Allocation of jobs and identical resources with two pooling centers

We examine the resource allocation problem of partitioning identical servers into two parallel pooling centers, and simultaneously assigning job types to pooling centers. Each job type has a distinct Poisson arrival rate and a distinct holding cost per unit time. Each pooling center becomes a queueing system with an exponential service time distribution. The goal is to minimize the total holding cost. The problem is shown to be polynomial if a job type can be divided between the pooling centers, and NP-hard if dividing job types is not possible. When there are two servers and jobs cannot be divided, we demonstrate that the two pooling center configuration is rarely optimal. A heuristic which checks the single pooling center has an upper bound on the relative error of 4/3. The heuristic is extended for the multiple server problem, where relative error is bounded above by the number of servers.      

The computational complexity of the pooling problem

The pooling problem is an extension of the minimum cost flow problem defined on a directed graph with three layers of nodes, where quality constraints are introduced at each terminal node. Flow entering the network at the source nodes has a given quality, at the internal nodes (pools) the entering flow is blended, and then sent to the terminal nodes where all entering flow streams are blended again. The resulting flow quality at the terminals has to satisfy given bounds. The objective is to find a cost-minimizing flow assignment that satisfies network capacities and the terminals’ quality specifications. Recently, it was proved that the pooling problem is NP-hard, and that the hardness persists when the network has a unique pool. In contrast, instances with only one source or only one terminal can be formulated as compact linear programs, and thereby solved in polynomial time. In this work, it is proved that the pooling problem remains NP-hard even if there is only one quality constraint at each terminal. Further, it is proved that the NP-hardness also persists if the number of sources and the number of terminals are no more than two, and it is proved that the problem remains hard if all in-degrees or all out-degrees are at most two. Examples of special cases in which the problem is solvable by linear programming are also given. Finally, some open problems, which need to be addressed in order to identify more closely the borderlines between polynomially solvable and NP-hard variants of the pooling problem, are pointed out.     

The FEXP estimator for potentially non-stationary linear time series

We consider semiparametric fractional exponential (FEXP) estimators of the memory parameter d for a potentially non-stationary linear long-memory time series with additive polynomial trend. We use differencing to annihilate the polynomial trend, followed by tapering to handle the potential non-invertibility of the differenced series. We propose a method of pooling the tapered periodogram which leads to more efficient estimators of d than existing pooled, tapered estimators. We establish asymptotic normality of the tapered FEXP estimator in the Gaussian case with or without pooling. We establish asymptotic normality of the estimator in the linear case if pooling is used. Finally, we consider minimax rate-optimality and feasible nearly rate-optimal estimators in the Gaussian case.     

A Multi-agent Based Self-adaptive Genetic Algorithm for the Long-term Car Pooling Problem

Rising vehicles number and increased use of private cars have caused significant traffic congestion, noise and energy waste. Public transport cannot always be set up in the non-urban areas. Car pooling, which is based on the idea that sets of car owners having the same travel destination share their vehicles has emerged to be a viable possibility to reduce private car usage around the world. In this paper, we present a multi-agent based self-adaptive genetic algorithm to solve long-term car pooling problem. The system is a combination of multi-agent system and genetic paradigm, and guided by a hyper-heuristic dynamically adapted by a collective learning process. The aim of our research is to solve the long-term car pooling problem efficiently with limited exploration of the search space. The proposed algorithm is tested using large scale instance data sets. The computational results show that the proposed method is competitive with other known approaches for solving long-term car pooling problem.     

Accelerating Branch-and-Bound through a Modeling Language Construct for Relaxation-Specific Constraints

In the tradition of modeling languages for optimization, a single model is passed to a solver for solution. In this paper, we extend BARON’s modeling language in order to facilitate the communication of problem-specific relaxation information from the modeler to the branch-and-bound solver. This effectively results into two models being passed from the modeling language to the solver. Three important application areas are identified and computational experiments are presented. In all cases, nonlinear constraints are provided only to the relaxation constructor in order to strengthen the lower bounding step of the algorithm without complicating the local search process. In the first application area, nonlinear constraints from the reformulation–linearization technique (RLT) are added to strengthen a problem formulation. This approach is illustrated for the pooling problem and computational results show that it results in a scheme that makes global optimization nearly as fast as local optimization for pooling problems from the literature. In the second application area, we communicate with the relaxation constructor the first-order optimality conditions for unconstrained global optimization problems. Computational experiments with polynomial programs demonstrate that this approach leads to a significant reduction of the size of the branch-and-bound search tree. In the third application, problem-specific nonlinear optimality conditions for the satisfiability problem are used to strengthen the lower bounding step and are found to significantly expedite the branch-and-bound algorithm when applied to a nonlinear formulation of this problem.     

On the Jump Number Problem in Hereditary Classes of Bipartite Graphs

We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying this condition, we propose polynomial algorithms to solve the jump number problem.     

On the computational complexity of a merge recognition problem

A problem arising from the work of C.A.R. Hoare on parallel programming is that of deciding whether a given string ? is a “merge” of two other given strings σ and τ. We describe a polynomial time algorithm for this problem. This algorithm can easily be extended to check, in polynomial time, whether ? is a merge of any fixed number of strings. The problem for an arbitrary number of strings is shown to be NP-complete and so is unlikely to have a polynomial time algorithm.     

N-fold integer programming and nonlinear multi-transshipment

The multi-transshipment problem is NP-hard already for two commodities over bipartite networks. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the polynomial time solvability of the problem in two broad situations. First, for any fixed number of commodities and number of suppliers, we solve the problem over bipartite networks with variable number of consumers in polynomial time. This is very natural in operations research applications where few facilities serve many customers. Second, for every fixed network, we solve the problem with variable number of commodities in polynomial time.     

