Solvable Cases Research Articles

Bi-criterion single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times

This paper addresses single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times. Due-window assignment with common flow allowances means that each job has a job-dependent due window, the start time and finish time of which are equal to its actual processing time plus individual job-independent parameters shared by all the jobs, respectively. The processing time of each job can be controlled by extra resource allocation as a linear function of the amount of a common continuously divisible resource allocated to the job. Two criteria are considered, where one criterion is an integrated cost consisting of job earliness, weighted number of tardy jobs, and due-window assignment cost, while the other criterion is the resource consumption cost. Four different models are considered for treating the two criteria. It is shown that the problem under the model where the two criteria are integrated into a single criterion is polynomially solvable, while the problems under the other three models are all NP-hard and an optimal solution procedure is developed for them. Two polynomially solvable cases are also identified and investigated. Finally, numerical studies with randomly generated instances are conducted to assess the performance of the proposed algorithms.

Numeric Calculation of Antiferromagnetic Resonance Frequencies for the Noncollinear Antiferromagnet

We present an algorithm for the numeric calculation of antiferromagnetic resonance frequencies for the non-collinear antiferromagnets of general type. This algorithm uses general exchange symmetry approach \cite{andrmar} and is applicable for description of low-energy dynamics of an arbitrary noncollinear spin structure in weak fields. Algorithm is implemented as a MatLab and C++ program codes, which are available for download. Program codes are tested against some representative analytically solvable cases.

A unified approach to recognize squares of split graphs

The square of a graph G, denoted by G2, is obtained from G by putting an edge between two distinct vertices whenever their distance is two. Then G is called a square root of G2. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique.We give a wide range of polynomial time solvable cases for the problem of recognizing if a given graph is the square of some special kind of split graph. To the best of our knowledge, our result properly contains all previously known such cases. Our polynomial time algorithms are built on a structural investigation of graphs that admit a split square root that is 3-sun-free, and may pave the way toward a dichotomy theorem for recognizing squares of (3-sun-free) split graphs.

Polynomially solvable cases of the bipartite traveling salesman problem

Given two sets, R and B, consisting of n cities each, in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of R and B, returning to the city of origin. This problem is known to be NP-hard for arbitrary sets R and B. In this paper we provide an O(n6) algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds. In particular, this algorithm can be applied to solve in O(n6) time the bipartite traveling salesman problem in the following cases: S=R∪B is a convex point set in the plane, S=R∪B is the set of vertices of a simple polygon and V=R∪B is the set of vertices of a circular graph. For this last case, we also describe another algorithm which runs in O(n2) time.

The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock's models

We consider the propagation of acoustic and thermal waves in classical perfect gases under a coupled, weakly-nonlinear system first derived by Blackstock. Our primary aim is to ascertain the usefulness of Blackstock's system as an approximate model of nonlinear acoustic phenomena. Working in the context of the piston problem, and using a solvable special case of the Navier--Stokes--Fourier system as our benchmark, we compare Blackstock's system against a simpler weakly-nonlinear model whose constitute equations are not coupled. In particular, traveling wave solutions (TWS)s are determined, the structure of the solution profiles is analyzed, numerical comparisons are presented, and follow-on studies are suggested.

Partitioning Perfect Graphs into Stars

AbstractThe partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same‐size stars, a problem known to be NP‐complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial‐time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP‐complete cases, for example, on grid graphs and chordal graphs.

A polynomially solvable case of the pooling problem

Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview of known complexity results and remaining open problems to further characterize the border between (strongly) NP-hard and polynomially solvable cases of the pooling problem.

Stable Marriage and Roommates problems with restricted edges: Complexity and approximability

In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to NP-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an NP-hard but (under some cardinality assumptions) 2-approximable problem. In the case of NP-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs.

Progress in many-body theory with the equation of motion method: Time-dependent density matrix meets self-consistent RPA and applications to solvable models

The Bogoliubov-Born-Green-Kirkwood-Yvon or Time-Dependent Density Matrix (TDDM) hierarchy of equations for higher density matrices is truncated at the three body level in approximating the three body correlation function by a quadratic form of two body ones, closing the equations in this way. The procedure is discussed in detail and it is shown in non-trivial model cases that the approximate inclusion of three body correlation functions is very important to obtain precise results. A small amplitude approximation of this time dependent nonlinear equation for the two body correlation function is performed (STDDM*-b) and it is shown that the one body sector of this generalised non-linear second RPA equation is equivalent to the Self-Consistent RPA (SCRPA) approach which had been derived previously by different techniques. It is discussed in which way SCRPA also contains the three body correlations. TDDM and SCRPA are tested versus exactly solvable model cases.

The two-level economic lot sizing problem with perishable items

We present an economic lot sizing model of a supply chain for the procurement and distribution of a perishable item. We assume that the consumers always buy the item that lasts longer. We show that determining optimal procurement and transfer plan is NP-hard, and present polynomial time solvable special cases.

