Solvable Cases Research Articles

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.

Numerical calculation of the quasinormal frequencies for the Dirac field in a Lifshitz black brane

In the zero momentum limit we numerically calculate the quasinormal frequencies of the massive Dirac field propagating in a Lifshitz black brane. We focus on the non-exactly solvable cases for the fermionic perturbations, so that our results are an extension of the examples already reported for the massive Klein–Gordon and Dirac fields in the zero momentum limit. Based on our numerical results, we propose an analytical approximation of the obtained quasinormal frequencies of the Dirac field and compare their behavior with those of the Klein–Gordon field. We extend the results on the Klein–Gordon quasinormal frequencies already published. Furthermore, by imposing the Dirichlet boundary condition at the asymptotic region, we are able to find more general results for the fermionic exactly solvable case previously studied.

Semiclassical work and quantum work identities in Weyl representation

We derive a semiclassical nonequilibrium work identity by applying the Wigner–Weyl quantization scheme to the Jarzynski identity for a classical Hamiltonian. This allows us to extend the concept of work to the leading order in ℏ. We propose a geometric interpretation of this semiclassical Jarzynski relation in terms of trajectories in a complex phase space and illustrate it with the exactly solvable case of the quantum harmonic oscillator.

Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories

We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulate Noether's theorem for global, local, continuous or discrete Abelian symmetries: for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classical approximations for classical solutions with periodic boundary conditions in two solvable cases. We show the correspondence of their weak coupling limit with the tensor formulation after Poisson summation. We briefly discuss connections with other approaches and implications for quantum computing.

Full nonuniversality of the symmetric 16-vertex model on the square lattice.

We consider the symmetric two-state 16-vertex model on the square lattice whose vertex weights are invariant under any permutation of adjacent edge states. The vertex-weight parameters are restricted to a critical manifold which is self-dual under the gauge transformation. The critical properties of the model are studied numerically with the Corner Transfer Matrix Renormalization Group method. Accuracy of the method is tested on two exactly solvable cases: the Ising model and a specific version of the Baxter eight-vertex model in a zero field that belong to different universality classes. Numerical results show that the two exactly solvable cases are connected by a line of critical points with the polarization as the order parameter. There are numerical indications that critical exponents vary continuously along this line in such a way that the weak universality hypothesis is violated.

Integrable multistate Landau–Zener models with parallel energy levels

We discuss solvable multistate Landau–Zener (MLZ) models whose Hamiltonians have commuting partner operators with ∼1/τ-time-dependent parameters. Many already known solvable MLZ models belong precisely to this class. We derive the integrability conditions on the parameters of such commuting operators, and demonstrate how to use such conditions in order to derive new solvable cases. We show that MLZ models from this class must contain bands of parallel diabatic energy levels. The structure of the scattering matrix and other properties are found to be the same as in the previously discussed completely solvable MLZ Hamiltonians.

Subgraph Isomorphism on Graph Classes that Exclude a Substructure

We study Subgraph Isomorphism on graph classes defined by a fixed forbidden graph. Although there are several ways for forbidding a graph, we observe that it is reasonable to focus on the minor relation since other well-known relations lead to either trivial or equivalent problems. When the forbidden minor is connected, we present a near dichotomy of the complexity of Subgraph Isomorphism with respect to the forbidden minor, where the only unsettled case is \(P_{5}\), the path of five vertices. We then also consider the general case of possibly disconnected forbidden minors. We show fixed-parameter tractable cases and randomized XP-time solvable cases parameterized by the size of the forbidden minor H. We also show that by slightly generalizing the tractable cases, the problem becomes NP-complete. All unsettle cases are equivalent to \(P_{5}\) or the disjoint union of two \(P_{5}\)’s. As a byproduct, we show that Subgraph Isomorphism is fixed-parameter tractable parameterized by vertex integrity. Using similar techniques, we also observe that Subgraph Isomorphism is fixed-parameter tractable parameterized by neighborhood diversity.

Optimization-Driven Scenario Grouping

Scenario decomposition algorithms for stochastic programs compute bounds by dualizing all nonanticipativity constraints and solving individual scenario problems independently. We develop an approach that improves on these bounds by reinforcing a carefully chosen subset of nonanticipativity constraints, effectively placing scenarios into groups. Specifically, we formulate an optimization problem for grouping scenarios that aims to improve the bound by optimizing a proxy metric based on information obtained from evaluating a subset of candidate feasible solutions. We show that the proposed grouping problem is NP-hard in general, identify a polynomially solvable case, and present two formulations for solving the problem: a matching formulation for a special case and a mixed-integer programming formulation for the general case. We use the proposed grouping scheme as a preprocessing step for a particular scenario decomposition algorithm and demonstrate its effectiveness in solving standard test instances of two-stage 0–1 stochastic programs. Using this approach, we are able to prove optimality for all previously unsolved instances of a standard test set. Additionally, we implement this scheme as a preprocessing step for PySP, a publicly available and widely used implementation of progressive hedging, and compare this grouping approach with standard grouping approaches on large-scale stochastic unit commitment instances. Finally, the idea is extended to propose a finitely convergent algorithm for two-stage stochastic programs with a finite feasible region.

