On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

The Kidney Exchange Problem (KEP) is a combinatorial optimization problem and has attracted the attention from the community of integer programming/combinatorial optimisation in the past few years. Defined on a directed graph, the KEP has two variations: one concerns cycles only, and the other, cycles as well as chains on the same graph. We call the former a Cardinality Constrained Multi-cycle Problem (CCMcP) and the latter a Cardinality Constrained Cycles and Chains Problem (CCCCP). The cardinality for cycles is restricted in both CCMcP and CCCCP. As for chains, some studies in the literature considered cardinality restrictions, whereas others did not. The CCMcP can be viewed as an Asymmetric Travelling Salesman Problem that does allow subtours, however these subtours are constrained by cardinality, and that it is not necessary to visit all vertices. In existing literature of the KEP, the cardinality constraint for cycles is usually considered to be small (to the best of our knowledge, no more than six). In a CCCCP, each vertex on the directed graph can be included in at most one cycle or chain, but not both. The CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights, particularly due to their similarities to some travelling salesman- and vehicle routing-family of problems. In this paper, our main focus is to review the existing mathematical programming models and solution methods in the literature, analyse the performance of these models, and identify future research directions. Further, we propose a polynomial-sized and an exponential-sized mixed-integer linear programming model, discuss a number of stronger constraints for cardinality-infeasible-cycle elimination for the latter, and present some preliminary numerical results.     

Polynomially solvable special cases of the quadratic bottleneck assignment problem

Quadratic bottleneck assignment problems (QBAP) are obtained by replacing the addition of cost terms in the objective function of a quadratic (sum) assignment problem by taking their maximum. Since the QBAP is an NP\mathcal{NP}-hard problem, polynimially solvable special cases of the QBAP are of interest. In this paper we specify conditions on the cost matrices of QBAP leading to special cases which can be solved to optimality in polynomial time. In particular, the following three cases are discussed: (i) any permutation is optimal (constant QBAP), (ii) a certain specified permutation is optimal (constant permutation QBAP) and (iii) the solution can be found algorithmically by a polynomial algorithm. Moreover, the max-cone of bottleneck Monge matrices is investigated, its generating matrices are identified and it is used as a tool in proving polynomiality results.     

On global integer extrema of real-valued box-constrained multivariate quadratic functions

Determining global integer extrema of an real-valued box-constrained multivariate quadratic functions is a very difficult task. In this paper, we present an analytic method, which is based on a combinatorial optimization approach in order to calculate global integer extrema of a real-valued box-constrained multivariate quadratic function, whereby this problem will be proven to be as NP-hard via solving it by a Travelling Salesman instance. Instead, we solve it using eigenvalue theory, which allows us to calculate the eigenvalues of an arbitrary symmetric matrix using Newton’s method, which converges quadratically and in addition yields a Jordan normal form with \(1 \times 1\)-blocks, from which a special representation of the multivariate quadratic function based on affine linear functions can be derived. Finally, global integer minimizers can be calculated dynamically and efficiently most often in a small amount of time using the Fourier–Motzkin- and a Branch and Bound like Dijkstra-algorithm. As an application, we consider a box-constrained bivariate and multivariate quadratic function with ten arguments.     

On residual approximation in solution extension problems

The solution extension variant of a problem consists in, being given an instance and a partial solution, finding the best solution comprising the given partial solution. Many problems have been studied with a similar approach. For instance the Pre-Coloring Extension problem, the clustered variant of the Travelling Salesman problem, or the General Routing Problem are in a way typical examples of solution extension variant problems. Motivated by practical applications of such variants, this work aims to explore different aspects around extension on classical optimization problems. We define residue-approximations as algorithms whose performance ratio on the non-prescribed part can be bounded, and corresponding complexity classes. Using residue-approximation, we classify problems according to their residue-approximability, exhibit distinct behaviors and give several examples and first interesting results.     

