Exact Solution of Graph Coloring Problems via Constraint Programming and Column Generation

A column generation (CG) approach for the solution of timetabling problems is presented. This methodology could be used for various instances of the timetabling problem, although in this paper the solution of the high-school situation in Greece is presented. The results obtained show clearly that the CG approach that has been extremely successful in recent years in the solution of airline crew scheduling problems could also be very efficient and robust for the solution of timetabling problems. Several large timetabling problems corresponding to real problems have been successfully solved, with the solutions obtained feasible and of very high quality in accordance with the problem definition. In addition, none of the solutions contained any idle hour for any of the teachers, which was one of the main goals of this optimization effort.

Nowadays, large-scale failures in elastic optical inter-data center networks (EO-DCNs) are critical issues for network survivability. Once the original connection or the primary DC fails, an alternative data center (DC) can serve the cloud services with replicated content. In this paper, we study disaster-resilient cloud services provisioning with distance-adaptive modulation formats, considering content placement, routing with modulation format adaption, protection of path and content, and spectrum allocation simultaneously. Both dedicated end-to-content backup path protection (DEBPP) and shared end-to-content backup path protection (SEBPP) are studied. First, integer linear program (ILP) models are formulated with the objective of minimum spectrum usage. For the sake of scalability, a column generation (CG) approach is developed to decompose the ILP model into a master problem and a pricing problem. The CG approach is shown to achieve comparable results to ILP models. Numerical simulations are conducted to investigate the spectrum usage of DEBPP and SEBPP with different traffic parameters and different network topologies.

A column generation approach to capacitated p-median problems

The problem of scheduling and tree routing in WiMAX/802.16 based mesh networks were not defined in the standard and are thus subject to extensive research. In this paper, we consider the problem of joint routing and scheduling in 802.16-based wireless mesh network, with the objective of determining a minimum length schedule that satisfies a given (uplink/downlink) end-to-end traffic demand. Minimizing the schedule length amounts to maximizing the spectrum spatial reuse by concurrently transmitting on as many links as possible, which we refer to as a transmission configuration (a group of links that can simultaneously transmit without violating the signal-to-interferenceplus- noise ratio (SINR) requirement). Our model is referred to as maximum spatial reuse (MSR). Since there is an overwhelming number of possible transmission configurations to be assigned to time slots, we adopt the column generation technique to construct our MSR model. We present two formulations for modeling MSR, namely the link-based column generation (CGLink) formulation and the path-based column generation (CGPath) formulation. These two formulations differ mainly in the number of routing decision variables. Our experimental results indicate that the path-based formulation needs much less computational (CPU) time than the link-based formulation in order to determine the (same) optimized solution with the same spatial reuse gain.

In intensity-modulated radiation therapy (IMRT), a network flow is adopted to solve the pricing problem of the generic column generation approach in order to obtain a deliverable aperture. However, excessive computation results from the direct use of a network flow. In addition, a decline in plan quality may result from the direct determination of the leaf position using the gradient information. To overcome these problems, a column generation approach based on region growth is proposed. The proposed method is designed to reduce the computational cost of solving the pricing problem and improve the IMRT plan quality. First, the gradients of the beamlets are obtained by an objective function constructed under the constraint conditions of the organs. Second, the gradients are transformed nonlinearly. Third, the positions of the continuous negative gradient regions in each row of the aperture are determined and stored. Fourth, these gradients are taken as a whole and added to the aperture network flow, which is solved as a shortest-path problem. Finally, the deliverable aperture is obtained and added to the treatment plan. To verify the effectiveness of the proposed method, experiments involving five five-field prostate cancer cases and five nine-field head and neck cancer cases were conducted. Compared with the generic column generation method, the dose distribution of the target is ensured by the proposed method, which also effectively protects organs at risk and reduces the running time. Specifically, in ten groups of comparative experiments, the normal tissue complication probability of the proposed method is reduced by up to 3.37%, and the maximum acceleration rate is 20.44%. According to the experimental results, the proposed method is more consistent with clinical requirements compared with the generic column generation method.

We present a complementary column generation feature that produces tight upper bounds, thereby enhancing heuristic and exact column generation approaches for (minimization) set partitioning formulations that possess dense column structures. We also introduce a duality-based lower bound that prompts a useful termination criterion, which can be utilized to mitigate the tailing-off effect induced by column generation approaches. Computations are presented for the one-dimensional bin packing problem and a joint vehicle assembly-routing problem.

