Totally Unimodular Research Articles

New cases of integrable ninth-order conservative and dissipative dynamical systems

New cases of integrable dynamical systems of the ninth order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Extended Special Linear group ESL2(F) and matrix equations in SL2(F), SL2(Z) and GL2(Fp)

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

Polynomials Counting Nowhere-Zero Chains Associated with Homomorphisms

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ZE orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let Aψ[N] be the set of vectors g from (A−0)E, such that ∑e∈Eg(e)·f(e)=ψ(f) for each f∈N (where · is a scalar multiplication). We show that |Aψ[N]| can be evaluated by a polynomial function of |A|. In particular, if ψ(f)=0 for each f∈N, then the corresponding assigning polynomial is the classical characteristic polynomial of M.

Reliable Frequency Regulation Through Vehicle-to-Grid: Encoding Legislation with Robust Constraints

Problem definition: Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to electricity grids. We formulate a robust optimization problem that maximizes a vehicle owner’s expected profit from selling primary frequency regulation to the grid and guarantees that market commitments are met at all times for all frequency deviation trajectories in a functional uncertainty set that encodes applicable legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem nonconvex. Methodology/results: By exploiting a total unimodularity property of the uncertainty set and an exact linear decision rule reformulation, we prove that this nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. Managerial implications: We find that the prevailing penalties for nondelivery of promised regulation power are too low to incentivize vehicle owners to honor the delivery guarantees given to grid operators. Funding: This work was supported by the Institut Vedecom. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0154 .

A Lagrangian relaxation algorithm for stochastic fixed interval scheduling problem with non-identical machines and job classes

This paper deals with operational fixed interval scheduling problems under uncertainty caused by random delays. This stochastic programming problem has a deterministic reformulation based on network flow under the assumption that the machines are identical and the multivariate distribution of random delays follows an Archimedean copula. In this paper, we focus on the problem with heterogeneous machines and jobs classes. The deterministic problem with no delays and more than one machine type has been shown to be NP-hard. We generalize the deterministic reformulation of the stochastic problem with random delays for non-identical (heterogeneous) machines. This formulation for more than one machine type loses an important property of total unimodularity that holds for identical machines reformulation using the network flow problem. Therefore, an algorithm based on the Lagrangian relaxation is derived and implemented together with an efficient upper bounding heuristic. Its efficiency is confirmed by solving a large number of simulated instances.

Total Unimodularity and Strongly Polynomial Solvability of Constrained Minimum Input Selections for Structural Controllability: An LP-Based Method

This paper investigates several cost-sparsity induced optimal input selection problems for structured systems. Given an autonomous system and a prescribed set of input links, where each input link has a non-negative cost, the problems include selecting the minimum cost of input links and selecting the input links with the smallest possible cost while bounding their cardinality to achieve system structural controllability. Current studies show that in the dedicated input case (i.e., each input can actuate only one state variable), the former problem is polynomially solvable by some graph-theoretic algorithms, while the general nontrivial constrained case is largely unexplored. We show that these problems can be formulated as equivalent integer linear programming (ILP) problems. Subject to the 'source strongly-connected component grouped input constraint', which contains the dedicated input one as a special case, we demonstrate that the constraint matrices of these ILPs are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">totally unimodular</i> . This property allows us to solve these ILPs efficiently via their linear programming (LP) relaxations, leading to a unifying algebraic method for these problems with polynomial time complexity. We further show that these problems can be solved in strongly polynomial time, independent of the size of the costs and cardinality bounds. Finally, we provide an example to illustrate the power of the proposed method.

Optimizing Path-Planning Solutions Obtained by Using Petri Nets Models

This paper addresses the multirobot path planning problem using Petri net (PN) models, integrating insights from linear integer programming with a focus on totally unimodular matrices. Initially, we establish theoretical foundations linking optimal solutions of linear integer problems to their relaxed counterparts. Subsequently, we adapt these findings to robot path planning. The core contribution lies in the introduction of an innovative algorithm designed to iteratively converge on the optimal solution by solving successive continuous linear problems. A key feature of this algorithm is its proven efficiency, which converges to the optimal solution of the integer problem within a theoretically established finite number of iterations. Empirical validation is achieved through the resolution of 300 different path planning scenarios, which include varying numbers of robots, maps, and missions. These cases, solved using both the proposed method and existing techniques, demonstrate the superiority of the proposed method in performance.

INVARIANTS OF FIVE-ORDER HOMOGENEOUS DYNAMICAL SYSTEMS WITH DISSIPATION

New cases of integrable dynamical systems of the fifth order homogeneous in terms of variables are obtained, in which a system on a tangent bundle to a two-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

Hard problems on box-totally dual integral polyhedra

In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. (2008) who proved that deciding whether a given system is box-TDI is co-NP-complete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete.To derive these complexity results, we exhibit new classes of totally equimodular matrices – a generalization of totally unimodular matrices – by characterizing the total equimodularity of incidence matrices of graphs.

