Totally Unimodular Research Articles

Polynomial Upper Bounds on the Number of Differing Columns of Δ-Modular Integer Programs

We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IPs) with bounded determinants. For example, an IP can be solved in strongly polynomial time if the constraint matrix is bimodular: that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the [Formula: see text] distance between IP solutions and solutions of its linear relaxation. One of the first to quantify the complexity of IPs with bounded determinants was Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller’s bound to general determinants has been superpolynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first [Formula: see text] distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest. Funding: J. Lee was supported in part by the Office of Naval Research [Grant N00014-21-1-2135] and the Air Force Office of Scientific Research [Grant FA9550-19-1-0175]. J. Paat was supported by a Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant [Grant RGPIN-2021-02475].

Two-Zero Textures Based on A4 Symmetry and Unimodular Mixing Matrix

We propose a phenomenological model of two-zeros Majorana neutrino mass matrix based on the A4 symmetry, where the structure of mixing matrix is a unimodular second scheme of trimaximal TM2, and the charged lepton mass matrix is diagonal. We show that, among seven possible two-zero textures with A4 symmetry, only two textures, namely the texture with Mee=0 and Meμ=0 and its permutation, are acceptable in the non-perturbation method, since the results associated with these two textures are consistent with the experimental data. We obtain a unique relation between our phases, namely ρ+σ=ϕ±π, and an effective equation sin2θ13=23Rν where Rν=δm2Δm2. Then, only by using the experimental ranges of Rν, we obtain the allowable range of the unknown parameter ϕ as the phase of TM2 mixing matrix, which leads to obtaining not only the ranges of all neutrino oscillation parameters of the model (which agree well with experimental data) but also with the masses of neutrinos, the Dirac and Majorana phases and the Jarlskog parameter, and to predict the normal neutrino mass hierarchy. Finally, we show that all the predictions regarding our two specific textures agree with the corresponding data reported from neutrino oscillation, cosmic microwave background and neutrinoless double beta decay.

Phase Retrieval from Linear Canonical Transforms

The classical phase retrieval problem aims to recover an unknown function from the Fourier magnitudes. The linear canonical transform has a more generalized form of the well-known (fractional) Fourier transform and a wide range of engineering applications such as optics and quantum mechanism. In this paper, we consider the linear canonic phase retrieval problem of determining a function from the magnitudes of the linear canonic transforms. We show that a compactly supported function f can be determined, up to a global phase, from the magnitudes of multiple linear canonic transforms, where is a class of real unimodular matrices. It generalizes the results of phase retrieval from multiple fractional Fourier transforms. On the other hand, we show that a compactly supported function f can be determined, up to a global phase, from the interference linear canonic magnitudes and where Moreover, if the ambiguity of conjugate reflection is taken into account, the compactly supported function f can be determined, up to a rotation and conjugate reflection, from the linear canonic magnitudes and

A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A, that is, unimodular matrices U and V such that AV=US, with S the Smith normal form of A. The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as A. Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of V, the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear system solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.

Dispersive waves in magneto-electro-elastic periodic waveguides

Magneto-electro-elastic (MEE) materials are attracting an increasing amount of scientific interest in physics, for their valuable potential in the functional management of mechanical-to-magnetoelectric energy conversions. The present paper focuses on heterogeneous MEE media – artfully realized by virtue of the periodic repetition of a multi-phase elementary cell – which represent promising solutions to passively control the propagation of multifield Bloch waves. Specifically, the linear continuum model governing the coupled free dynamics of a cellular non-dissipative MEE material is formulated for the unbounded three-dimensional domain. Then, the governing equations are specialized for a periodic layered waveguide with perfect inter-layer interfaces. The Floquet–Bloch decomposition is employed to analytically determine the uni-modular transfer matrix of the heterogeneous multi-layered cell. The related eigenproblem, governed by a palindromic characteristic polynomial in the Floquet multipliers, is solved in closed form to analyze the dispersion properties in the real-valued frequency domain and complex-valued wavenumber domain. Alternatively, a perturbation scheme is outlined to asymptotically approximate the exact eigensolutions. The dispersion spectra, composed of propagation and attenuation branches, are parametrically investigated for a two-layered periodic waveguide. The corresponding band structures are assessable a priori by leveraging the formal analogy with the stability analysis for non-autonomous dynamical systems. As main findings, slow-propagating quasi-shear-elastic waves and fast-propagating quasi-magneto-electric waves are found to dominate the low-frequency and high-frequency ranges, respectively, although some crossing points between the respective spectral branches can be detected. Quantitatively different but qualitatively similar behaviors of propagation and/or attenuation are observed for the quasi-shear-elastic waves and quasi-magneto-electric waves. Consequently, pass–pass, pass–stop and stop–stop bands of frequencies can be recognized and tuned by varying the key physical parameters. This rich spectral scenario opens the way for the functional customization of MEE-based metafilters or metapropagators, purposely designed to govern the transfer, localization, conversion, harvesting of energy across large frequency ranges.

