Solvable Cases Research Articles

Bose–Hubbard models with on-site and nearest-neighbor interactions: exactly solvable case

We study the discrete spectrum of the two-particle Schrödinger operator , , associated to the Bose–Hubbard Hamiltonian of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice with interaction magnitudes and , respectively. We completely describe the spectrum of and establish the optimal lower bound for the number of eigenvalues of outside its essential spectrum for all values of . Namely, we partition the (μ, λ)-plane such that in each connected component of the partition the number of bound states of below or above its essential spectrum cannot be less than the corresponding number of bound states of below or above its essential spectrum.

On the Complexity of Sum-of-Products Problems over Semirings

Many important problems in AI, among them SAT, #SAT, and probabilistic inference, amount to Sum-of-Products Problems, i.e. evaluating a sum of products of values from some semiring R. While efficiently solvable cases are known, a systematic study of the complexity of this problem is missing. We characterize the latter by NP(R), a novel generalization of NP over semiring R, and link it to well-known complexity classes. While NP(R) is unlikely to be contained in FPSPACE(poly) in general, for a wide range of commutative (resp. in addition idempotent) semirings, there are reductions to #P (resp. NP) and solutions are thus only mildly harder to compute. We finally discuss NP(R)-complete reasoning problems in well-known semiring formalisms, among them Semiring-based Constraint Satisfaction Problems, obtaining new insights into their computational properties.

Finding Diverse Trees, Paths, and More

Mathematical modeling is a standard approach to solve many real-world problems and diversity of solutions is an important issue, emerging in applying solutions obtained from mathematical models to real-world problems. Many studies have been devoted to finding diverse solutions. Baste et al. (Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse solutions of combinatorial problems from the perspective of fixed-parameter tractability. They considered problems of finding r solutions that maximize some diversity measures (the minimum or sum of the pairwise Hamming distances among them) and gave some fixed-parameter tractable algorithms for the diverse version of several well-known problems, such as Vertex Cover, Feedback Vertex Set, d-Hitting Set}, and problems on bounded-treewidth graphs. In this work, we further investigate the (fixed-parameter) tractability of problems of finding diverse spanning trees, paths, and several subgraphs. In particular, we show that, given a graph G and an integer r, the problem of computing r spanning trees of G maximizing the sum of the pairwise Hamming distances among them can be solved in polynomial time. To the best of the authors' knowledge, this is the first polynomial-time solvable case for finding diverse solutions of unbounded size.

Ethical Dilemmas in the Therapeutic Management of Patients with Sars-Cov-2 Infection

The new coronavirus pandemic has brought into the light-on top of the strain it put on the medical services around the world-a variety of ethical issues in relation to how the treatment is being administered to the patients. Understanding the priority of treating COVID-19 patients, we still ask ourselves how effective these antiviral/immunomodulatory molecules are recommended by national/international protocols and which is the benefit/risk ratio in different categories of patients. To solve these dilemmas, we present the case of a 36-year-old patient, admitted to our clinic in April 2020, with mild symptomatic SARS-CoV-2 infection. Given the suggestive clinical and paraclinical elements, we recommended treatment with lopinavir/ ritonavir according to national protocol, and we explained to patient the benefits of this treatment, as well as the possible side effects. The patient refused this treatment, but later accepted an alternative therapy, hydroxychloroquine. The evolution of clinical and paraclinical parameters allowed the patient to be discharged after 19 days. This apparently simple and solvable medical case becomes complicated when the patient complained about the violation of her rights and of certain articles from deontological code. Beyond the elements of subjectivism, it is necessary an ethical approach of this problem. After 9 months of pandemics, we can say that some anti-COVID -19 therapies have proven a practical effectiveness and others have been partially invalidated by clinical trials and removed from the guidelines, but can we say every information regarding anti - SARS-CoV-2 medication is absolutely clear or that ethical aspects are solved?

Compacting oblivious agents on dynamic rings.

