System Of Linear Constraints Research Articles

Simplicial techniques for operator solutions of linear constraint systems

A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Zd.

Linear constraints on variables in influence diagrams for causal models

The paper considers some ways of representing probabilistic causal models using Bayesian network theory (hereafter referred to as BN). These models describe well problems with different types of uncertainty. The theory of BN extended with some additional types of nodes called influence diagrams or IDs. Influence diagrams make it possible to consider a number of solution options, to evaluate them quantitatively, and to select the best of the considered options. However, it is practically impossible to find an optimal solution in an ID. It is not even possible to create a system of linear constraints on some variables in an ID, although there is a large class of practical problems with such constraints. The paper describes the idea of extending ID to describe linear constraints on some variables of the BN. In the future, it will help to use the ideas of linear programming in ID to find an optimal solution in the sense of LP for problems with different types of uncertainties and causal relationships between some variables. This work has been done under grant AP19679142 "Search for optimal solutions in Bayesian networks in models with linear constraints and linear functionals. Development of algorithms and programs " (2023-2025) of MES RK. This project will develop the theory for finding optimal solutions in Bayesian networks. Optimality will be understood in the sense of linear programming - a system of linear constraints, extremum of a linear functional. The theory will be implemented in a software product.

Constrained read-once refutations in UTVPI constraint systems: A parallel perspective

AbstractIn this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$ , where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A \cdot x \le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.

Discrete optimizations’ problems of deliveries of heterogeneous products

The article touches upon issues related to the tasks of the national project «Smart City». The paper analyzes the problem of supplying a conditional consumer with heterogeneous products in accordance with his demand for deliveries from several suppliers in a situation with fixed surcharges, in addition to the cost of purchasing each unit of production. For the situation under study, a model of a reduced transport type with a discontinuous piecewise linear objective function of minimized total costs and with a system of linear constraints is constructed. A method for finding the optimal solution one of the many such solutions, based on the ideas of the Hungarian algorithm is proposed, the justification of which is given on the basis of the corresponding lemma. A refinement of the method is presented in the presence of some restrictions related to suppliers. The polynomial complexity of the method, i.e. the problem is quickly solvable, and the significant limitation of the applicability of the method within the framework of the model are noted. Further possible research directions of a stochastic or fuzzy nature are indicated.

Configuration design of the emission control areas for coastal ships: A Stackelberg game model

Several countries and regions have announced sulphur emission control areas (SECAs) to regulate sulphur emissions from ships. Incorporating environmental benefits and economic impacts, a bi-level programming model following the Stackelberg game regime is proposed to capture the SECA design problem for a given coastal area, in which the irregular polygon of the SECA is assumed to be a convex hull and described by a system of linear constraints. The government, as the leader, minimizes total sulphur emissions of ships over the entire coastal area with the voyage cost constraint. The ships, as followers, minimize individual voyage costs by jointly choosing the sailing trajectories and speed profile inside and outside the SECA between the associated port pairs with the given geometric configuration of the SECA. We first derive the optimal trajectory and speed profile for each ship, which can be simplified to a shortest path problem. Subsequently, a heuristic algorithm based on the steepest descent method is proposed to obtain the Pareto optimal solution in the sense of simultaneous minimization of total emissions and voyage costs. Finally, two announced SECAs are adopted to illustrate the validity of the proposed model and algorithm.

Two-Stage Transportation Problem with Unknown Consumer Demands

The work investigates a mathematical model of a two-stage transportation problem for finding the most economical plan for the transportation of homogeneous products from suppliers to consumers, where the demands of consumers are unknown, taking into account constraints on their lower and upper bounds. It is an extension of the classic two-stage transportation problem, where products are transported from suppliers to consumers only through intermediate points. Intermediary firms and various storage facilities (warehouses) can be such intermediate points.The relationship of the developed mathematical model with the two-stage continuous-discrete problem of optimal partitioning-distribution, which is characterized by the presence of two stages, is investigated. The problem consists in determining the areas of collection of the continuously distributed resource (raw material) by enterprises of the first stage and the volumes of transportation of the processed product from the enterprises of the first stage to consumers (points of the second stage), in order to minimize the total costs of transportation of the resource from suppliers to consumers through processing points (collection points, storage points).The material of the article is presented in two sections. Section 1 describes the mathematical model of the two-stage transportation problem with unknown consumer demands and provides the necessary and sufficient conditions for the compatibility of the system of linear constraints. It is shown that its special case coincides with the classic two-stage transportation problem.Section 2 provides a description of the model problem of optimal partitioning-distribution for the continuous area Ω and the discrete analog of the model problem. The results of computational experiments for a rectangular area Ω = {x = (x(1), x(2)) : 0 ≤ x(1) ≤ 1, 0 ≤ x(2) ≤ 1} with discretizations by grids 31 × 31 and 500 × 500 are presented. Optimal plans for transportation of processed product from points of the first stage to points of the second stage for both grids were found. The average time spent by the Gurobi solver to solve problems for the second grid, where the number of variables equals 250018 and the number of constraints equals 250009, is a few seconds on modern PCs.