Parameter choices on Guruswami–Sudan algorithm for polynomial reconstruction

Guruswami–Sudan algorithm for polynomial reconstruction problem plays an important role in the study of error-correcting codes. In this paper, we study new better parameter choices in Guruswami–Sudan algorithm for the polynomial reconstruction problem. As a consequence, our result gives a better upper bound for the number of solutions for the polynomial reconstruction problem comparing with the original algorithm.     

不确定二维多项式族的稳定性检验集合

不确定二维多项式族,包括多胞形二维多项式族,区间二维多项式族,钻石形二维多项式族,其鲁棒稳定性可由其棱边多项式的稳定性确定.如果二维多项式族的不确定参数较多,被检验棱边多项式的数目可能出现所谓组合爆炸问题.为解决这一问题,提出一检验集合,该集合根据棱边二维多项式的凸方向来构造,仅对属于该检验集合棱边二维多项式进行稳定性检验,从而大大地减少被检验的棱边二维多项式数目.给出一例来说明这种新方法的可行性.     

A bilinear approach to the pooling problem†

In this paper we present an algorithm for the pooling problem in refinery optimization based on a bilinear programming approach. The pooling problem occurs frequently in process optimization problems, especially refinery planning models. The main difficulty is that pooling causes an inherent nonlinearity in the otherwise linear models. We shall define the problem by formulating an aggregate mathematical model of a refinery, comment on solution methods for pooling problems that have been presented in the literature, and develop a new method based on convex approximations of the bilinear terms. The method is illustrated on numerical examples     

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].     

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.     

A class of nonlinear nonseparable continuous knapsack and multiple-choice knapsack problems

This paper considers a general class of continuous, nonlinear, and nonseparable knapsack problems, special cases of which arise in numerous operations and financial contexts. We develop important properties of optimal solutions for this problem class, based on the properties of a closely related class of linear programs. Using these properties, we provide a solution method that runs in polynomial time in the number of decision variables, while also depending on the time required to solve a particular one-dimensional optimization problem. Thus, for the many applications in which this one-dimensional function is reasonably well behaved (e.g., unimodal), the resulting algorithm runs in polynomial time. We next develop a related solution approach to a class of continuous, nonlinear, and nonseparable multiple-choice knapsack problems. This algorithm runs in polynomial time in both the number of variables and the number of variants per item, while again dependent on the complexity of the same one-dimensional optimization problem as for the knapsack problem. Computational testing demonstrates the power of the proposed algorithms over a commercial global optimization software package.     

A strongly polynomial minimum cost circulation algorithm

A new algorithm is presented for the minimum cost circulation problem. The algorithm is strongly polynomial, that is, the number of arithmetic operations is polynomial in the number of nodes, and is independent of both costs and capacities.     

带有链优先序的分批排序问题

本文首次就带有优先序的分批排序问题进行了讨论，目标函数为最大完工时间．当优先序为链，一条链上的工件个数为饨，而其它链的工件个数为常数，分批的容量B大于等于链的条数，在这种情况下，问题为多项式可解的．文中并讨论了几种特殊情况的多项式算法．     

一个双目标单机调度问题及其几个多项式可解情形

研究了一个双目标单机调度问题及其几个多项式可解的情形。问题的主目标是延误工件数最小,在此条件下,最小化各工件加权提前期的总和,由于该问题是NP一难的,故研究求解它的一个启发式算法及问题的几个多项式可解的特殊情形。     

Does inventory pooling improve customer service levels?

We study the risk pooling problem in a two-location inventory system, and analytically demonstrate that pooling improves the Type-I service levels at both locations. We also present an example showing that pooling may reduce the Type-II service levels at both locations.     

Decomposition strategy for the stochastic pooling problem

The stochastic pooling problem is a type of stochastic mixed-integer bilinear program arising in the integrated design and operation of various important industrial networks, such as gasoline blending, natural gas production and transportation, water treatment, etc. This paper presents a rigorous decomposition method for the stochastic pooling problem, which guarantees finding an ${\epsilon}$ -optimal solution with a finite number of iterations. By convexification of the bilinear terms, the stochastic pooling problem is relaxed into a lower bounding problem that is a potentially large-scale mixed-integer linear program (MILP). Solution of this lower bounding problem is then decomposed into a sequence of relaxed master problems, which are MILPs with much smaller sizes, and primal bounding problems, which are linear programs. The solutions of the relaxed master problems yield a sequence of nondecreasing lower bounds on the optimal objective value, and they also generate a sequence of integer realizations defining the primal problems which yield a sequence of nonincreasing upper bounds on the optimal objective value. The decomposition algorithm terminates finitely when the lower and upper bounds coincide (or are close enough), or infeasibility of the problem is indicated. Case studies involving two example problems and two industrial problems demonstrate the dramatic computational advantage of the proposed decomposition method over both a state-of-the-art branch-and-reduce global optimization method and explicit enumeration of integer realizations, particularly for large-scale problems.     

Parametric simplex algorithms for a class of NP-Complete problems whose average number of steps is polynomial

We will show that the average number of steps of parametric simplex algorithms for obtaining global minima of rank-one and rank-two bilinear-programming problems are lower-order polynomial functions of the problem size under the standard assumptions on the distribution of the data imposed in the probabilistic analysis of the simplex method. This means that there exist algorithms for some special class of NP-complete problems, whose average number of arithmetics are polynomial order of the problem size.