On polynomially solvable constrained input selections for fixed and switched linear structured systems

This paper investigates two related optimal input selection problems for fixed (non-switched) and switched structured (or structural) systems. More precisely, we consider selecting the minimum cost of inputs from a prior set of inputs, and selecting the inputs of the smallest possible cost with a bound on their cardinality, all to ensure system structural controllability. Those problems have attracted much attention recently; unfortunately, they are NP-hard in general. In this paper, it is found that, if the input structure satisfies certain ’regularizations’, which are characterized by the proposed restricted total unimodularity notion, those problems can be solvable in polynomial time via linear programming (LP) relaxations. Particularly, the obtained characterizations depend only on the incidence matrix relating the inputs and the source strongly connected components (SCC) of the system structure, irrespective of how the inputs actuate states within the same SCC. They cover all the currently known polynomially solvable cases (such as the dedicated input case), and contain many new cases unexplored in the past, among which the source-SCC separated input (SSSI) constraint is highlighted. Further, these results are extended to switched systems, and a polynomially solvable condition, namely the joint SSSI constraint, is obtained that does not require each of the subsystems to satisfy the SSSI constraint. We achieve these by first formulating those problems as equivalent integer linear programmings (ILPs), and then proving the total unimodularity of the corresponding constraint matrices. We also study solutions obtained via LP-relaxation and LP-rounding in the general case, resulting in some lower and upper bounds. Several examples are given to illustrate the obtained theoretical results.

Congruency-Constrained TU Problems Beyond the Bimodular Case

A long-standing open question in integer programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs [Formula: see text] with a totally unimodular constraint matrix T. Such problems are shown to be polynomial-time solvable for m = 2, which led to an efficient algorithm for integer programs with bimodular constraint matrices, that is, full-rank matrices whose n × n subdeterminants are bounded by two in absolute value. Whereas these advances heavily rely on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, that is, for m &gt; 2. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m = 3 using a randomized algorithm. Furthermore, for general m, our techniques also allow for identifying flat directions of infeasible problems and deducing bounds on the proximity between solutions of the problem and its relaxation. Funding: This project received funding from the Swiss National Science Foundation [Grants 200021_184622 and P500PT_206742], the European Research Council under the European Union’s Horizon 2020 research and innovation program [Grant 817750], and the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy–GZ 2047/1 [Grant 390685813].

An intrinsic Bayesian bound for estimators on the Lie groups [formula omitted] and [formula omitted

In the past decade, estimation on manifolds has been paid an increased attention for making it possible to intrinsically account for constraints on the unknown parameters. Applications are widespread from computer vision to target tracking. To evaluate the performance of the algorithms, it is of interest to derive bounds on the estimation error. This issue has been mostly investigated for parametric estimation on Riemannian manifolds. This paper addresses the problem of computing a Bayesian lower bound on the norm of the estimation error when both the unknown variables and the observations belong to specific manifolds called Lie groups (LGs). For that purpose, a metric is considered that preserves the geometric properties of the latter. We firstly derive an inequality on the correlation matrix of this intrinsic error under the mild assumption of unimodular matrix LGs. Then, we obtain a closed-form expression of the bound for the special orthogonal group SO(3) and the special Euclidean group SE(3). Finally, we detail its derivation in the case when both the prior distribution and the likelihood are concentrated Gaussian distributions. The proposed Bayesian bound is implemented and tested on two estimation problems involving both unknown parameters and observations on SE(3): the inference of the pose of a camera and that of the centroid of a cluster of space debris.

Irreducibility of Binomials

In this paper, we prove that the family of binomials $x_1^{a_1} \cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1, \ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying the connection between the irreducibility of a binomial in ${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix $U_i$ with integer entries such that $(a_1, \ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots, m+n\}$, where $\be_i$ is the standard basis vector.

Odd-Order Integrable Dynamical Systems with Dissipation

In this paper, we prove the integrability of some classes of odd-order dynamical systems (namely, systems of order 3, 5, and 7), which are homogeneous in some variables and contain a system on the tangent bundle of a smooth manifolds. In this case, we separate force fields into internal (conservative) and external, which has sign-alternating dissipation. External fields are introduced by using some unimodular transformations and generalize fields considered earlier.

Linearizations of matrix polynomials viewed as Rosenbrock's system matrices

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP.

The Audio Encryption Approach uses a Unimodular Matrix and a Logistic Function

The encryption algorithm is very important to guard the confidentiality of audio information. One of the most effective audio encryption algorithms is to use a unimodular matrix and Bernoulli function. In this research, we develop an audio encryption algorithm that uses a unimodular matrix and a Bernoulli function. This algorithm is implemented using the language Python programming. The audio data encrypted using this algorithm is capable produce audio data that is secure and difficult for unauthorized parties to decipher authorized. In addition, this algorithm also has good performance in terms of speed and efficiency.

Asymptotic Forms of Solutions to System of Nonlinear Partial Differential Equations

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 the 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 eight asymptotic forms, presented explicitly.

Unimodular Matrix On Compound Mathematical Structures Applications and Results

Unimodular Matrix On Compound Mathematical Structures Applications and Results

COMPUTING UNIMODULAR MATRICES OF POWER TRANSFORMATIONS

An algorithm for solving the following problem is described. Let m n integer vectors in the n-dimensional real space be given. Their linear span forms a linear subspace L in R. It is required to find a unimodular matrix such that the linear transformation defined by it takes the subspace L into a coordinate subspace. Computer programs implementing the proposed algorithms and the power transforms for which they are designed are described.

Stable matching: An integer programming approach

This paper develops an integer programming approach to two‐sided many‐to‐one matching by investigating stable integral matchings of a fictitious market where each worker is divisible. We show that a stable matching exists in a discrete matching market when the firms' preference profile satisfies a total unimodularity condition that is compatible with various forms of complementarities. We provide a class of firms' preference profiles that satisfy this condition.