Polynomial Time Reachability Analysis in Discrete State Chemical Reaction Networks Obeying Conservation Laws

In this paper the reachability problem of discrete state Chemical Reaction Networks (d-CRNs) is studied. We consider sub-classes of sub-and superconservative d-CRN network structures and prove that the reachability relation can be decided in polynomial time. We make use of the result that in the studied d-CRN sub-classes, the reachability relation is equivalent to the existence of a non-negative integer solution of the d-CRN state equation. The equivalence implies the reformulation of the reachability problem as integer linear programming decision problem. We show that in the studied classes of d-CRN structures, the state equation has a totally unimodular coefficient matrix. As the reachability relation is equivalent to the non-negative integer solution of the state equation, the resulting integer programming decision program can be relaxed to a simple linear program having polynomial time complexity. Hence, in the studied sub-classes of sub and superconservative reaction network structures, the reachability relation can be decided in polynomial time and the number of continuous decision variables is equal to the number of reactions of the d-CRN.

Counting basis extensions in a lattice

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T T , producing an asymptotic estimate as T → ∞ T \to \infty . This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2 2 -dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in R n \mathbb R^n . Our main result on counting basis extensions also generalizes to arbitrary lattices in R n \mathbb R^n . Finally, we establish some basic properties of sparse representations of integers by multilinear forms.

Optimizing and Factorizing the Wilson Matrix

The Wilson matrix, W, is a 4 × 4 unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940’s, owing to its mild ill-conditioning. We ask how close W is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations with Z having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that W has an integer factor Z and two rational factors, up to signed permutations. This little matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results.

Exploiting complete linear descriptions for decentralized power market problems with integralities

It is well known that linear prices supporting a competitive equilibrium exist in the case of convex markets, however, in the presence of integralities this is open and hard to decide in general. We present necessary and sufficient conditions for the existence of such prices for decentralized market problems where market participants have integral decision variables and their feasible sets are given in complete linear description. We utilize total unimodularity and the aforementioned conditions to show that such linear prices exist and present some applications. Furthermore, we compute competitive equilibria for two classes of decentralized market problems arising in energy markets and show that competitive equilibria may exist regardless of integralities.

Unimodular matrix on shallow water wave theory. Unimodularity through matrix method

Unimodular matrix on shallow water wave theory. Unimodularity through matrix method

Вычисление унимодулярных матриц

Here we give an algorithm for solving the following problem. Let m<n integer vectors be given in the n-dimensional real space. Their linear span forms a linear subspace L in R<sup>n</sup>. It is required to calculate such an unimodular matrix that a linear transformation with it transforms the subspace L into a coordinate one. Also, programs that implement the algorithms and power transformations, for which they are needed, are given.

Novel Secure Outsourcing of Modular Inversion for Arbitrary and Variable Modulus

In cryptography and algorithmic number theory, modular inversion is viewed as one of the most common and time-consuming operations. It is hard to be directly accomplished on resource-constrained clients (e.g., mobile devices and IC cards) since modular inversion involves a great amount of operations on large numbers in practice. To address the above problem, this paper proposes a novel unimodular matrix transformation technique to realize secure outsourcing of modular inversion. This technique makes our algorithm achieve several amazing properties. First, to the best of our knowledge, it is the first secure outsourcing computation algorithm that supports arbitrary and variable modulus, which eliminates the restriction in previous work that the protected modulus has to be a fixed composite number. Second, our algorithm is based on the single untrusted program model, which avoids the non-collusion assumption between multiple servers. Third, for each given instance of modular inversion, it only needs one round interaction between the client and the cloud server, and enables the client to verify the correctness of the results returned from the cloud server with the (optimal) probability 1. Furthermore, we propose an extended secure outsourcing algorithm that can solve modular inversion in multi-variable case. Theoretical analysis and experimental results show that our proposed algorithms achieve remarkable local-client’s computational savings. At last, as two important and helpful applications of our algorithms, the outsourced implementations of the key generation of RSA algorithm and the Chinese Reminder Theorem are given.

Вычисление асимптотик решений системы нелинейных уравнений в частных производных

Here we considerably develop the methods of power geometry for a system of partial differential equations and apply them to two different fluid dynamics problems: computing the boundary layer on a needle in the first approximation and computing the asymptotic forms of solutions to problem of evolution of the turbulent flow. For each equation of the system, its Newton polyhedron and its hyperfaces with their normals and truncated equations are calculated. To simplify the truncated systems, power-logarithmic transformations are used and the truncated systems are further extracted. Here we propose algorithms for computing unimodular matrices of power transformations for differential equations. Results: 1) the boundary layer on the needle is absent in liquid, while in gas it is described in the first approximation; 2) the solutions to the problem of evolution of turbulent flow have 8 asymptotic forms presented explicitly.