In this paper we investigate dynamic networks populated by autonomous mobile agents. Dynamic networks are networks whose topology can change continuously, at unpredictable locations and at unpredictable times. These changes are not considered to be faults, but rather an integral part of the nature of the system. The agents can autonomously move on the network, with the goal of solving cooperatively an assigned common task. Here, we focus on a specific network: the unoriented ring. More specifically, we study 1-interval connected dynamic rings (i.e., at any time, at most one of the edges might be missing). The agents move according to the widely used Look–Compute–Move life cycle, and can be homogenous (thus identical) or heterogenous (agents are assigned colors from a set of c > 1 colors). For identical agents, their goal is to form a compact segment, where agents occupy a continuous part of the ring and no two agents occupy the same node: we call this the Compact Configuration Problem. In the case of agents with colors, called the Colored Compact Configuration Problem, the goal is to group agents such that each group is formed by all agents having the same color, it occupies a continuous segment of the network, and groups of agents having different colors occupy distinct areas of the network. In this paper we determine the necessary conditions to solve both proposed problems. For all solvable cases, we provide algorithms for both the monochromatic and the colored version of the compact configuration problem. All our algorithms work even for the simplest model where agents have no persistent memory, no communication capabilities and do not agree on a common orientation within the network. To the best of our knowledge this is the first work on the compaction problem in a dynamic network.

On Optimistic and Pessimistic Bilevel Optimization Models for Demand Response Management

This paper investigates bilevel optimization models for demand response management, and highlights the often overlooked consequences of a common modeling assumption in the field. That is, the overwhelming majority of existing research deals with the so-called optimistic variant of the problem where, in case of multiple optimal consumption schedules for a consumer (follower), the consumer chooses an optimal schedule that is the most favorable for the electricity retailer (leader). However, this assumption is usually illegitimate in practice; as a result, consumers may easily deviate from their expected behavior during realization, and the retailer suffers significant losses. One way out is to solve the pessimistic variant instead, where the retailer prepares for the least favorable optimal responses from the consumers. The main contribution of the paper is an exact procedure for solving the pessimistic variant of the problem. First, key properties of optimal solutions are formally proven and efficiently solvable special cases are identified. Then, a detailed investigation of the optimistic and pessimistic variants of the problem is presented. It is demonstrated that the set of optimal consumption schedules typically contains various responses that are equal for the follower, but bring radically different profits for the leader. The main procedure for solving the pessimistic variant reduces the problem to solving the optimistic variant with slightly perturbed problem data. A numerical case study shows that the optimistic solution may perform poorly in practice, while the pessimistic solution gives very close to the highest profit that can be achieved theoretically. To the best of the authors’ knowledge, this paper is the first to propose an exact solution approach for the pessimistic variant of the problem.

Kundt geometries and memory effects in the Brans\u2013Dicke theory of gravity

Memory effects are studied in the simplest scalar–tensor theory, the Brans–Dicke (BD) theory. To this end, we introduce, in BD theory, novel Kundt spacetimes (without and with gyratonic terms), which serve as backgrounds for the ensuing analysis on memory. The BD parameter omega and the scalar field (phi ) profile, expectedly, distinguishes between different solutions. Choosing specific localised forms for the free metric functions H'(u) (related to the wave profile) and J(u) (the gyraton) we obtain displacement memory effects using both geodesics and geodesic deviation. An interesting and easy-to-understand exactly solvable case arises when omega =-2 (with J(u) absent) which we discuss in detail. For other omega (in the presence of J or without), numerically obtained geodesics lead to results on displacement memory which appear to match qualitatively with those found from a deviation analysis. Thus, the issue of how memory effects in BD theory may arise and also differ from their GR counterparts, is now partially addressed, at least theoretically, within the context of this new class of Kundt geometries.

Solutions of Sylvester equation in -modular operators

UDC 517.9 We study the solvability of the Sylvester equation and the operator equation in the general setting of the adjointable operators between Hilbert -modules. Based on the Moore – Penrose inverses of the associated operators, we propose necessary and sufficient conditions for the existence of solutions to these equations, and obtain the general expressions of the solutions in the solvable cases. We also provide an approach to the study of the positive solutions for a special case of Lyapunov equation.