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

The objective of the maximum weighted set of disjoint submatrices problem is to discover K disjoint submatrices that together cover the largest sum of entries of an input matrix. It has many practical data-mining applications, as the related biclustering problem, such as gene module discovery in bioinformatics. It differs from the maximum-weighted submatrix coverage problem introduced in [6] by the explicit formulation of disjunction constraints: submatrices must not overlap. In other words, all matrix entries must be covered by at most one submatrix. The particular case of \(K=1\), called the maximal-sum submatrix problem, was successfully tackled with constraint programming in [5]. Unfortunately, the case of \(K > 1\) is more challenging to solve as the selection of rows cannot be decided in polynomial time solely from the selection of K sets of columns. It can be proved to be \(\mathcal {NP}\)-hard. We introduce a hybrid column generation approach using constraint programming to generate columns. It is compared to a standard mixed integer linear programming (MILP) through experiments on synthetic datasets. Overall, fast and valuable solutions are found by column generation while the MILP approach cannot handle a large number of variables and constraints.

A column generation approach for the unconstrained binary quadratic programming problem

To develop a novel aperture-based algorithm for volumetric modulated are therapy (VMAT) treatment plan optimization with high quality and high efficiency. The VMAT optimization problem is formulated as a large-scale convex programming problem solved by a column generation approach. The authors consider a cost function consisting two terms, the first enforcing a desired dose distribution and the second guaranteeing a smooth dose rate variation between successive gantry angles. A gantry rotation is discretized into 180 beam angles and for each beam angle, only one MLC aperture is allowed. The apertures are generated one by one in a sequential way. At each iteration of the column generation method, a deliverable MLC aperture is generated for one of the unoccupied beam angles by solving a subproblem with the consideration of MLC mechanic constraints. A subsequent master problem is then solved to determine the dose rate at all currently generated apertures by minimizing the cost function. When all 180 beam angles are occupied, the optimization completes, yielding a set of deliverable apertures and associated dose rates that produce a high quality plan. The algorithm was preliminarily tested on five prostate and five head-and-neck clinical cases, each with one full gantry rotation without any couch/collimator rotations. High quality VMAT plans have been generated for all ten cases with extremely high efficiency. It takes only 5-8 min on CPU (MATLAB code on an Intel Xeon 2.27 GHz CPU) and 18-31 s on GPU (CUDA code on an NVIDIA Tesla C1060 GPU card) to generate such plans. The authors have developed an aperture-based VMAT optimization algorithm which can generate clinically deliverable high quality treatment plans at very high efficiency.

Branch-and-price and constraint programming for solving a real-life technician dispatching problem

GloptLab is an easy-to-use testing and development platform for solving quadratic constraint satisfaction problems, written in Matlab. The algorithms implemented in GloptLab are used to reduce the search space: scaling, constraint propagation, linear relaxations, strictly convex enclosures, conic methods, and branch and bound. All these methods are rigorous; hence, it is guaranteed that no feasible point is lost. Finding and verifying feasible points complement the reduction methods. From the method repertoire custom-made strategies can be built, with a user-friendly graphical interface. GloptLab was tested on a large test set of constraint satisfaction problems, and the results show the importance of composing a clever strategy.

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms.

Constraint satisfaction and search problems generally fall into the class of problems for which a direct algorithmic solution does not exist. The solution of these problems requires the examination of state spaces. A problem solver (inference engine) alone is not able to organize and maintain state space consistently. For this purpose a so‐called truth maintenance system is required. Our truth maintenance system (MEKON) organizes data within a data abstraction called a context. Each context corresponds to one problem state and contains currently believed data. The truth maintenance system provides believed data retrieval, belief revision, contradiction handling and non‐monotonicity handling, the features that help a problem solver to examine state spaces. MEKON represents an ATMS‐like system implemented within BEST (Blackboard‐based Expert System Toolkit). However, some special MEKON features such as state variables, context sensitive generation of assumptions and explicit context generation, that are not present in standard ATMSs, facilitate not only the solution of constraint satisfaction problems, but also the solution of search problems (not provided by standard ATMSs). Being deeply integrated with Prolog/Rex, BEST’s knowledge representation language, and BEST’s inference engine, MEKON provides a simple and efficient means for the examination of state spaces. Facts, hypotheses (assumptions), and concepts (frames) are used to describe a problem state, contexts are used to represent points in the state spaces, while rules are used to perform state transitions. MEKON is the only truth maintenacne system that provides truth maintenance capabilities on local blackboards thus enabling the solution of complex problems which include different kinds of real constraint satisfaction and search problems concerning diagnosis, allocation tasks, classification tasks, planning or scenario making.

Constraint satisfaction problems (CSPs) occur widely in artificial intelligence. In the last twenty years, many algorithms and heuristics were developed to solve CSP. Recently, a constraint-graph based evolutionary algorithm was proposed to solve CSP, [17]. It shown that it is advantageous to take into account the knowledge of the constraint network to design genetic operators. On the other hand, recent publications indicate that parallel genetic algorithms (PGA's) with isolated evolving subpopulations (that exchange individuals from time to time) may offer advantages over sequential approaches, [1]. In this paper we examine the gain of the performance obtained using multiple populations - that evolve in parallel - of the constraint-graph based evolutionary algorithm with a migration policy. We show that a multiple populations approach outperforms a single population implementation when applying it to the 3-coloring problem.