Improving an exact approach for solving separable integer quadratic knapsack problems

We consider the specially structured (pure) integer Quadratic Multi-Knapsack Problem (QMKP) tackled in the paper “Exact solution methods to solve large scale integer quadratic knapsack problems” by D. Quadri, E. Soutif and P. Tolla (2009), recently appeared on this journal, where the problem is solved by transforming it into an equivalent 0–1 linearized Multi-Knapsack Problem (MKP). We show that, by taking advantage of the structure of the transformed (MKP), it is possible to derive an effective variable fixing procedure leading to an improved branch-and-bound approach. This procedure reduces dramatically the resulting linear problem size inducing an impressive improvement in the performances of the related branch and bound approach when compared to the results of the approach proposed by D. Quadri, E. Soutif and P. Tolla.     

Market‐Based Allocation with Indivisible Bids

We study multiunit double auctions accepting bids with indivisibility constraints. Modeling the auction problem as a Multiple Choice Knapsack Problem and using dynamic programming, we show that incremental computations during bid processing can speed the handling of key auction operations such as clearing and quoting. We propose different price‐quote policies and study their influence on the efficiency of market‐based allocation. Using a reconfigurable manufacturing scenario where agents trade large quantities of multiple goods, we demonstrate potential benefits of supporting indivisibility constraints in bidding. These benefits are highly sensitive to the form of price quote provided, indicating interesting tradeoffs in communication and allocation efficiency.     

车辆路径问题的混合蚁群算法设计与实现

蚁群算法是一种新型的模拟进化算法,具有许多优良的性质,可以很好地解决TSP问题.在分析车辆路径问题(VRP)与TSP区别的基础上,论文将蚁群算法应用于VRP的求解,针对VRP的具体特点,构造了具有自适应功能的混合蚁群算法.该算法对基本规则作了进一步改进,并有机结合了爬山法、节约法等方法,以减少计算时间,避免算法停滞.指出可行解问题是蚁群算法的关键问题,提出了大蚂蚁数、近似解可行化等四个解决策略.计算机仿真结果表明,自适应混合蚁群算法性能优良,能够有效地求解VRP.     

Generalized Steiner Problems and Other Variants

In this paper, we examine combinatorial optimization problems by considering the case where the set N (the ground set of elements) is expressed as a union of a finite number of m nonempty distinct subsets N ₁,...,N _m. The term we use is the generalized Steiner problems coined after the Generalized Traveling Salesman Problem. We have collected a short list of classical combinatorial optimization problems and we have recast each of these problems in this broader framework in an attempt to identify a linkage between these “generalized” problems. In the literature one finds generalized problems such as the Generalized Minimum Spanning Tree (GMST), Generalized Traveling Salesman Problem (GTSP) and Subset Bin-packing (SBP). Casting these problems into the new problem setting has important implications in terms of the time effort required to compute an optimal solution or a “good” solution to a problem. We examine questions like “is the GTSP “harder” than the TSP?” for a number of paradigmatic problems starting with “easy” problems such as the Minimal Spanning Tree, Assignment Problem, Chinese Postman, Two-machine Flow Shop, and followed by “hard” problems such as the Bin-packing, and the TSP.     

Polynomially solvable cases of the constant rank unconstrained quadratic 0-1 programming problem

In this paper we consider the constant rank unconstrained quadratic 0-1 optimization problem, CR-QP01 for short. This problem consists in minimizing the quadratic function 〈x, Ax〉 + 〈c, x〉 over the set {0,1} ⁿ where c is a vector in ℝ ⁿ and A is a symmetric real n × n matrix of constant rank r. We first present a pseudo-polynomial algorithm for solving the problem CR-QP01, which is known to be NP-hard already for r = 1. We then derive two new classes of special cases of the CR-QP01 which can be solved in polynomial time. These classes result from further restrictions on the matrix A. Finally we compare our algorithm with the algorithm of Allemand et al. (2001) for the CR-QP01 with negative semidefinite A and extend the range of applicability of the latter algorithm. It turns out that neither of the two algorithms dominates the other with respect to the class of instances which can be solved in polynomial time.     

