Biclique completion problems for multicast network design

This paper considers the problem of aggregating several multicast sessions. A multicast session is defined as a subset of clients requiring the same information. Besides, each client can require several multicast sessions. A telecommunication network cannot manage many multicast sessions at the same time. It is hence necessary to group the sessions into a limited number of clusters. The problem then consists in aggregating the sessions into clusters to limit the number of unnecessary information sent to clients. The strong relationship of the problems with biclique problems in bipartite graph is established. We then model the problems using integer quadratic and linear programming formulations. We investigate some properties to strengthen the models. Several algorithms are provided and compared with a series of numerical experiments.     

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number. Research supported in part by NSF grant number DMS-0317323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

Constraint programming-based column generation

This paper surveys recent applications and advances of the constraint programming-based column generation framework, where the master subproblem is solved by traditional OR techniques, while the pricing subproblem is solved by constraint programming (CP). This framework has been introduced to solve crew assignment problems, where complex regulations make the pricing subproblem demanding for traditional techniques, and then it has been applied to other contexts. The main benefits of using CP are the expressiveness of its modeling language and the flexibility of its solvers. Recently, the CP-based column generation framework has been applied to many other problems, ranging from classical combinatorial problems such as graph coloring and two dimensional bin packing, to application oriented problems, such as airline planning and resource allocation in wireless ad hoc networks.      

Comparison of wavelength requirements between two wavelength assignment methods in survivable WDM networks

We consider the routing and wavelength assignment (RWA) in survivable WDM network. A path protection scheme assumed and two different wavelength assignment methods for protection paths are considered. Integer programming formulations of RWA under two wavelength assignment methods are proposed and we devised algorithms to solve them. Test results show that the difference of wavelength requirements between two wavelength assignment methods is 5–30–     

A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].     

A column generation approach for the split delivery vehicle routing problem

A column generation approach is presented for the split delivery vehicle routing problem with large demand. Columns include route and delivery amount information. Pricing sub-problems are solved by a limited-search-with-bound algorithm. Feasible solutions are obtained iteratively by fixing one route once. Numerical experiments show better solutions than in the literature.     

Scheduling inspired models for two-dimensional packing problems

We propose two exact algorithms for two-dimensional orthogonal packing problems whose main components are simple mixed-integer linear programming models. Based on the different forms of time representation in scheduling formulations, we extend the concept of multiple time grids into a second dimension and propose a hybrid discrete/continuous-space formulation. By relying on events to continuously locate the rectangles along the strip height, we aim to reduce the size of the resulting mathematical problem when compared to a pure discrete-space model, with hopes of achieving a better computational performance. Through the solution of a set of 29 test instances from the literature, we show that this was mostly accomplished, primarily because the associated search strategy can quickly find good feasible solutions prior to the optimum, which may be very important in real industrial environments. We also provide a comprehensive comparison to seven other conceptually different approaches that have solved the same strip packing problems.     

A column generation based heuristic for sensor placement,activity scheduling and data routing in wireless sensor networks

A wireless sensor network is a network consisting of distributed autonomous electronic devices called sensors. In this work, we develop a mixed-integer linear programming model to maximize the network lifetime by optimally determining locations of sensors and sinks, sensor-to-sink data flows, and activity schedules of the deployed sensors subject to coverage, flow conservation, energy consumption and budget constraints. Since solving this model is difficult except for very small instances, we propose a heuristic method which works on a reformulation of the problem. In the first phase of this heuristic, the linear programming relaxation of the reformulation is solved by column generation. The second phase consists of constructing a feasible solution for the original problem using the columns obtained in the first phase. Computational experiments conducted on a set of test instances indicate that both the accuracy and the efficiency of the proposed heuristic is quite promising.     

Equivalence between intersection cuts and the corner polyhedron

Intersection cuts were introduced by Balas and the corner polyhedron by Gomory. Balas showed that intersection cuts are valid for the corner polyhedron. In this paper we show that, conversely, every nontrivial facet-defining inequality for the corner polyhedron is an intersection cut.     

Exact and heuristic methods to maximize network lifetime in wireless sensor networks with adjustable sensing ranges

Wireless sensor networks involve many different real-world contexts, such as monitoring and control tasks for traffic, surveillance, military and environmental applications, among others. Usually, these applications consider the use of a large number of low-cost sensing devices to monitor the activities occurring in a certain set of target locations. We want to individuate a set of covers (that is, subsets of sensors that can cover the whole set of targets) and appropriate activation times for each of them in order to maximize the total amount of time in which the monitoring activity can be performed (network lifetime), under the constraint given by the limited power of the battery contained in each sensor. A variant of this problem considers that each sensor can be activated in a certain number of alternative power levels, which determine different sensing ranges and power consumptions. We present some heuristic approaches and an exact approach based on the column generation technique. An extensive experimental phase proves the advantage in terms of solution quality of using adjustable sensing ranges with respect to the classical single range scheme.     