Comparing temporal graphs using dynamic time warping

Within many real-world networks, the links between pairs of nodes change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a proximity measure to compare different temporal graphs. To this end, we propose to study dynamic time warping on temporal graphs. We define the dynamic temporal graph warping (dtgw) distance to determine the dissimilarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (in general) NP-hard optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. In experiments on real-world data, we show that the heuristic performs very well and that our dtgw-distance performs favorably in de-anonymizing networks compared to other approaches.

Steady states of lattice population models with immigration

ABSTRACT In a lattice population model where individuals evolve as subcritical branching random walks subject to external immigration, the cumulants are estimated and the existence of the steady state is proved. The resulting dynamics are Lyapunov stable in that their qualitative behavior does not change under suitable perturbations of the main parameters of the model. An explicit formula of the limit distribution is derived in the solvable case of no birth. Monte Carlo simulation shows the limit distribution in the solvable case.

A class of exponential neighbourhoods for the quadratic travelling salesman problem

The quadratic travelling salesman problem (QTSP) is to find a least-cost Hamiltonian cycle in an edge-weighted graph, where costs are defined on all pairs of edges such that each edge in the pair is contained in the Hamiltonian cycle. This is a more general version than the one that appears in the literature as the QTSP, denoted here as the adjacent quadratic TSP, which only considers costs for pairs of adjacent edges. Major directions of research work on the linear TSP include exact algorithms, heuristics, approximation algorithms, polynomially solvable special cases and exponential neighbourhoods (Gutin G, Punnen A (eds): The traveling salesman problem and its variations, vol 12. Combinatorial optimization. Springer, New York, 2002) among others. In this paper we explore the complexity of searching exponential neighbourhoods for QTSP, the fixed-rank QTSP, and the adjacent quadratic TSP. The fixed-rank QTSP is introduced as a restricted version of the QTSP where the cost matrix has fixed rank p. It is shown that fixed-rank QTSP is solvable in pseudopolynomial time and admits an FPTAS for each of the special cases studied, except for the case of matching edge ejection tours. The adjacent quadratic TSP is shown to be polynomially-solvable in many of the cases for which the linear TSP is polynomially-solvable. Interestingly, optimizing over the matching edge ejection tour neighbourhood is shown to be pseudopolynomial for the rank 1 case without a linear term in the objective function, but NP-hard for the adjacent quadratic TSP case. We also show that the quadratic shortest path problem on an acyclic digraph can be solved in pseudopolynomial time and by an FPTAS when the rank of the associated cost matrix is fixed.

The Goursat and Cauchy Problems for Three-Dimensional Bianchi Equation

We apply new approach for the Goursat and Cauchy problems with additional recovering of coefficients of equation for selection of the cases of solvability of these problems in quadratures. Instead of introduction of additional boundary value conditions we offer restrictions on the structure of the equation connected with possibility of its factorization.

Three-body problem for Langevin dynamics with different temperatures.

A mixture of Brownian particles at different temperatures has been a useful model for studying the out-of-equilibrium properties of systems made up of microscopic components with differing levels of activity. This model was previously studied analytically for two-particle interactions in the dilute limit, yielding a Boltzmann-like two-particle distribution with an effective temperature. Like the Newtonian two- and three-body problems, we ask here whether the two-particle results can be extended to three-particle interactions to get the three-particle distributions. By considering the special solvable case of pairwise quadratic interactions, we show that, unlike the two-particle distribution, the three-particle distribution cannot in general be Boltzmann-like with an effective temperature. We instead find that the steady-state distribution of any two particles in a triplet depends on the properties of and interactions with the third particle, leading to some unexpected behaviors not present in equilibrium.

The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem

In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.