Trilevel Mixed Integer Optimization for Day-Ahead Spinning Reserve Management of Electric Vehicle Aggregator With Uncertainty

This paper studies a trilevel profit maximization problem of electric vehicle (EV) aggregator participating in the day-ahead reserve market, considering the uncertain EV connectivity to the grid. At the upper level (UL), the aggregator purchases reserve from individual EVs and trades it in the reserve market. It determines the optimal reserve purchasing price and reserve trading amount to maximize profit. Responding to the reserve purchasing price, at the middle level (ML), each EV owner maximizes his/her own utility by scheduling the battery usage for charging/discharging, reserve or transportation. As the connectivity of EV to the grid is uncertain due to transportation randomness, we characterize the worst-case connectivity at the lower level (LL) such that energy consumption for transportation tasks can be guaranteed. The proposed trilevel optimization problem is challenging because of its multi-level structure and binary variables at ML and LL. Firstly, total unimodularity property, primal-dual and value-function methods are used to convert this problem into a single-level mixed integer nonlinear program (MINLP). Then, a sample-based algorithm is developed to solve the single-level MINLP and the convergence is proved. In addition, an acceleration strategy is proposed to facilitate the computation. Case studies validate the effectiveness of our proposed solution method.

Unimodular matrix and bernoulli map on text encryption algorithm using python

One of the encryption algorithms is the Hill Cipher. The square key matrix in the Hill Cipher method must have an inverse modulo. The unimodular matrix is one of the few matrices that must have an inverse. A unimodular matrix can be utilized as a key in the encryption process. This research aims to demonstrate that there is another approach to protect text message data. Symmetric cryptography is the sort of encryption utilized. A Bernoulli Map is used to create a unimodular matrix. To begin, the researchers use an identity matrix to generate a unimodular matrix. The Bernoulli Map series of real values in (0,1) is translated to integers between 0 and 255. The numbers are then inserted into the unimodular matrix's top triangular entries. To acquire the full matrix as the key, the researchers utilize Elementary Row Operations. The data is then encrypted using modulo matrix multiplication.

A characterization of circle graphs in terms of total unimodularity

A graph G has an associated multimatroid Z3(G), which is equivalent to the isotropic system of G studied by Bouchet. In previous work it was shown that G is a circle graph if and only if for every field F, the rank function of Z3(G) can be extended to the rank function of an F-representable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this total-unimodularity analogue.

Cost Allocation for Less-Than-Truckload Collaboration via Shipper Consortium

We study the problem of collaborative less-than-truckload (LTL) transportation in the form of a shipper consortium, which is operated by a third-party logistics provider (3PL) through a cross-dock/pooling network. The 3PL has responsibility for planning the combined loads prior to actual shipments, hiring and routing carriers to execute shipping, and allocating the cost to the shippers in the consortium. Shippers receive substantial cost savings from combined truckload shipments. However, achieving consolidation and realizing this benefit requires addressing two essential issues: (i) how to find an approximately optimal consolidation solution in a large network with many freights and (ii) how to determine a fair cost allocation rule among the shippers’ consolidated freights that ensures budget balance while minimally violating coalitional stability. Our work resolves these two issues. We formulate a time-space network flow model of the problem under both incremental and all-unit discount structures of LTL rates and propose a computationally efficient algorithm based on local search heuristics. We model the problem of allocating cost to the shippers as a cooperative game. We decompose and linearize the Lagrangian dual problem by using total unimodularity and concavity. We propose an efficiently computable cost allocation rule from the linearized dual models. The dual rule ensures stable cooperation but may have underallocation equal to a duality gap. To cover the underallocation, we further develop a budget covering procedure and define an [Formula: see text]-core allocation with desirable properties. Through extensive computational experiments, we find that the shipper consortium reduces total shipping costs by more than 40% in most cases; meanwhile, the [Formula: see text]-core allocation is typically in the core for small-scale networks while violating stability by at most 5% for large-scale networks and provides consolidated freights with more than 50% individual cost savings on average.

A Discrete Convex Min-Max Formula for Box-TDI Polyhedra

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which the minimum is taken over the set of integral elements of a box total dual integral polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.

Notes on {a,b,c}-Modular Matrices

Let A in mathbb {Z}^{m times n} be an integral matrix and a, b, c in mathbb {Z} satisfy a ≥ b ≥ c ≥ 0. The question is to recognize whether A is {a,b,c}-modular, i.e., whether the set of n × n subdeterminants of A in absolute value is {a,b,c}. We will succeed in solving this problem in polynomial time unless A possesses a duplicative relation, that is, A has nonzero n × n subdeterminants k1 and k2 satisfying 2 ⋅|k1| = |k2|. This is an extension of the well-known recognition algorithm for totally unimodular matrices. As a consequence of our analysis, we present a polynomial time algorithm to solve integer programs in standard form over {a,b,c}-modular constraint matrices for any constants a, b and c.

Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.