Perturbative calculations of entanglement entropy

This paper is an attempt to extend the recent understanding of the Page curve for evaporating black holes to more general systems coupled to a heat bath. Although calculating the von Neumann entropy by the replica trick is usually a challenge, we have identified two solvable cases. For the initial section of the Page curve, we sum up the perturbation series in the system-bath coupling κ; the most interesting contribution is of order 2s, where s is the number of replicas. For the saturated regime, we consider the effect of an external impulse on the entropy at a later time and relate it to OTOCs. A significant simplification occurs in the maximal chaos case such that the effect may be interpreted in terms of an intermediate object, analogous to the branching surface of a replica wormhole.

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to the problem of finding an r-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of r = 2, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of r and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT-algorithm.

АЛГОРИТМЫ РЕШЕНИЯ СИСТЕМ УРАВНЕНИЙ НАД РАЗЛИЧНЫМИ КЛАССАМИ КОНЕЧНЫХ ГРАФОВ

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n > 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.

Some solvable cases of optimal control for Volterra integral systems

Some solvable cases of optimal control for Volterra integral systems

On Efficiently Solvable Cases of Quantum k-SAT

Estimating ground state energies of local Hamiltonian models is a central problem in quantum physics. The question of whether a given local Hamiltonian is frustration-free, meaning the ground state is the simultaneous ground state of all local interaction terms, is known as the Quantum k-SAT (k-QSAT) problem. In analogy to its classical Boolean constraint satisfaction counterpart, the NP-complete problem k-SAT, Quantum k-SAT is hbox {QMA}_1-complete (for kge 3, and where hbox {QMA}_1 is a quantum generalization of NP with one-sided error), and thus likely intractable. But whereas k-SAT has been well-studied for special tractable cases, as well as from a “parameterized complexity” perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a “dimer covering” or “matching”; such systems are known to be frustration-free, but it remains open whether one can efficiently compute a ground state. Our results fall into three directions, all of which relate to the “dimer covering” setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two interaction terms or clauses. (2) We give a “parameterized algorithm” for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases. This is achieved by reducing the problem to solving for a single root of a single univariate polynomial. An explicit family of hypergraphs, denoted Crash, for which such a speedup is obtained is introduced. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a “dimer covering”. We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry.

The Rank-One Quadratic Assignment Problem

In this paper, we study the quadratic assignment problem with a rank-one cost matrix (QAP-R1). Four integer-programming formulations are introduced of which three are assumed to have partial integer data. Unlike the standard quadratic assignment problem, some of our formulations can solve reasonably large instances of QAP-R1 with impressive running times and are faster than some metaheuristics. Pairwise relative strength of the LP relaxations of these formulations are also analyzed from theoretical and experimental points of view. Finally, we present a new metaheuristic algorithm to solve QAP-R1 along with its computational analysis. Our study offers the first systematic experimental analysis of integer-programming models and heuristics for QAP-R1. The benchmark instances with various characteristics generated for our study are made available to the public for future research work. Some new polynomially solvable special cases are also introduced. Summary of Contribution: This paper aims to advance our knowledge and ability in solving an important special case of the quadratic assignment problem. It shows how to exploit inherent properties of an optimization problem to achieve computational advantages, a strategy that was followed by researchers in model building and algorithm developments for decades. Our computational results attest to this time-tested general philosophy. The paper presents the first systematic computational study of the rank one quadratic assignment problem, along with new mathematical programming models and complexity analysis. We believe the theoretical and computational results of this paper will inspire further research on the topic and will be of significant value to practitioners using rank one quadratic assignment models.