Solution group representations as quantum symmetries of graphs

In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex- and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first known example of a graph that has quantum symmetry and finite quantum automorphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry.

Averaged Controllability Of Transversely Isotropic Ambartsumyan Plate

In this paper we study the averaged controllability of Ambartsumyan plate made of a transversely isotropic material. Governing equations are based on assumptions of particular theory of anisotropic plates developed by S. A. Ambartsumyan specifically for describing the deformation of anisotropic plates made of material with isotropy plane which, at each point of the plate, is parallel to its middle plane. The Green’s function approach is applied to express the plate normal displacement by means of material parameters (density, Young moduli in both directions of isotropy) which are assumed to be uniformly distributed random variables. Eventually, the averaged controllability condition is reduced to an infinite system of linear constraints with respect to the control action. Three types of particular (heuristic) solutions of the truncated version of the infinite system are discussed. Determination of control parameters is reduced to the solution of a problem of nonlinear programming.

Черненко О. О., Чілікіна Т. В., Кошова О. П., Ольховська О. В., Олексійчук Ю. Ф., Оріхівська О. Г. ОЦІНКА ЕКОЛОГІЧНОЇ БЕЗПЕКИ РЕГІОНУ У ВИГЛЯДІ МОДЕЛІ БАГАТОКРИТЕРІАЛЬНОЇ ЗАДАЧІ ОПТИМІЗАЦІЇ

The purpose of the article is to analyze the latest research and publications on environmental quality management; identifying previously unresolved issues and long-term forecasting challenges. Research methodology. Mathematical modeling is a powerful tool for studying the interaction of human society and the natural environment within a single economic and ecological system. Results. For the convenience of perception of the material, the concepts of environmental safety, environmental assessment were introduced and a system of criteria for such assessment was developed. The article clarifies the understanding of the concepts of “air pollution”, “specialization of industrial enterprises”, “specialization of agriculture”. on the use (pollution) of natural and extracted resources. In particular, the activity of the region on the use of the following resources: air, water and energy was assessed. Restrictions on the use of natural resources are presented in the region as a whole. The model takes into account the integrated indicator of degradation of the region and environmental damage of the region from soil pollution. Within the framework of the problem statement and the constructed model the ecological and economic efficiency of regional nature management is estimated. The constructed model is a model of the multicriteria optimization problem with a system of linear constraints. In the simplest case, it can be solved by reducing the criteria to dimensionless values using the method of constraints. Practical significance of research results. The constructed model can be used to model more complex, for example, ecological, economic and social systems, or applied to the needs of practice.

Read-once refutations in Horn constraint systems: an algorithmic approach

Abstract In this paper, we discuss exact and parameterized algorithms for the problem of finding a read-once refutation (ROR) in an unsatisfiable Horn constraint system (HCS). Recall that a linear constraint system $\mathbf {A \cdot x \ge b}$ is said to be an HCS if each entry in $\textbf {A}$ belongs to the set $\{0,1,-1\}$ and at most one entry in each row of $\textbf {A}$ is positive. In this paper, we examine the importance of constraints in which more variables have negative coefficients than positive coefficients. In particular, we study the impact of the proportion of these ‘net-negative’ constraints has on the difficulty of finding RORs. There exist several algorithms for checking whether an HCS is feasible. To the best of our knowledge, these algorithms are not certifying, i.e. they do not provide a certificate of infeasibility. Our work is concerned with providing a specialized class of certificates called ‘read-once refutations’. In an ROR, each constraint defining the HCS may be used at most once in the derivation of a refutation. The problem of checking if an HCS has an ROR has been shown to be NP-hard. We analyse the HCS ROR problem from three different algorithmic perspectives, viz., parameterized algorithms, exact exponential algorithms and approximation algorithms. In particular, we show that the HCS ROR problem is fixed-parameter tractable (FPT) with respect to the number of constraints in the system that have more variables with negative coefficient than variables with positive coefficient. Additionally, we show that the HCS ROR problem becomes easy when this parameter is both small and large. We also derive an algorithm that runs in time $O(1.66^{m})$, where $m$ is the number of constraints in the HCS. On the lower-bound side, we derive a lower bound on the algorithmic resources needed for this problem using the Exponential Time Hypothesis. We also establish that the HCS ROR problem does not have a polynomial kernel when the number of constraints with three or more variables in a refutation is used as a parameter. Finally, we show that the problem of approximating the length of the shortest ROR in an HCS is NPO PB-complete1.