Holographic Wilson loops, Hamilton-Jacobi equation, and regularizations

The minimal area for surfaces whose borders are rectangular and circular loops are calculated using the Hamilton-Jacobi (HJ) equation. This amounts to solving the HJ equation for the value of the minimal area, without calculating the shape of the corresponding surface. This is done for bulk geometries that are asymptotically anti--de Sitter (AdS). For the rectangular contour, the HJ equation, which is separable, can be solved exactly. For the circular contour an expansion in powers of the radius is implemented. The HJ approach naturally leads to a regularization which consists in locating the contour away from the border. The results are compared with the $\ensuremath{\epsilon}$-regularization which leaves the contour at the border and calculates the area of the corresponding minimal surface up to a diameter smaller than the one of the contour at the border. The results for the circular loop do not coincide if the expansion parameter is taken to be the radius of the contour at the border. It is shown that using this expansion parameter the $\ensuremath{\epsilon}$-regularization leads to incorrect results for certain solvable non-AdS cases. However, if the expansion parameter is taken to be the radius of the minimal surface whose area is computed, then the results coincide with the HJ scheme. This is traced back to the fact that in the HJ case the expansion parameter for the area of a minimal surface is intrinsic to the surface; however, the radius of the contour at the border is related to the way one chooses to regularize in the $\ensuremath{\epsilon}$-scheme the calculation of this area.

Escape rate of Brownian particles from a metastable potential well under time derivative Ornstein-Uhlenbeck noise

We investigate the escape rate of Brownian particles that move in a cubic metastable potential subjected to an internal time derivative Ornstein-Uhlenbeck noise (DOUN). This noise can induce the ballistic diffusion of force-free Brownian particles. Some new features are found. The escape rate for DOUN shows qualitative different dependence on potential well width compared with OUN which induces normal diffusion. As the potential barrier height decreases, the escape rate of DOUN deviates from Arrhenius law considerably earlier than that of Ornstein-Uhlenbeck noise (OUN). The Brownian particles escape faster under DOUN than that under OUN. A quasi-periodic oscillation occurs in transient state. A solvable case is presented to demonstrate the significant cancellation behavior in the barrier region that governs most of these phenomena. The physical mechanism of the findings can be clarified by the noise features. These characteristics should be common for internal noises that induce superdiffusion, especially the ballistic diffusion.

Quantum cosmology: From hidden symmetries towards a new (supersymmetric) perspective

We review pedagogically some of the basic essentials regarding recent results intertwining boundary conditions, the algebra of constraints and hidden symmetries in quantum cosmology. They were extensively published in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], where complete discussions and full details can be found. More concretely, in Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014) and S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439] it has been shown that specific boundary conditions can be related to the algebra of Dirac observables. Moreover, a process afterwards associated to the algebra of existent hidden symmetries, from which the boundary conditions can be selected, was introduced. On the other hand, in Ref. [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212] it was subsequently argued that some factor ordering choices can be extracted from the hidden symmetries structure of the minisuperspace model.In Refs. [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 89 (2014), S. Jalalzadeh, T. Rostami and P. V. Moniz, Eur. Phys. J. C 75 (2015) 38, arXiv:gr-qc/1412.6439 and T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212], we proceeded gradually towards less simple models, ranging from a FLRW model with a perfect fluid [S. Jalalzadeh, S. M. M. Rasouli and P. V. Moniz, Phys. Rev. D 90 (2014) 023541] up to a conformal scalar field content [T. Rostami, S. Jalalzadeh and P. V. Moniz, Phys. Rev. D 92 (2015) 023526, arXiv:gr-qc/1507.04212]. We envisage that we could extend this framework towards a class of shape invariant potentials, which could include well known analytically solvable cosmological cases. Provided, we identify integrability in terms of the shape invariance conditions, we could eventually consider to import features of supersymmetric quantum mechanics towards quantum cosmology [P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 1: Fundamentals, Lecture Notes in Physics, Vol. 803 (Springer-Verlag, Berlin, 2010), P. V. Moniz, Quantum Cosmology-the Supersymmetric Perspective-Vol. 2: Advanced Topics, Lecture Notes in Physics, Vol. 804 (Springer, New York, 2010)], which we will also discuss in this review.Another point to emphasize is that by means of a hidden symmetry and then an algebra of Dirac observables, boundary conditions are extracted (and not ad hoc formulated) within a framework intrinsic to each model dynamics. Therefore, meeting DeWitt’s conjecture [B. S. DeWitt, Phys. Rev. 160 (1967) 1113] that “the constraints are everything” and nothing else but the constraints should be needed.

