Solvable Cases Research Articles

Squeezing the Efimov effect

The quantum mechanical three-body problem is a source of continuing interest due to its complexity and not least due to the presence of fascinating solvable cases. The prime example is the Efimov effect where infinitely many bound states of identical bosons can arise at the threshold where the two-body problem has zero binding energy. An important aspect of the Efimov effect is the effect of spatial dimensionality; it has been observed in three dimensional systems, yet it is believed to be impossible in two dimensions. Using modern experimental techniques, it is possible to engineer trap geometry and thus address the intricate nature of quantum few-body physics as function of dimensionality. Here we present a framework for studying the three-body problem as one (continuously) changes the dimensionality of the system all the way from three, through two, and down to a single dimension. This is done by considering the Efimov favorable case of a mass-imbalanced system and with an external confinement provided by a typical experimental case with a (deformed) harmonic trap.

On the complexity of instationary gas flows

We study a simplistic model of instationary gas flows consisting of a sequence of k stationary gas flows. We present efficiently solvable cases and NP-hardness results, establishing complexity gaps between stationary and instationary gas flows (already for k=2) as well as between instationary gas s-t-flows and instationary gas b-flows.

On integer images of max-plus linear mappings

Let us extend the pair of operations ⊕,⊗=max,+ over real numbers to matrices in the same way as in conventional linear algebra.We study integer images of mappings x→A⊗x, where A∈Rm×n and x∈Rn. The question whether A⊗x is an integer vector for at least one x∈Rn has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems.We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case.

Convergence and covering on graphs for wait-free robots

The class of robot convergence tasks has been shown to capture fundamental aspects of fault-tolerant computability. A set of asynchronous robots that may fail by crashing, start from unknown places in some given space, and have to move towards positions close to each other. In this article, we study the case where the space is uni-dimensional, modeled as a graph G. In graph convergence, robots have to end up on one or two vertices of the same edge. We consider also a variant of robot convergence on graphs, edge covering, where additionally, it is required that not all robots end up on the same vertex. Remarkably, these two similar problems have very different computability properties, related to orthogonal fundamental issues of distributed computations: agreement and symmetry breaking. We characterize the graphs on which each of these problems is solvable, and give optimal time algorithms for the solvable cases. Although the results can be derived from known general topology theorems, the presentation serves as a self-contained introduction to the algebraic topology approach to distributed computing, and yields concrete algorithms and impossibility results.

Effective equilibrium states in mixtures of active particles driven by colored noise.

We consider the steady-state behavior of pairs of active particles having different persistence times and diffusivities. To this purpose we employ the active Ornstein-Uhlenbeck model, where the particles are driven by colored noises with exponential correlation functions whose intensities and correlation times vary from species to species. By extending Fox's theory to many components, we derive by functional calculus an approximate Fokker-Planck equation for the configurational distribution function of the system. After illustrating the predicted distribution in the solvable case of two particles interacting via a harmonic potential, we consider systems of particles repelling through inverse power-law potentials. We compare the analytic predictions to computer simulations for such soft-repulsive interactions in one dimension and show that at linear order in the persistence times the theory is satisfactory. This work provides the toolbox to qualitatively describe many-body phenomena, such as demixing and depletion, by means of effective pair potentials.

The quadratic M-convexity testing problem

M-convex functions, which are a generalization of valuated matroids, play a central role in discrete convex analysis. Quadratic M-convex functions constitute a basic and important subclass of M-convex functions, which has a close relationship with phylogenetics as well as valued constraint satisfaction problems. In this paper, we consider the quadraticM-convexity testing problem (QMCTP), which is the problem of deciding whether a given quadratic function on {0,1}n is M-convex. We show that QMCTP is co-NP-complete in general, but is polynomial-time solvable under a natural assumption. Furthermore, we propose an O(n2)-time algorithm for solving QMCTP in the polynomial-time solvable case.

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach

We introduce a stochastic multi-stage fixed charge transportation problem, in which a producer has to satisfy an uncertain demand within a deadline. At each time period, a fixed transportation cost can be paid to buy a transportation capacity. If the transportation capacity is used, the supplier also pays an uncertain unit transportation cost. A unit inventory cost is charged for the unsatisfied demand. The aim is to determine the transportation capacities to buy and the quantity to send at each time period in order to minimize the expected total cost. We prove that this problem is NP-hard, we propose a multi-stage stochastic optimization model formulation, and we determine optimal policies for particular cases, with deterministic unit transportation costs or demand and zero fixed costs. Furthermore, we provide the worst–case analysis of the rolling horizon approach, a classical heuristic approach for solving multi-stage stochastic programming models, applied to this NP-hard problem and to polynomially solvable particular cases. Worst–case results show that the rolling horizon approach can be very suboptimal. We also provide experimental results.