Variationally Improved Bézier Surfaces with Shifted Knots

The Plateau-Bézier problem with shifted knots is to find the surface of minimal area amongst all the Bézier surfaces with shifted knots spanned by the admitted boundary. Instead of variational minimization of usual area functional, the quasi-minimal Bézier surface with shifted knots is obtained as the solution of variational minimization of Dirichlet functional that turns up as the sum of two integrals and the vanishing condition gives us the system of linear algebraic constraints on the control points. The coefficients of these control points bear symmetry for the pair of summation indices as well as for the pair of free indices. These linear constraints are then solved for unknown interior control points in terms of given boundary control points to get quasi-minimal Bézier surface with shifted knots. The functional gradient of the surface gives possible candidate functions as the minimizers of the aforementioned Dirichlet functional; when solved for unknown interior control points, it results in a surface of minimal area called quasi-minimal Bézier surface. In particular, it is implemented on a biquadratic Bézier surface by expressing the unknown control point P 11 as the linear combination of the known control points in this case. This can be implemented to Bézier surfaces with shifted knots of higher degree, as well if desired.

AutoECART: Automatic energy conservation analysis of rovibronic transitions

Despite decades of diligent work on their development, line-by-line (LBL) spectroscopic information systems, widely employed by scientists and engineers, may still contain a number of incorrect rovibronic lines. A novel heuristic protocol, relying on cycle bases of spectroscopic networks (SN) and a system of linear constraints, is proposed to unravel incorrect transitions present in the database. The algorithm is named autoECART, standing for automatic Energy Conservation Analysis of Rovibronic Transitions. The autoECART method is tested on synthetic SNs constructed from spectroscopic databases of nine water isotopologues. The systematic numerical tests demonstrate the outstanding capability of the autoECART procedure to identify almost all of the outliers generated randomly in these synthetic SNs. As a ‘real-life’ example, the GEISA-2019-H218O database is scrutinized, revealing 17 rough outlier lines. Developers of spectroscopic information systems are encouraged to utilize autoECART for cleansing the huge number of transitions deposited in LBL databases.

A complexity analysis and algorithms for two-machine shop scheduling problems under linear constraints

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time.

Synchronous linear constraint system games

Mathematical models of quantum mechanics can be studied and distinguished using nonlocal games. We discuss a class of nonlocal games called synchronous linear constraint system (syncLCS) games. We unify two algebraic approaches to studying syncLCS games and relate these games to nonlocal games played on graphs, known as graph isomorphism games. In more detail, syncLCS games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated with these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here, we show that these objects are essentially the same, i.e., the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that syncLCS games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.

Constant Depth Decision Rules for multistage optimization under uncertainty

In this paper, we introduce a new class of decision rules, referred to as Constant Depth Decision Rules (CDDRs), for multistage optimization under linear constraints with uncertainty-affected right-hand sides. We consider two uncertainty classes: discrete uncertainties which can take at each stage at most a fixed number d of different values, and polytopic uncertainties which, at each stage, are elements of a convex hull of at most d points. Given the depthμ of the decision rule, the decision at stage t is expressed as the sum of t functions of μ consecutive values of the underlying uncertain parameters. These functions are arbitrary in the case of discrete uncertainties and are poly-affine in the case of polytopic uncertainties. For these uncertainty classes, we show that when the uncertain right-hand sides of the constraints of the multistage problem are of the same additive structure as the decision rules, these constraints can be reformulated as a system of linear inequality constraints where the numbers of variables and constraints is O(1)(n+m)dμN2 with n the maximal dimension of control variables, m the maximal number of inequality constraints at each stage, and N the number of stages.As an illustration, we discuss an application of the proposed approach to a Multistage Stochastic Program arising in the problem of hydro-thermal production planning with interstage dependent inflows. For problems with a small number of stages, we present the results of a numerical study in which optimal CDDRs show similar performance, in terms of optimization objective, to that of Stochastic Dual Dynamic Programming (SDDP) policies, often at much smaller computational cost.

