Solvable Cases Research Articles

A case study of the widely applicable Bayesian information criterion and its optimality

In Bayesian statistics, the marginal likelihood (evidence) is one of the key factors that can be used as a measure of model goodness. However, for many practical model families it cannot be computed analytically. An alternative solution is to use some approximation method or time-consuming sampling method. The widely applicable Bayesian information criterion (WBIC) was developed recently to have a marginal likelihood approximation that works also with singular models. The central idea of the approximation is to select a single thermodynamic integration term (power posterior) with the (approximated) optimal temperature $$\beta ^*=1/\log (n)$$βÂ?=1/log(n), where $$n$$n is the data size. We apply this new approximation to the analytically solvable Gaussian process regression case to show that the optimal temperature may depend also on data itself or other variables, such as the noise level. Moreover, we show that the steepness of a thermodynamic curve at the optimal temperature indicates the magnitude of the error that WBIC makes.

Generalized Langevin equation: An efficient approach to nonequilibrium molecular dynamics of open systems

The Generalized Langevin Equation (GLE) has been recently suggested to simulate the time evolution of classical solid and molecular systems when considering general non-equilibrium processes. In this approach, a part of the whole system (an open system), which interacts and exchanges energy with its dissipative environment, is studied. Because the GLE is derived by projecting out exactly the harmonic environment, the coupling to it is realistic, while the equations of motion are non-Markovian. Although the GLE formalism has already found promising applications, e.g., in nanotribology and as a powerful thermostat for equilibration in classical molecular dynamics simulations, efficient algorithms to solve the GLE for realistic memory kernels are highly non-trivial, especially if the memory kernels decay non-exponentially. This is due to the fact that one has to generate a colored noise and take account of the memory effects in a consistent manner. In this contribution, we present a simple, yet efficient, algorithm for solving the GLE for practical memory kernels and we demonstrate its capability for the exactly solvable case of a harmonic oscillator coupled to a Debye bath.

A polyhedral study of a two level facility location model

We study an uncapacitated facility location model where customers are served by facilities of level one, then each level one facility that is opened must be assigned to an opened facility of level two. We identify a polynomially solvable case, and study some valid inequalities and facets of the associated polytope.

Complexity and approximation for Traveling Salesman Problems with profits

In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for the polynomially solvable cases and fully polynomial time approximation schemes (FPTAS) for some NP-hard cases.

A Polynomially Solvable Case for Solving Crane Scheduling Problem in Batch Annealing Process

This paper investigates a special case for scheduling of the multi-crane operations in batch annealing process in an iron and steel enterprise so that the completion time of the last annealed coil (makespan) is minimized. Additional no-delay and no-buffer constraints for taking fully advantage of machines are considered. No-delay constraints for machine unloading means technological requires fixed heating time and cooling time that once the heating (cooling) is completed, the furnace (cooler) must be unloaded immediately without any delay by crane. No-buffer constraints for machine unloading means the time gap between the unloading time of one machine for one job and its start time on the next job is exactly equal to the corresponding transportation time. We prove that this problem can be solved optimally in polynomial time where the starting time and ending time of every possible crane movement are formulated.

A note on efficient domination in a superclass of [formula omitted]-free graphs

In a graph G=(V,E), an efficient dominating set is a subset D⊆V such that D is an independent set such that each vertex outside D has exactly one neighbor in D. E stands for the graph with vertices a,b,c,d,e,f and edges ab,bc,cd,de and cf, while xNet stands for the graph with vertices a,b,c,d,e,f,g and edges ad,be,cf,de,ef,fd and cg. This note shows that, given an (E,xNet)-free graph, a minimum efficient dominating set (if any) can be found in polynomial time, extending two polynomially solvable cases for efficient dominating sets recently found by Milanič (2013) [4], and Brandstädt et al. (2013) [2].

Practical method to obtain a lower bound to the three-tangle

The quantitative assessment of the entanglement in multipartite quantum states is, apart from its fundamental importance, a practical problem. Recently there has been significant progress in developing new methods to determine certain entanglement measures. In particular, there is a method---in principle, analytical---to compute a certified lower bound for the three-tangle. The purpose of this work is to provide a manual for the implementation of this approach and to explicitly discuss several analytically solvable cases in order to gauge the numerical tools. Moreover, we derive a simple analytical bound for the mixed state three-tangle.

