Constrained Inverse Minimum Spanning Tree Problems under the Bottleneck-Type Hamming Distance

In this paper, we consider the inverse minimum spanning tree problem under the bottleneck-type Hamming distance, where the weights of edges can be modified only within given intervals. We further consider the constrained case in which the total modification cost cannot exceed a given upper bound. It is shown that these inverse problems can be transformed into a minimum node cover problem on a bipartite graph, and we give a strongly polynomial time algorithm to solve this type of node cover problems. This work is supported by The National Natural Science Foundation of China (60021201), The Hong Kong Research Grant Council under the grant CERG 9040883 (CITYU 103003), and the Doctoral Foundation of Hohai University (2005-02).     

Inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow x⁰, an inverse minimum cost flow problem is to modify the cost vector as little as possible to make x⁰ form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the inverse minimum cost flow problems, where the modification of the arcs is measured by the weighted Hamming distance. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in O(n · m²).     

Capacity inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow \(x^0\), the inverse minimum cost flow problem is to modify the capacity vector or the cost vector as little as possible to make \(x^0\) form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the capacity inverse minimum cost flow problems under the weighted Hamming distance, where we use the weighted Hamming distance to measure the modification of the arc capacities. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in \(O(n\cdot m^2)\) time.     

哈明距离下的网络逆问题研究综述

逆优化问题研究的是如何改变原问题中的权参数,使得某些给定的解是问题在新的权参数下的最优解,且使总的改造费用尽可能少.作为逆优化问题中相对较新的一个分支,哈明距离下的网络逆问题具有较大的理论研究及实际应用价值.此文首先介绍了逆优化问题和哈明距离下的网络逆问题以及它们的应用,然后详细介绍了哈明距离下的网络逆问题的研究动态及使用的研究方法.最后给出了该领域中的一些值得研究的问题.     

Constrained inverse min–max spanning tree problems under the weighted Hamming distance

In this paper, we consider the constrained inverse min–max spanning tree problems under the weighted Hamming distance. Three models are studied: the problem under the bottleneck-type weighted Hamming distance and two mixed types of problems. We present their respective combinatorial algorithms that all run in strongly polynomial times. This research is supported by the National Natural Science Foundation of China (Grant No. 10601051).     

动态最小费用路在L1模下的逆问题讨论

考察动态最小费用路在L_1模下的逆问题,其中在弧费用的定义中,将弧(i,j)上的运行时间d_(ij)(t)分成最小可能运行时间d_(ij)~*和超出的运行时间(excess time)e_(ij)(t)两部分,弧(i,j)上费用即为两者赋权之和.在逆问题的讨论中考虑先将动态网络中的问题通过时间扩张网络G~T转化为静态问题,然后再利用解线性规划的逆问题的方法来解该动态最短路问题的逆问题.     

Weighted inverse maximum perfect matching problems under the Hamming distance

Given an undirected network G(V, E, c) and a perfect matching M ⁰, the inverse maximum perfect matching problem is to modify the cost vector as little as possible such that the given perfect matching M ⁰ can form a maximum perfect matching. The modification can be measured by different norms. In this paper, we consider the weighted inverse maximum perfect matching problems under the Hamming distance, where we use the weighted Hamming distance to measure the modification of the edges. We consider both of the sum-type and the bottleneck-type problems. For the general case of the sum-type case, we show it is NP-hard. For the bottleneck-type, we present a strongly polynomial algorithm which can be done in O(m · n ³).     

An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance

Shortest path problems play important roles in computer science, communication networks, and transportation networks. In a shortest path improvement problem under unit Hamming distance, an edge-weighted graph with a set of source–terminal pairs is given. The objective is to modify the weights of the edges at a minimum cost under unit Hamming distance such that the modified distances of the shortest paths between some given sources and terminals are upper bounded by the given values. As the shortest path improvement problem is NP-hard, it is meaningful to analyze the complexity of the shortest path improvement problem defined on rooted trees with one common source. We first present a preprocessing algorithm to normalize the problem. We then present the proofs of some properties of the optimal solutions to the problem. A dynamic programming algorithm is proposed for the problem, and its time complexity is analyzed. A comparison of the computational experiments of the dynamic programming algorithm and MATLAB functions shows that the algorithm is efficient although its worst-case complexity is exponential time.     

Inverse 1-median problem on trees under weighted Hamming distance

The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l _?? norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is ${\mathcal{NP}}$ -hard.     