The quadratic minimum spanning tree problem and its variations

The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict pair constraints are useful in modeling various real life applications. All these problems are known to be NP-hard. In this paper, we investigate these problems to obtain additional insights into the structure of the problems and to identify possible demarcation between easy and hard special cases. New polynomially solvable cases have been identified, as well as NP-hard instances on very simple graphs. As a byproduct, we have a recursive formula for counting the number of spanning trees on a (k,n)-accordion and a characterization of matroids in the context of a quadratic objective function.

Efficient Implicit Solvation Method for Full Potential DFT.

With the advent of efficient electronic structure methods, effective continuum solvation methods have emerged as a way to, at least partially, include solvent effects into simulations without the need for expensive sampling over solvent degrees of freedom. The multipole moment expansion (MPE) model, while based on ideas initially put forward almost 100 years ago, has recently been updated for the needs of modern electronic structure calculations. Indeed, for an all-electron code relying on localized basis sets and-more importantly-a multipole moment expansion of the electrostatic potential, the MPE method presents a particularly cheap way of solving the macroscopic Poisson equation to determine the electrostatic response of a medium surrounding a solute. In addition to our implementation of the MPE model in the FHI-aims electronic structure theory code [ Blum , V. ; Comput. Phys. Commun. 2009 , 180 , 2175 - 2196 , DOI: 10.1016/j.cpc.2009.06.022 ], we describe novel algorithms for determining equidistributed points on the solvation cavity-defined as a charge density isosurface-and the determination of cavity surface and volume from just this collection of points and their local density gradients. We demonstrate the efficacy of our model on an analytically solvable test case, against high-accuracy finite-element calculations for a set of ≈140000 2D test cases, and finally against experimental solvation free energies of a number of neutral and singly charged molecular test sets [ Andreussi , O. ; J. Chem. Phys. 2012 , 136 , 064102 , DOI: 10.1063/1.3676407 ; Marenich , A. V. ; Minnesota Solvation Database , Version 2012; University of Minnesota : Minneapolis, MN, USA , 2012 . ]. In all test cases we find that our MPE approach compares very well with given references at computational overheads < 20% and sometimes much smaller compared to a plain self-consistency cycle.

Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset \(E'\) of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of \(E'\). We say that \(E'\) is a perfect edge dominating set of G, if every edge not in \(E'\) is dominated by exactly one edge of \(E'\). The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most \(d \ge 3\) and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a \(P_5\)-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a \(P_5\).

Clustering with Lower-Bounded Sizes

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios, however, the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for $$k=2$$k=2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.

Tidal Love numbers and moment–Love relations of polytropic stars

The physical significance of tidal deformation in astronomical systems has long been known. The recently discovered universal I-Love-Q relations, which connect moment of inertia, quadrupole tidal Love number, and spin-induced quadrupole moment of compact stars, also underscore the special role of tidal deformation in gravitational wave astronomy. Motivated by the observation that such relations also prevail in Newtonian stars and crucially depend on the stiffness of a star, we consider the tidal Love numbers of Newtonian polytropic stars whose stiffness is characterised by a polytropic index $n$. We first perturbatively solve the Lane-Emden equation governing the profile of polytropic stars through the application of the scaled delta expansion method and then formulate perturbation series for the multipolar tidal Love number about the two exactly solvable cases with $n=0$ and $n=1$, respectively. Making use of these two series to form a two-point Pad\'e approximant, we find an approximate expression of the quadrupole tidal Love number, whose error is less than $2.5 \times 10^{-5}$ per cent (0.39 per cent) for $n\in[0,1]$ ($n\in[0,3]$). Similarly, we also determine the mass moments for polytropic stars accurately. Based on these findings, we are able to show that the I-Love-Q relations are in general stationary about the incompressible limit irrespective of the equation of state (EOS) of a star. Moreover, for the I-Love-Q relations, there is a secondary stationary point near $n \approx 0.4444$, thus showing the insensitivity to $n$ for $n\in[0,1]$. Our investigation clearly tracks the universality of the I-Love-Q relations from their validity for stiff stars such as neutron stars to their breakdown for soft stars.

The induced motion of a probe coupled to a bath with random resettings

We consider a probe linearly coupled to the center of mass of a nonequilibrium bath. We study the induced motion on the probe for a model where a resetting mechanism is added to an overdamped bath dynamics with quadratic potentials. The fact that each bath particle is at random times reset to a fixed position is known for optimizing diffusive search strategies, but here stands for the nonequilibrium aspect of the bath. In the large bath scaling limit the probe is governed by an effective Langevin equation. Depending on the value of the parameters, there appear three regimes: (i) an equilibrium-like regime but with a reduced friction and an increased effective temperature; (ii) a regime where the noise felt by the probe is continuous but non-Gaussian and exhibits fat-tails; (iii) a regime with a non-Gaussian noise exhibiting power-law distributed jumps. The model thus represents an exactly solvable case for the origin of nonequilibrium probe dynamics.