The no-wait flow shop with rejection

We consider the no-wait flow shop scheduling problem with rejection. We first show how to augment the travelling salesman problem formulation for the no-wait flow shop minimum makespan problem to incorporate the rejection option. We then focus on polynomially solvable cases by considering problems with ordered jobs. We present a third-order polynomial-time dynamic programming algorithm to minimise the sum of makespan and total rejection cost and faster quadratic algorithms for two special cases. We also present a third-order polynomial-time dynamic programming algorithm to minimise the sum of total completion time and total rejection cost with ordered jobs. We also exploit the duality between no-wait and no-idle flow shops to determine which operations to outsource in a no-idle flow shop with an outsourcing option.

Optimal order picker routing in a conventional warehouse with two blocks and arbitrary starting and ending points of a tour

This paper investigates manual order picking, where workers travel through the warehouse to retrieve requested items from shelves. To minimise the completion time of orders, researchers have developed various routing procedures that guide order pickers through the warehouse. The paper at hand contributes to this stream of research and proposes an optimal order picker routing policy for a conventional warehouse with two blocks and arbitrary starting and ending points of a tour. The procedure proposed in this paper extends an earlier work of Löffler et al. (2018. Picker routing in AGV-assisted order picking systems, Working Paper, DPO-01/2018, Deutsche Post Chair-Optimization of Distribution Networks, RWTH Aachen University, 2018) by applying the concepts of Ratliff and Rosenthal (1983. “Order-picking in a Rectangular Warehouse: a Solvable Case of the Traveling Salesman Problem.” Operations Research 31 (3): 507–521) and Roodbergen and de Koster (2001a. “Routing Order Pickers in a Warehouse with a Middle Aisle.” European Journal of Operational Research 133 (1): 32–43) that used graph theory and dynamic programming for finding an optimal picker route. We also propose a routing heuristic, denoted S*-shape, for conventional two-block warehouses with arbitrary starting and ending points of a tour. In computational experiments, we compare the average order picking tour length in a conventional warehouse with a single block to the case of a conventional warehouse with two blocks to assess the impact of the middle cross aisle on the performance of the warehouse. Furthermore, we evaluate the performance of the S*-shape heuristic by comparing it to the exact algorithm proposed in this study.

Stable Matchings with Covering Constraints: A Complete Computational Trichotomy

Stable matching problems with lower quotas are fundamental in academic hiring and ensuring operability of rural hospitals. Only few tractable (polynomial-time solvable) cases of stable matching with lower quotas have been identified; most such problems are mathsf {NP}-hard and also hard to approximate (Hamada et al. in Algorithmica 74(1):440–465, 2016). We therefore consider stable matching problems with lower quotas under a relaxed notion of tractability, namely fixed-parameter tractability. By cloning hospitals we focus on the case when all hospitals have upper quota equal to 1, which generalizes the setting of “arranged marriages” first considered by Knuth (Mariages stables et leurs relations avec d’autres problèmes combinatoires, Les Presses de l’Université de Montréal, Montreal, 1976). We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy: for each choice of parameters we either provide a polynomial-time algorithm, or an mathsf {NP}-hardness proof and fixed-parameter algorithm, or mathsf {NP}-hardness proof and mathsf {W}[1]-hardness proof. As corollary, we negatively answer a question by Hamada et al. (Algorithmica 74(1):440–465, 2016) by showing fixed-parameter intractability parameterized by optimal solution size. We also classify all cases of one-sided constraints where only women may be distinguished.

Optimal Makespan for the Coupled Task Ordered Flow-Shop

The ordered flow-shop scheduling problem deals with the case where processing times follow a specific structure. We extend the ordered flow-shop scheduling to the coupled tasks setting, i.e., the coupled task ordered flow-shop, in which an exact delay exists between the consecutive tasks of each job. The aim is to minimize the makespan. The coupled task flow-shop with distinct delays between the consecutive tasks of the jobs is known to be strongly NP-hard, even for two machines and the unit execution time tasks. We investigate the problem when delays follow the ordered structure and show that the optimal permutation schedule has the pyramidal-shaped property. We also introduce two polynomially solvable cases.

Two-Machine Ordered Flow Shop Scheduling with Generalized Due Dates

We consider a two-machine flow shop scheduling with two properties. The first is that each due date is assigned for a specific position different from the traditional definition of due dates, and the second is that a consistent pattern exists in the processing times within each job and each machine. The objective is to minimize maximum tardiness, total tardiness, or total number of tardy jobs. We prove the strong NP-hardness and inapproximability, and investigate some polynomially solvable cases. Finally, we develop heuristics and verify their performances through numerical experiments.

Control sets of linear systems on semi-simple Lie groups

In this paper we study the main properties of control sets with nonempty interior of linear systems on semisimple Lie groups. We show that, unlike the solvable case, linear systems on semisimple Lie groups may have more than one control set with nonempty interior and that they are contained in right translations of the one around the identity.