A best first search exact algorithm for the Multiple-choice Multidimensional Knapsack Problem

In this paper, we propose an optimal algorithm for the Multiple-choice Multidimensional Knapsack Problem MMKP. The main principle of the approach is twofold: (i) to generate an initial feasible solution as a starting lower bound, and (ii) at different levels of the search tree to determine an intermediate upper bound obtained by solving an auxiliary problem called MMKP_aux and perform the strategy of fixing items during the exploration. The approach which we develop is of best-first search strategy. The method was able to optimally solve the MMKP. The performance of the exact algorithm is evaluated on a set of small and medium instances, some of them are extracted from the literature and others are randomly generated. This algorithm is parallelizable and it is one of its important feature.     

The QAP-polytope and the graph isomorphism problem

In this paper we propose a geometric approach to solve the Graph Isomorphism (GI in short) problem. Given two graphs \(G_1, G_2\), the GI problem is to decide if the given graphs are isomorphic i.e., there exists an edge preserving bijection between the vertices of the two graphs. We propose an Integer Linear Program (ILP) that has a non-empty solution if and only if the given graphs are isomorphic. The convex hull of all possible solutions of the ILP has been studied in literature as the Quadratic Assignment Problem (QAP) polytope. We study the feasible region of the linear programming relaxation of the ILP and show that the given graphs are isomorphic if and only if this region intersects with the QAP-polytope. As a consequence, if the graphs are not isomorphic, the feasible region must lie entirely outside the QAP-polytope. We study the facial structure of the QAP-polytope with the intention of using the facet defining inequalities to eliminate the feasible region outside the polytope. We determine two new families of facet defining inequalities of the QAP-polytope and show that all the known facet defining inequalities are special instances of a general inequality. Further we define a partial ordering on each exponential sized family of facet defining inequalities and show that if there exists a common minimal violated inequality for all points in the feasible region outside the QAP-polytope, then we can solve the GI problem in polynomial time. We also study the general case when there are k such inequalities and give an algorithm for the GI problem that runs in time exponential in k.     

Novel quadratic programming approach for time series clustering with biomedical application

Fundamental problems in data mining mainly involve discrete decisions based on numerical analyses of data (e.g., class assignment, feature selection, data categorization, identifying outlier samples). These decision-making problems in data mining are combinatorial in nature and can naturally be formulated as discrete optimization problems. One of the most widely studied problems in data mining is clustering. In this paper, we propose a new optimization model for hierarchical clustering based on quadratic programming and later show that this model is compact and scalable. Application of this clustering technique in epilepsy, the second most common brain disorder, is a case point in this study. In our empirical study, we will apply the proposed clustering technique to treatment problems in epilepsy through the brain dynamics analysis of electroencephalogram (EEG) recordings. This study is a proof of concept of our hypothesis that epileptic brains tend to be more synchronized (clustered) during the period before a seizure than a normal period. The results of this study suggest that data mining research might be able to revolutionize current diagnosis and treatment of epilepsy as well as give a greater understanding of brain functions (and other complex systems) from a system perspective. This work was partially supported by the NSF grant CCF 0546574 and Rutgers Research Council grant-202018.     

A Branch and Cut solver for the maximum stable set problem

This paper deals with the cutting-plane approach to the maximum stable set problem. We provide theoretical results regarding the facet-defining property of inequalities obtained by a known project-and-lift-style separation method called edge-projection, and its variants. An implementation of a Branch and Cut algorithm is described, which uses edge-projection and two other separation tools which have been discussed for other problems: local cuts (pioneered by Applegate, Bixby, Chvátal and Cook) and mod-k cuts. We compare the performance of this approach to another one by Rossi and Smiriglio (Oper. Res. Lett. 28:63–74, 2001) and discuss the value of the tools we have tested.     

Cardinality constraints and systems of restricted representatives

Cardinality constraints have received considerable attention from the Constraint Programming community as (so-called) global constraints that appear in the formulation of several real-life problems, while also having an interesting combinatorial structure. After discussing the relation of cardinality constraints with well-known combinatorial problems (e.g., systems of restricted representatives), we study the polytope defined by the convex hull of vectors satisfying two such constraints, in the case where all variables share a common domain. We provide families of facet-defining inequalities that are polytime separable, together with a condition for when these families of inequalities define a convex hull relaxation. Our results also hold for the case of a single such constraint.     