Finding a subdivision of a prescribed digraph of order 4

AbstractThe problem of when a given digraph contains a subdivision of a fixed digraph F is considered. Bang‐Jensen et al. [4] laid out foundations for approaching this problem from the algorithmic point of view. In this article, we give further support to several open conjectures and speculations about algorithmic complexity of finding F‐subdivisions. In particular, up to five exceptions, we completely classify for which 4‐vertex digraphs F, the F‐subdivision problem is polynomial‐time solvable and for which it is NP‐complete. While all NP‐hardness proofs are made by reduction from some version of the 2‐linkage problem in digraphs, some of the polynomial‐time solvable cases involve relatively complicated algorithms.

Algorithmic analysis for ridesharing of personal vehicles

The ridesharing problem is to share personal vehicles by individuals with similar itineraries. A trip in the ridesharing problem is an individual and his/her itinerary. To realize a trip is to deliver the individual to his/her destination by a vehicle satisfying the itinerary requirement. Major optimization goals in the ridesharing problem include minimizing the number of vehicles and minimizing the total travel distance of vehicles to realize all trips. There are many parameters in the minimization problems, making them complex and NP-hard. The problems can be simplified by considering only some of the parameters. We give an algorithmic analysis for the simplified minimization problems and explore a boundary between the NP-hard and polynomial time solvable cases. We prove that the simplified minimization problems, where only the source, destination, vehicle capacity, detour distance and preferred path parameters are considered, are still NP-hard. We show that the simplified problem of minimizing the number of vehicles becomes polynomial time solvable if the considered parameters satisfy certain conditions. These suggest a boundary between the NP-hard and polynomial time solvable cases.

Preemptive parallel‐machine scheduling with a common server to minimize makespan

AbstractWe consider parallel‐machine scheduling with a common server and job preemption to minimize the makespan. While the non‐preemptive version of the problem is strongly NP‐hard, the complexity status of the preemptive version has remained open. We show that the preemptive version is NP‐hard even if there is a fixed number of machines. We give a pseudo‐polynomial time algorithm to solve the case with two machines. We show that the case with an arbitrary number of machines is unary NP‐hard, analyze the performance ratios of some natural heuristic algorithms, and present several solvable special cases. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 388–398, 2017

Application of triple partition model to the proof of supply chain scheduling problem

This paper focuses on the two-stage supply chain scheduling problem. For the first stage, workpieces with difference weights are processed in a certain sequence by one machine; for the second stage, the finished workpieces are transported by batches to various prespecified destinations via certain modes of transportation. Recursive algorithm is designed to minimize the sum of weighted workpieces processing time and the transportation cost. We draw the problem complexity for consistent weights by polynominal algorithms for solvable cases. On the other hand we proved for the most general case to be NP-hard by translating triple partition problem into our supply chain problem.

Squeezing effects applied in nonclassical superposition states for quantum nanoelectronic circuits

Quantum characteristics of a driven series RLC nanoelectronic circuit whose capacitance varies with time are studied using an invariant operator method together with a unitary transformation approach. In particular, squeezing effects and nonclassical properties of a superposition state composed of two displaced squeezed number states of equal amplitude, but 180° out of phase, are investigated in detail. We applied our developments to a solvable specific case obtained from a suitable choice of time-dependent parameters. The pattern of mechanical oscillation of the amount of charges stored in the capacitor, which are initially displaced, has exhibited more or less distortion due to the influence of the time-varying parameters of the system. We have analyzed squeezing effects of the system from diverse different angles and such effects are illustrated for better understanding. It has been confirmed that the degree of squeezing is not constant, but varies with time depending on specific situations. We have found that quantum interference occurs whenever the two components of the superposition meet together during the time evolution of the probability density. This outcome signifies the appearance of nonclassical features of the system. Nonclassicality of dynamical systems can be a potential resource necessary for realizing quantum information technique. Indeed, such nonclassical features of superposition states are expected to play a key role in upcoming information science which has attracted renewed attention recently.

Exactly solvable cases in QED with t-electric potential steps

In this paper, we present in detail consistent QED (and scalar QED) calculations of particle creation effects in external electromagnetic field that correspond to three most important exactly solvable cases of t-electric potential steps: Sauter-like electric field, T-constant electric field, and exponentially growing and decaying electric fields. In all these cases, we succeeded to obtain new results, such as calculations in the modified configurations of the above-mentioned steps and detailed considerations of new limiting cases in already studied steps. As was recently discovered by us, the information derived from considerations of exactly solvable cases allows one to make some general conclusions about quantum effects in fields for which no closed form solutions of the Dirac (or Klein–Gordon) equation are known. In the present paper, we briefly represent such conclusions about universal behavior of vacuum mean values in slowly varying strong electric fields.