On modelling hard combinatorial optimisation problems as linear programs: refutations of the 'unconditional impossibility' claims

There has been a series of developments in the recent literature (by essentially a same 'circle' of authors) with the absolute/unconditioned (implicit or explicit) claim that there exists no abstraction of an NP-complete combinatorial optimisation problem in which the defining combinatorial configurations [such as 'tours' in the case of the travelling salesman problem (TSP) for example] can be modelled by a polynomial-sized system of linear constraints. The purpose of this paper is to provide general as well as specific refutations for these recent claims.

Causal Bounds for Outcome-Dependent Sampling in Observational Studies

Outcome-dependent sampling designs are common in many different scientific fields including epidemiology, ecology, and economics. As with all observational studies, such designs often suffer from unmeasured confounding, which generally precludes the nonparametric identification of causal effects. Nonparametric bounds can provide a way to narrow the range of possible values for a nonidentifiable causal effect without making additional untestable assumptions. The nonparametric bounds literature has almost exclusively focused on settings with random sampling, and the bounds have often been derived with a particular linear programming method. We derive novel bounds for the causal risk difference, often referred to as the average treatment effect, in six settings with outcome-dependent sampling and unmeasured confounding for a binary outcome and exposure. Our derivations of the bounds illustrate two approaches that may be applicable in other settings where the bounding problem cannot be directly stated as a system of linear constraints. We illustrate our derived bounds in a real data example involving the effect of vitamin D concentration on mortality. Supplementary materials for this article are available online.

Polynomial time algorithms for optimal length tree-like refutations of linear infeasibility in UTVPI constraints

In this paper, we propose several algorithms for determining an optimal length tree-like refutation of linear feasibility in systems of Unit Two Variable Per Inequality (UTVPI) constraints. Given an infeasible UTVPI constraint system (UCS), a refutation certifies its infeasibility. The problem of finding refutations in a UCS finds applications in domains such as program verification and operations research. In general, there exist several types of refutations of feasibility in constraint systems. In this paper, we focus on a specific type of refutation called a tree-like refutation. Tree-like refutations are complete, in the sense that if a system of linear constraints is infeasible, then it must have a tree-like refutation. Associated with a refutation is its length, which equals the total number of constraints (accounting for the reuse of constraints) that are used to establish the infeasibility of the corresponding linear constraint system. Our goal in this paper is to find an optimal length tree-like refutation (OTLR) of an infeasible UCS. We describe two deterministic algorithms that find an OTLR of a UCS. If m is the number of constraints, n is the number of variables in the system, and k is the length of an OTLR, then our two algorithms run in O(n3⋅logk) time and O(m⋅n⋅k) time respectively. We also propose a true-biased, randomized algorithm for this problem. This algorithm runs in O(m⋅n⋅logn) time, and returns the length of an OTLR with probability (1−1e). We extend our work to weighted UTVPI constraint systems (WUCS), where a positive weight is associated with each UTVPI constraint. The weight of a tree-like refutation is defined as the sum of the weights of the constraints used by the refutation. The problem of finding a minimum weight refutation in a WUCS is called the weighted optimal length tree-like refutation (WOTLR) problem and is known to be NP-hard. We propose a pseudo-polynomial time algorithm for the WOTLR problem and convert it into a fully polynomial time approximation scheme (FPTAS).

A generalization of CHSH and the algebraic structure of optimal strategies

Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick (FOCS 2018), iterated compression by Fitzsimons et al.\ (STOC 2019), and NEEXP in MIP* due to Natarajan and Wright (FOCS 2019). The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of LCS games that self-test non-Pauli operators resolving an open question posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012). Additionally, our games have 1 bit question and log⁡n bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the solution group of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 2017) to the non-pseudo-telepathic regime. Finally, we give a game that is not a self-test by "gluing" together two copies of the magic square game. Our results suggest a richer landscape of self-testing phenomena than previously considered.

Classical vs quantum satisfiability in linear constraint systems modulo an integer

A system of linear constraints can be unsatisfiable and yet admit a solution in the form of quantum observables whose correlated outcomes satisfy the constraints. Recently, it has been claimed that such a satisfiability gap can be demonstrated using tensor products of generalized Pauli observables in odd dimensions. We provide an explicit proof that no quantum-classical satisfiability gap in any linear constraint system can be achieved using these observables. We prove a few other results for linear constraint systems modulo d > 2. We show that a characterization of the existence of quantum solutions when d is prime, due to Cleve et al, holds with a small modification for arbitrary d. We identify a key property of some linear constraint systems, called phase-commutation, and give a no-go theorem for the existence of quantum solutions to constraint systems for odd d whenever phase-commutation is present. As a consequence, all natural generalizations of the Peres–Mermin magic square and pentagram to odd prime d do not exhibit a satisfiability gap.