Co-Clustering under the Maximum Norm

Co-clustering, that is partitioning a numerical matrix into “homogeneous” submatrices, has many applications ranging from bioinformatics to election analysis. Many interesting variants of co-clustering are NP-hard. We focus on the basic variant of co-clustering where the homogeneity of a submatrix is defined in terms of minimizing the maximum distance between two entries. In this context, we spot several NP-hard, as well as a number of relevant polynomial-time solvable special cases, thus charting the border of tractability for this challenging data clustering problem. For instance, we provide polynomial-time solvability when having to partition the rows and columns into two subsets each (meaning that one obtains four submatrices). When partitioning rows and columns into three subsets each, however, we encounter NP-hardness, even for input matrices containing only values from {0, 1, 2}.

Equitable coloring of corona products of cubic graphs is harder than ordinary coloring

A graph is equitably k -colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = ( G ) . In this paper the problem of determinig χ = for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas of cubic graphs is solvable in polynomial time, the problem of equitable coloring becomes NP-hard for these graphs. We provide polynomially solvable cases of coronas of cubic graphs and prove the NP-hardness in a general case. As a by-product we obtain a simple linear time algorithm for equitable coloring of such graphs which uses χ = ( G ) or χ = ( G ) + 1 colors. Our algorithm is best possible, unless P=NP. Consequently, cubical coronas seem to be the only known class of graphs for which equitable coloring is harder than ordinary coloring.

Quantization of charged fields in the presence of critical potential steps

QFT approaches elaborated for treating quantum effects in time-dependent external electric fields are not directly applicable to time-independent nonuniform electric fields that are given by a step potential and their generalization for the such potentials was not sufficiently developed. Such fields can also create particles from the vacuum, the Klein paradox being closely related to this process. We believe that the present work presents a consistent solution of the latter problem. Quantizing the Dirac and scalar fields with time independent backgrounds, we have found in- and out-creation and annihilation operators that allow one to have particle interpretation of the physical system under consideration. To justify the proposed identification, we have performed a detailed mathematical and physical analysis of solutions of the corresponding relativistic wave equations with a subsequent QFT analysis. We elaborated a nonperturbative technique that allows one to calculate all characteristics of zero-order processes such scattering, reflection, and electron-positron pair creation, and to calculate Feynman diagrams that describe all characteristics of processes with interaction between the in-, out-particles and photons. These diagrams have formally the usual form, but contain special propagators. Expressions for these propagators in terms of in- and out-solutions are presented. We apply the elaborated approach to two popular exactly solvable cases: to the Sauter potential and to the Klein step.

On the exact results of a non-linear model arising in catalytic reaction in a flat particle with power kinetics reaction rate

The latest study of a model of heat and mass transfer for a catalytic reaction within a porous catalyst flat particle (Journal of the Taiwan Institute of Chemical Engineers 48 (2015) 49–55 [7]) shows that this model can be exactly solved when the reaction rate is taken linear term as f(y)=y, where y is heat within porous catalyst particle. In this paper, the mentioned model is revisited by considering the power kinetics nth order reaction rate instead of a linear reaction term. It is shown that the problem is still exactly tractable. Furthermore, it is revealed that previous results can be recovered from this exactly solvable generalization case. It is also proved that the problem might have multiple stationary solutions (unique, dual and triple solutions) depending on the values of the parameters of the model.

Solvable Cases in the Problem of Motion of a Heavy Rotationally Symmetric Ellipsoid on a Perfectly Rough Plane

We consider a classical problem of nonholonomic system dynamics – the problem of motion of a rotationally symmetric body on a fixed perfectly rough plane in a case when the moving body is a rotationally symmetric ellipsoid. Using the Kovacic algorithm we found several conditions under which equations of motion of the ellipsoid can be completely solved in quadratures.

Solvable structures of a simple model of earthquakes

This paper propose an extension of Random Domino Automaton by introducing extra parameter related to a very short-distance interactions between clusters which allows to isolate influence two mechanisms of clusters grow, namely enlarging and coalescence. In this setting statistically stationary state under assumption of independence of clusters is investigated and respective set of equations is derived.Moreover, a special solvable case is studied in detail and it is shown how to derive Motzkin numbers out of the automaton for a family of parameters without considering a limit.

Cyclic inventory routing in a line-shaped network

The inventory routing problem (IRP) is a very challenging optimization task that couples two of the most important components of supply chain management, i.e., inventory control and transportation. Routes of vehicles are to be determined to repeatedly resupply multiple customers with constant demand rates from a single depot. We alter this basic IRP setting by two aspects: (i) only cyclic tours are allowed, i.e., each vehicle continuously tours its dedicated route, and (ii) all customers are located along a line. Both characteristics occur, for instance, in liner shipping (when feeder ships service inland ports along a stream) and in facility logistics (when tow trains deliver part bins to the stations of an assembly line). We formalize the resulting problem setting, identify NP-hard as well as polynomially solvable cases, and develop suited solution procedures.