The quadratic balanced optimization problem

We introduce the quadratic balanced optimization problem (QBOP) which can be used to model equitable distribution of resources with pairwise interaction. QBOP is strongly NP-hard even if the family of feasible solutions has a very simple structure. Several general purpose exact and heuristic algorithms are presented. Results of extensive computational experiments are reported using randomly generated quadratic knapsack problems as the test bed. These results illustrate the efficacy of our exact and heuristic algorithms. We also show that when the cost matrix is specially structured, QBOP can be solved as a sequence of linear balanced optimization problems. As a consequence, we have several polynomially solvable cases of QBOP.

Modeling a thermionic energy converter using finite-difference time-domain particle-in-cell simulations

A thermionic energy converter (TEC) is a static device that converts heat directly into electricity by boiling electrons off a hot emitter surface across a small inter-electrode gap to a cooler collector surface. The main challenge in TECs is overcoming the space charge limit, which limits the current transmitted across a gap of a given voltage and width. We have verified the feasibility of studying and developing a TEC using a bounded finite-difference time-domain particle-in-cell plasma simulation code, OOPD1, developed by Plasma Theory and Simulation Group, formerly at UC Berkeley and now at Michigan State University. In this preliminary work, a TEC has been modeled kinetically using OOPD1, and the accuracy has been verified by comparing with an analytically solvable case, giving good agreement. With further improvement of the code, one will be able to quickly and cheaply analyze space charge effects, and seek designs that mitigate the space charge effect, allowing TECs to become more efficient and cost-effective.

Dynamic Container Deployment: Two-Stage Robust Model, Complexity, and Computational Results

Containers are widely used in the shipping industry mainly because of their capability to facilitate multimodal transportation. How to effectively reposition the nonrevenue empty containers is the key to reduce the cost and improve the service in the liner shipping industry. In this paper, we propose a two-stage robust optimization model that takes into account the laden containers routing as well as the empty container repositioning, and define the robustness for this model with uncertainties in the supply and demand of the empty containers. Based on this definition, we present the robust formulations for the uncertainty sets corresponding to the ℓp-norm, where p = 1, 2, and ∞, and analyze the computational complexities for all of these formulations. The only polynomial-time solvable case corresponds to the ℓ1-norm, which we use to conduct the numerical study. We compare our approach with both the deterministic model and the stochastic model for the same problem in the rolling horizon simulation environment. The computational results establish the potential practical usefulness of the proposed approach.

Dedicated Transportation Subnetworks: Design, Analysis, and Insights

A dedicated subnetwork (DSN) refers to a subset of lanes, with associated loads, in a shipper's transportation network, for which resources—trucks, drivers, and other equipment—are exclusively assigned to accomplish shipping requirements. The resources assigned to a DSN are not shared with the rest of the shipper's network. Thus, a DSN is an autonomously operated subnetwork and, hence, can be subcontracted. We address a novel problem of extracting a DSN for outsourcing to one or more subcontractors, with the objective of maximizing the shipper's savings. In their pure form, the defining conditions of a DSN are often too restrictive to enable the extraction of a sizable subnetwork. We consider two notions— deadheading and lane‐sharing—that aid in improving the size of the DSN. We show that all the optimization problems involved are both strongly NP‐hard and APX‐hard, and demonstrate several polynomially solvable special cases arising from topological properties of the network and parametric relationships. Next, we develop a network‐flow‐based heuristic that provides near‐optimal solutions to practical instances in reasonable time. Finally, using a test bed based on data obtained from a national 3PL company, we demonstrate the substantial monetary impact of subcontracting a DSN and offer useful managerial insights.

Lefschetz coincidence numbers of solvmanifolds with Mostow conditions

For any two continuous maps $f,g$ between two solvmanifolds of same dimension satisfying the Mostow condition, we give a technique of computation of the Lefschetz coincidence number of $f,g$. This result is an extension of the result of Ha, Lee and Penninckx for completely solvable case.

Blockers for the Stability Number and the Chromatic Number

Given an undirected graph G = (V, E) and two positive integers k and d, we are interested in finding a set of edges (resp. non-edges) of size at most k to delete (resp. to add) in such a way that the chromatic number (resp. stability number) in the resulting graph will decrease by at least d compared to the original graph. We investigate these two problems in various classes of graphs (split graphs, threshold graphs, bipartite graphs and their complements) and determine their computational complexity. In some of the polynomial-time solvable cases, we also give a structural description of a solution.

