference to the iterative zero forcing process) is presented, highlighting the different behavior of the distinct forcing rules. Moreover, we obtain the families of minimal forcing and minimal immune sets in the case of complete hypergraphs and we provide a full characterization of forcing and immune uniform clutters, both in the graph and in the hypergraph case.

Approximate tensor decompositions: Disappearance of many separations

It is well known that tensor decompositions show separations, that is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here, we show that many of these separations disappear in the approximate case. Specifically, for every approximation error ɛ and norm, we define the approximate rank as the minimum rank of an element in the ɛ-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten p-norms. For non-negative tensors, we show that the separations between rank, positive semidefinite rank, and non-negative rank disappear for all ℓp-norms with p > 1. For the trace norm (p = 1), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of the Carathéodory theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.

Performance Evaluation of IPv6 Routing Protocol for Low Power and Lossy Networks Based on Objective Functions for Media Technology

The Internet Engineering Task Force (IETF) has proposed the use of the IPv6 Routing Protocol for the Low-Power and Lossy Networks (RPL). RPL has been seen to be a major routing protocol related to the Internet of Things (IoT) that could be used for wireless networks with lower power consumption and liable to a higher packet loss. The RPL construct-related network topology uses the Objective Functions (OF) for choosing the route from the source to a destination node, which has been based on constraints and metrics. The IETF designed two OFs for the RPL protocol, i.e., Objective Function Zero (OF0) and Minimum Rank with Hysteresis Objective Function (MRHOF). In this study, the researchers have investigated the RPL performance with the help of the OFs in a medium density. This performance has been judged using different network parameters like Packet Delivery Ratio (PDR) and Power Consumption (PC). Furthermore, in this study, the number of nodes has been maintained as 40 and 50. The RX ratio has varied between 20, 60, and 100%, in the grid, and random topologies. The results of the study have also indicated that the OF0 protocol has showed a better performance concerning the power consumption compared to the MRHOF, while both the OFs have showed a similar PDR performance.

Minimal rank completions for overlapping blocks

We consider the multi-objective optimization problem of choosing the bottom left block-entry of a block lower triangular matrix to minimize the ranks of all block sub-matrices. We provide a proof that there exists a simultaneous rank-minimizer by constructing the complete set of all minimizers.

A rank question for homogeneous polynomials

A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 > 4 n+1>4 that, while not sharp, is non-trivial.

Swap dynamics in single-peaked housing markets

This paper focuses on the problem of fairly and efficiently allocating resources to agents. We consider a specific setting, usually referred to as a housing market, where each agent must receive exactly one resource (and initially owns one). In this framework, in the domain of linear preferences, the Top Trading Cycle (TTC) algorithm is the only procedure satisfying Pareto-optimality, individual rationality and strategy-proofness. Under the restriction of single-peaked preferences, Crawler enjoys the same properties. These two centralized procedures might however involve long trading cycles. In this paper we focus instead on procedures involving the shortest cycles: bilateral swap-deals. In such swap dynamics, the agents perform pairwise mutually improving deals until reaching a swap-stable allocation (no improving swap-deal is possible). We prove that in the single-peaked domain every swap-stable allocation is Pareto-optimal, showing the efficiency of the swap dynamics. In fact, this domain turns out to be maximal when it comes to guaranteeing this property. Besides, both the outcome of TTC and Crawler can always be reached by sequences of swaps. However, some Pareto-optimal allocations are not reachable through improving swap-deals. We further analyze the outcome of swap dynamics through social welfare notions, in our context the average or minimum rank of the resources obtained by agents in the final allocation. We start by providing a worst-case analysis of these procedures. Finally, we present an extensive experimental study in which different versions of swap dynamics are compared to other existing allocation procedures. We show that they exhibit good results on average in this domain, under different cultures for generating synthetic data.