Quadratic Assignment Problems and M/G/C/C/ State Dependent Network Flows

One of the most notorious network design problems is the Quadratic Assignment Problem (QAP). We develop an heuristic algorithm for QAPs along with an M/G/C/C state dependent queueing model for capturing congestion in the traffic system interconnecting the nodes in the network. Computational results are also presented.     

Column generation approach for the point-feature cartographic label placement problem

This paper proposes a column generation approach for the Point-Feature Cartographic Label Placement problem (PFCLP). The column generation is based on a Lagrangean relaxation with clusters proposed for problems modeled by conflict graphs. The PFCLP can be represented by a conflict graph where vertices are positions for each label and edges are potential overlaps between labels (vertices). The conflict graph is decomposed into clusters forming a block diagonal matrix with coupling constraints that is known as a restricted master problem (RMP) in a Dantzig-Wolfe decomposition context. The clusters’ sub-problems are similar to the PFCLP and are used to generate new improved columns to RMP. This approach was tested on PFCLP instances presented in the literature providing in reasonable times better solutions than all those known and determining optimal solutions for some difficult large-scale instances.     

On the <Emphasis Type="Italic">m</Emphasis>-clique free interval subgraphs polytope: polyhedral analysis and applications

In this paper we study the m-clique free interval subgraphs. We investigate the facial structure of the polytope defined as the convex hull of the incidence vectors associated with these subgraphs. We also present some facet-defining inequalities to strengthen the associated linear relaxation. As an application, the generalized open-shop problem with disjunctive constraints (GOSDC) is considered. Indeed, by a projection on a set of variables, the m-clique free interval subgraphs represent the solution of an integer linear program solving the GOSDC presented in this paper. Moreover, we propose exact and heuristic separation algorithms, which are exploited into a Branch-and-cut algorithm for solving the GOSDC. Finally, we present and discuss some computational results.     

A solution framework for the multi-mode resource-constrained cross-dock scheduling problem

In this paper we propose a framework for shift-level container scheduling and resource allocation decisions at a cross-dock facility. The Multi-Mode Resource-Constrained Cross-Dock Scheduling Problem (MRCDSP) approach minimizes material flow and schedules inbound and outbound containers to dock-doors such that the total processing time is minimized subject to the resource constraints at the cross-dock. While container scheduling and resource allocation problems at cross-dock facilities have been studied previously in isolation, our work is the first to consider a complete view of cross-dock operations providing optimal container to dock-door allocation, and a makespan minimizing schedule of containers to the cross-dock. We present a comprehensive framework that includes identification of container clusters to reduce the problem size, a container-to-dock-door assignment algorithm, and a container clusters scheduling model that is solvable for practically sized problems. In a comparative numeric study based on data simulating a cross-dock facility, our approach is shown to outperform current practice, reducing the average time required for processing a set of containers by 37% and reducing the weighted-distance material traveled within the cross-dock by 45%.     

From Linear to Semidefinite Programming: An Algorithm to Obtain Semidefinite Relaxations for Bivalent Quadratic Problems

In this paper, we present a simple algorithm to obtain mechanically SDP relaxations for any quadratic or linear program with bivalent variables, starting from an existing linear relaxation of the considered combinatorial problem. A significant advantage of our approach is that we obtain an improvement on the linear relaxation we start from. Moreover, we can take into account all the existing theoretical and practical experience accumulated in the linear approach. After presenting the rules to treat each type of constraint, we describe our algorithm, and then apply it to obtain semidefinite relaxations for three classical combinatorial problems: the K-CLUSTER problem, the Quadratic Assignment Problem, and the Constrained-Memory Allocation Problem. We show that we obtain better SDP relaxations than the previous ones, and we report computational experiments for the three problems.     

应用内点算法求解允许买空/卖空情况下的投资组合问题

本文着重讨论了允许买空卖空条件下的投资组合求解问题。由于在实际的经济活动中存在买空卖空的现象,本文讨论了这种情况下的最优化模型,并用内点算法结合Matlab编程,给出了问题的最优解。同时,对投资者对风险的偏好系数A进行了数值分析,可以得到最优的A值A^*,即当投资者对风险的偏好系数A=A^*时,投资者可以得到最大的投资效用。