Non-monotonic dependence of reaction rate on rate constants: Role of time-averaging

• Non-monotonic dependence of rate on rate constant(s) is explicitly shown in a few solvable cases. • Intuitively expected behaviors are found only after a long-time averaging. • The topic possesses a very general appeal in enriching chemical wisdom. • Deterministic enzyme kinetics case is noted to be plagued with similar problems. • A simpler solvable model is used to rationalize numerical findings of the enzymatic problems. The rate of a reaction may not always show a monotonic dependence on some specific rate constant. In the context of stochastic single-substrate enzyme kinetics, such a behavior has recently [PNAS 111 (2014) 4391] been demonstrated; specifically, rate is noted to increase with increasing dissociation rate constant of the enzyme-substrate complex, contrary to standard expectation. In this work, we identify this counter-intuitive trait in several solvable and deterministic situations too. However, these anomalies are also analytically shown to disappear after a time-averaging. Numerical results of the deterministic enzyme kinetics scheme reveal similar features. The age-old intuitive chemistry thus retains its significance as an average.

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval $[a,b]$, $0<a<b<1$. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This in turn will allow us to find the asymptotics of the norms $h_n$ of the corresponding orthogonal polynomials and, in fact, the large $n$ asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkh\"auser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end we will discuss the prospects of future work and outline several technical, as well as conceptual, issues which we are going to address next within the $4\times 4$ Riemann-Hilbert framework introduced in this paper.

Hedonic Diversity Games Revisited

A hedonic diversity game (HDG) is a coalition formation problem, where the set of agents is partitioned into two types of agents (say red and blue agents), and each agent has preferences over the relative number (fraction) of agents of her own type in her coalition. In a dichotomous hedonic diversity game (DHDG) each agent partitions the set of possible fractions into a set of approved and a set of disapproved fractions. The solution concepts for these games considered in the literature so far are concerned with stability notions such as core and Nash stability. We add to the existing literature by providing NP-completeness results for the decision problems whether a DHDG admits (i) a Nash stable outcome and (ii) a strictly core stable outcome respectively, in restricted settings with only two (and three, respectively) approved fractions per agent. In addition, we consider the criterion of social welfare maximization in HDG and DHDG. In that context we provide an NP-completeness result for HDG under the use of Borda scores. For DHDG and approval scores, we draw the separation line between polynomially solvable and NP-complete cases with respect to the number of approved fractions per agent.

Solvability in the sense of sequences for some fourth order non-Fredholm operators

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L2(Rd) of their right sides implies the existence and the convergence in H4(Rd) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory for Schrödinger type operators. We especially emphasize that here we deal with the fourth order operators in contrast to the second order operators in [29] and investigate the dependence of the solvability conditions on the dimension of our problem when the constant a=0. We also consider the case of solvability with a single potential in an arbitrary dimension.

Assembly line balancing with parallel workstations

ABSTRACT The simple assembly line balancing problem (SALBP) considers work division among different workstations of a serially arranged assembly process to maximise its efficiency under workload (cumulative) and technological (precedence) constraints. In this work, we consider a variant of the SALBP which allows parallel workstations. To study the effect of parallel stations, we propose a new problem (the parallel station assembly line balancing problem or PSALBP) in which the objective is to minimise the number of parallel stations required to obtain the maximum theoretical efficiency of the assembly process. We study the complexity of the problem and identify a polynomially solvable case. This result is then used as a building block for the development of a heuristic solution procedure. Finally, we carry out a computational experiment to identify the characteristics of assembly lines that may benefit from station paralleling and to evaluate the performance of the proposed heuristic.

Robust approach to restricted items selection problem

We consider the robust version of items selection problem, in which the goal is to choose representatives from a family of sets, preserving constraints on the allowed items’ combinations. We prove NP-hardness of the deterministic version, and establish polynomially solvable special cases. Next, we consider the robust version in which we aim at minimizing the maximum regret of the solution under interval parameter uncertainty. We show that this problem is hard for the second level of polynomial-time hierarchy. We develop exact solution algorithms for the robust problem, based on cut generation and mixed-integer programming, and present the results of computational experiments.