The Group Problems and Subadditive Functions

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

The paper presents the Integer Programming (IP) algorithms in mining a compact rough classification model. The algorithm is based on creating a 0--1 integer programming model from a rough discernibility relations of a decision system (DS) to find minimum selection of important attributes which is called reduct in rough set theory. A branch and bound search strategy that performs a non-chronological backtracking is proposed to solve the problem. The experimental result shows that the proposed IP algorithm has significantly reduced the number of rules generated from the obtained reducts with high percentage of classification accuracy. The branch and bound search strategy has shown reduction in certain amount of search.

Efficient train re-routing and rescheduling: Valid inequalities and reformulation of RECIFE-MILP

In recent years approximation algorithms based on primal-dual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primal-dual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design. The worst-case bound of the proposed algorithm is related to a fundamental relationship (called strength) between the set of valid inequalities and the set of minimal solutions to the covering problems. In this way, we can construct an approximation algorithm simply by constructing the required valid inequalities. We apply the proposed algorithm to several problems, such as covering problems related to totally balanced matrices, cyclic scheduling, vertex cover, general set covering, intersections of polymatroids, and several network design problems attaining (in most cases) the best worst-case bound known in the literature.

Given an initial marking and final marking for a Petri net model, an optimal firing sequence problem is defined as the problem to find an optimal transition firing sequence to minimize the objective function. For the purpose of analysis of general integer programming problems, we propose a Petri net representation and reachability analysis of integer programming problems. In the proposed method, an integer programming problem is converted into the optimal firing sequence problem of Petri nets. By utilizing the proposed algorithm, integer programming problems can be visualized and analyzed by the Petri net theory. We apply the proposed methodology to the scheduling problems of dual armed cluster tools. The valid inequalities are derived from the reachability analysis. Numerical results show that those valid inequalities can significantly reduce the computational time of the original integer programming problem.

The constrained shortest path tour problem (CSPTP) is an NP‐hard combinatorial optimization problem defined on a connected directed graph , where V is the set of nodes and A is the set of nonnegative weighted arcs. Given two distinct nodes , an integer value , and node disjoint subsets , , the CSPTP aims at finding the shortest trail from s to t while visiting at least one node in every subset , in this order. In this paper, we perform a comparative analysis between two integer programming (IP) models for the problem. We also propose valid inequalities and a Lagrangian‐based heuristic framework. Branch‐and‐bound algorithms from the literature, as well as a metaheuristic approach, are used for comparison. Extensive computational experiments carried out on benchmark data sets show the effective use of valid inequalities and the quality of bounds obtained by the Lagrangian framework. Because benchmark instances do not require a great computational effort of IP models in the sense that their optimality is reached at the root node of the CPLEX branch‐and‐cut search tree, we introduce new challenging CSPTP instances for which our solution approaches outperform existing ones for the problem.

Valid inequalities for the k-Color Shortest Path Problem

We consider the minimum connected dominating set problem. We present an integer programming formulation and new valid inequalities. A branch-and-cut algorithm based on the reinforced formulation is also provided. Computational results indicate that the reinforced lower bounds are always stronger than the bounds implied by the formulation from which resulted one of the best known exact algorithms for the problem. In some cases, the reinforced lower bounds are stronger than those implied by the strongest known formulation to date. For dense graphs, our algorithm provides the best results in the literature. For sparse instances, known to be harder, our method is outperformed by another one. We discuss reasons for that and how to improve our current computational results. One possible way to achieve such goals is to devise specific separation algorithms for some classes of valid inequalities introduced here.

A partitioning algorithm for the network loading problem

In integer linear programming (ILP), formulating a "good" model is of crucial importance to solving that model. In this paper, we begin with a mathematical analysis of the structure of the assignment, timing, and resource constraints in high-level synthesis, and then evaluate the structure of the scheduling polytope described by these constraints. We then show how the structure of the constraints can be exploited to develop a well-structured ILP formulation, which can serve as a solid theoretical foundation for future improvement. As a start in that direction, we also present two methods to further tighten the formulation. The contribution of this paper is twofold: 1) it provides the first in-depth formal analysis of the structure of the constraints, and it shows how to exploit that structure in a well-designed ILP formulation, and 2) it shows how to further improve a well-structured formulation by adding new valid inequalities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The Chinese postman problem with time-dependent travel time (CPPTDT) is a generalization of the classical Chinese postman problem (CPP), where the travel time on an arc depends on the time of beginning of travel along it. While CPP and its almost static variants can be solved by integer program successfully, there are very challenging time varying CPP variants, such as CPPTDT, which are difficult to be formulated directly. The first and the only integer programming formulation modeling the time varying CPP directly was presented in the pioneering work of Wang and Wen (Comput Math Appl 44:375---387, 2002), which was unfortunately based on a strong assumption that each basic cycle in the graph must visit the depot. In this work, we propose a new integer programming formulation for the CPPTDT without any unrealistic assumptions, namely, the arc-cycle formulation, which can be viewed as an extended version of the formulation given by Wang and Wen. The constraint set of this formulation can be divided into two parts. The first part has a strong combinatorial structure, which is linear and used to define the polytope of cycle-path alternation sequence (CPAS). We determine the dimension of the CPAS polytope and identify the facet defining inequalities which may be helpful to tighten the integer programming formulation. The second part is closely related to time-dependent travel time and is nonlinear. The linearization is provided to the case when all the travel times are piecewise functions of beginning time, and several strong, valid inequalities are also proposed. The computation results with a cutting plane algorithm using the new cuts are reported on several randomly generated instances.

We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s,z)R 1 + ×ℤ1:s≥z−b}. These sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs. For four such sets we give a complete description by valid inequalities and/or an integral extended formulation, and we also indicate what constraints can be added without destroying integrality. We apply these results to provide tight formulations for certain piecewise–linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot-sizing problems.

Lot sizing and job shop scheduling with compressible process times: A cut and branch approach

Integer programming approaches to the multiple team formation problem

An integer programming approach to scheduling the transshipment of products at cross-docks in less-than-truckload industries