对称的运输问题及其逆问题

本文对［１,２,６］中提出的运输问题进行了推广,并提出了一个强多项式算法,从而改进了原有的结果．同时对对称的运输问题的逆问题进行了研究,并借助于最小费用循环流技术得到了一个强多项式算法．     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).     

Combinatorial Algorithms for Some Variants of Inverse Obnoxious Median Location Problem on Tree Networks

This paper concerns with some variants of the inverse obnoxious median location problem on tree networks, where the aim is either to augment or to reduce the edge lengths at the minimum total cost such that a prespecified subset of vertices becomes an obnoxious multi-facility median location with respect to the perturbed edge lengths. For both augmentation and reduction models, under the rectilinear norm and the sum-type Hamming distance, we develop novel combinatorial algorithms with polynomial time complexities. Particularly, if the underlying tree is an extended star graph, then the problems can be solved in linear time.     

一类最小支撑树的逆问题及其求解方法

本文针对传统的基于边的最小支撑树逆问题，提出了一类基于点边更新策略的最小支撑树逆问题．更新一个点是指减少与此点相关联的某些边的权值．根据是否含有更新点的费用，考虑了两类模型，它们均可转化为森林上的最小(费用)点覆盖的求解问题，算法的复杂性都是O(mn)，其中m=|E|n=|V|。     

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.     

A Class of Inverse Dominant Problems under Weighted l ∞ Norm and an Improved Complexity Bound for Radzik’s Algorithm

In this paper, we first discuss a class of inverse dominant problems under weighted l_∞ norm, which is how to change the original weights of elements with bounds in a finite ground set so that a given set becomes a weakly dominant set with respect to a given collection of subsets under the new weights and the largest change of the weights is minimum. This model includes a large class of improvement problems in combinatorial optimization. We propose a Newton-type algorithm for the model. This algorithm can solve the model in strongly polynomial time if the subproblem involved is solvable in strongly polynomial time. In the second part of the paper, we improve the complexity bound for Radzik’s Newton-type method which is designed to solve linear fractional combinatorial optimization problems. As Radzik’s method is closely related to our algorithm, this bound also estimates the complexity of our algorithm. Supported by the Hong Kong Universities Grant Council (CERG CITYU 9040883 and 9041091). Xiaoguang Yang - The author is also grateful for the support by the National Key Research and Development Program of China (Grant No. 2002CB312004) and the National Natural Science Foundation of China (Grant No. 70425004).     

Some inverse optimization problems under the Hamming distance

Given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph, the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal. The aim is to optimize the difference between the initial cost vector and the disturbed one. This difference can be measured in several ways. We consider the Hamming distance measuring in how many components two vectors are different, where weights are associated to the components. General algorithms for the bottleneck or minimax criterion are described and (after modification) applied to the inverse minimum spanning tree problem, the inverse shortest path tree problem and the linear assignment problem.     

NEWTON-REGULARIZED METHOD BASED ON A BASIC FUNCTION FOR SOLVING THE INVERSE PROBLEM OF CONVECTION-DIFFUSION EQUATIONS

In this paper, a new iteration algorithm to solve the coefficient inverse problem is described by using a "basic function" which is specially defined and the idea of regularization. The method is simple and clear.The main advantage of the algorithm is that its computing cost is less than other current algorithms, such as PST and Purlerbalion Methods. Since it has uniform scheme, on the other hand, the method can be easily exleded to other kinds of inverse problems of different leal equations, multidimensional inverse problem and multiparameler inverse problems, etc.     

A penalty function method for solving inverse optimal value problem

In order to consider the inverse optimal value problem under more general conditions, we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently. Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the nonlinear bilevel programming into a normal nonlinear programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem, also analysis the convergence of the proposed algorithm. The numerical result shows that the algorithm can solve a wider class of inverse optimal value problem.     

基于计算机试验的均匀设计

本文在计算机试验的基础上，提出了最小相关准则和最小距离离差准则，并将信息论中的Hamming距离和Lee距离引入到计算机试验中，证明了均匀设计在Hamming距离下的最优性和部分好格子点均匀设计在Lee距离下的最优性．基于偏差的考虑，给出了一类新的好格子点均匀设计和一个学习算法，利用这个学习算法，给出了基于Lee距离的最小距离离差准则的均匀设计表的构造方法．通过与已有的好格子点均匀设计和循环拉丁方均匀设计作比较，证明了文中的均匀设计在距离和偏差意义下有更好的均匀性．     

A multi-population hybrid biased random key genetic algorithm for hop-constrained trees in nonlinear cost flow networks

Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.