Work fluctuations in a time-dependent harmonic potential: Rigorous results beyond the overdamped limit

We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail with a power-law prefactor, accompanied by an interesting oscillatory feature (multiple pseudo-locking-unlocking transitions) due to the inertial effect. Some exactly solvable cases are discussed in the overdamped limit.

Extension of the zero-range potential model onto the Hamiltonians with a singularity at the origin

We evaluate the short-range asymptotic behavior of the Green function for a Hamiltonian when its potential energy part has an inverse power singularity at the origin. The analytically solvable case of sharply screened Coulomb potential is considered firstly. For this potential the additional logarithmic singular term has been found in the short-range asymptote of the Green function as in the case of the pure Coulomb potential. The case of a short-range potential of an arbitrary form with inverse power singularity is treated on the basis of the integral Lippmann-Schwinger equation. It is shown that, if the singularity is weaker than the Coulomb one, the Green function has only standard singularity. For the case of rsingularity of the potential with 1 � � < 2 the additional singularity in the asymptotic behavior of the Green function appears. In the case of � = 1 the additional logarithmic singularity has the same form as in the case of the pure Coulomb potential. In the case of 1 < � < 2 the additional singularity of the Green function has the form of the polar singularity r �+1 . These results are applied for extending the zero-range potential formalism on Hamiltonians with singular potentials.

Large deviation rate functions for the partition function in a log-gamma distributed random potential

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the $1+1$-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dimensions.

Bicriteria hierarchical optimization of two-machine flow shop scheduling problem with time-dependent deteriorating jobs

We study a two-machine flowshop scheduling problem with time-dependent deteriorating jobs, i.e. the processing times of jobs are an increasing function of their starting time. The objective is to minimize the total completion time subject to minimum makespan. We propose a mixed integer programming model, and develop two pairwise interchange algorithms and a branch-and-bound procedure to solve the problem while using several dominance conditions to limit the size of the search tree. Several polynomial-time solvable special cases are discussed. Finally, numerical studies are performed to examine the effectiveness and the efficiency of the proposed algorithms.

Improved estimation of distribution algorithm for the problem of single-machine scheduling with deteriorating jobs and different due dates

This paper investigates single-machine scheduling problem, which is an NP-hard problem, with deteriorating jobs and different due dates to minimize total tardiness. First, two special polynomially solvable cases of the problem and a mixed-integer programming (MIP) model are proposed. Since the large-scale problem needs a long time when the MIP is solved using the CPLEX, the improved estimation of distribution algorithm (EDA) is proposed to solve the problem with a large size. EDA depends on the probabilistic model, which denotes the distribution of decision variables in the feasible region space. Meanwhile, EDA owns efficient search capability and convergence. To obtain an improved initial population, an efficient initialization scheme based on the feature of two special cases and a heuristic algorithm are adopted in the process of constructing the initial population. The probabilistic model is composited based on elite solutions from each generation. Simultaneously, mutation is embedded to maintain the diversity of the population. Compared with the results, numerical experiments show that the proposed algorithm can obtain good near-optimal solutions within a short period.

Economic lot sizing: The capacity reservation model

Capacity reservation contracts allow a consumer to purchase up to a certain capacity at a unit price lower than that of the spot market, while the consumer’s excess orders are realized at the spot price. In this paper, we consider a lot sizing problem where the consumer places orders following a capacity reservation contract. In particular, we study the general problem and the polynomial time solvable special cases of the problem and propose corresponding algorithms for them.

Scheduling the two-machine open shop problem under resource constraints for setting the jobs

This paper addresses the problem of open shop scheduling on two machines with resources constraints. In the context of our study, in order to be executed, a job requires first, for its preparation for a given period of time, a number of resources which cannot exceed a given resource capacity. Then, it goes onto its execution while the resources allocated to it become available again. We seek a schedule that minimizes the makespan. We first prove the $\mathcal{N}\mathcal{P}$ -hardness of several versions of this problem. Then, we present a well solvable case, lower bounds, and heuristic algorithms along with an experimental study.