Optimal receiver scheduling algorithms for a multicast problem

This paper studies a multicast problem arising in wavelength division multiplexing single-hop lightwave networks, as well as in Video-on-Demand systems. In this problem, the same message of duration Δ has to be transmitted to a set of n receivers which are not all available simultaneously. The receivers can be partitioned into subsets, each served by a different transmission, with the objective of minimizing their overall waiting cost. When there is a single data channel available for transmission, a dynamic programming algorithm is devised which finds an optimal solution in O(nlogn+min{n²,nΔ²}) time, improving over a previously known O(n³) time algorithm. When multiple data channels are available for transmission, an optimal O(n) time algorithm is proposed which finds an optimal solution if the message has constant transmission duration, whereas an NP-completeness proof is given if the message has arbitrary transmission duration.     

A branch-and-price approach for the partition coloring problem

This work proposes a new integer programming model for the partition coloring problem and a branch-and-price algorithm to solve it. Experiments are reported for random graphs and instances originating from routing and wavelength assignment problems arising in telecommunication network design. We show that our method largely outperforms previously existing approaches.     

Sequential pairing of mixed integer inequalities

We investigate a scheme, called pairing, for generating new valid inequalities for mixed integer programs by taking pairwise combinations of existing valid inequalities. The pairing scheme essentially produces a split cut corresponding to a specific disjunction, and can also be derived through the mixed integer rounding procedure. The scheme is in general sequence-dependent and therefore leads to an exponential number of inequalities. For some important cases, we identify combination sequences that lead to a manageable set of non-dominated inequalities. We illustrate the framework for some deterministic and stochastic integer programs and we present computational results showing the efficiency of adding the new generated inequalities as cuts.     

Improved state space relaxation for constrained two-dimensional guillotine cutting problems

Christofides and Hadjiconstantinou (1995) introduced a dynamic programming state space relaxation for obtaining upper bounds for the Constrained Two-dimensional Guillotine Cutting Problem. The quality of those bounds depend on the chosen item weights, they are adjusted using a subgradient-like algorithm. This paper proposes Algorithm X, a new weight adjusting algorithm based on integer programming that provably obtains the optimal weights. In order to obtain even better upper bounds, that algorithm is generalized into Algorithm X2 for obtaining optimal two-dimensional item weights. We also present a full hybrid method, called Algorithm X2D, that computes those strong upper bounds but also provides feasible solutions obtained by: (1) exploring the suboptimal solutions hidden in the dynamic programming matrices; (2) performing a number of iterations of a GRASP based primal heuristic; and (3) executing X2H, an adaptation of Algorithm X2 to transform it into a primal heuristic. Extensive experiments with instances from the literature and on newly proposed instances, for both variants with and without item rotation, show that X2D can consistently deliver high-quality solutions and sharp upper bounds. In many cases the provided solutions are certified to be optimal.     

Dantzig–Wolfe decomposition of the daily course pattern formulation for curriculum-based course timetabling

In this paper, we considered the problem of Curriculum-Based Course Timetabling, i.e., assigning weekly lectures to a time schedule and rooms. We developed a Column Generation algorithm based on a pattern formulation of the time scheduling part of the problem by Bagger et al. (2016). The pattern formulation is an enumeration of all schedules by which each course can be assigned on each day; it is a lower bounding model. Pattern enumeration has also been considered in Burke (2008), where the authors enumerated all schedules to which each curriculum can be assigned on each day. We applied the Dantzig–Wolfe reformulation, so each column corresponded to a schedule for an entire day.We solved the reformulation with the Column Generation algorithm, where each pricing problem generated a full schedule for a single day. We provided a pre-processing technique that, on average, removed approximately 45% of the pattern variables in the pricing problems. We then extended the pre-processing technique into inequalities that we added to the model. Lastly, we describe how we applied Local Branching to the pricing problem by using the columns generated in previous iterations.We compare the lower bounds we obtained, with other methods from literature, on 20 data instances of real-world applications. For 16 instances the optimal solutions are known, but the remaining four are still open. Our approach improved the best-known lower bound for all four open instances, and decreased the average gap from 24 to 11%.     

A cutting plane algorithm for graph coloring

We present an approach based on integer programming formulations of the graph coloring problem. Our goal is to develop models that remove some symmetrical solutions obtained by color permutations. We study the problem from a polyhedral point of view and determine some families of facets of the 0/1-polytope associated with one of these integer programming formulations. The theoretical results described here are used to design an efficient Cutting Plane algorithm.     

Inventory placement in acyclic supply chain networks

By adding a set of redundant constraints, and by iteratively refining the approximation, we show that a commercial solver is able to routinely solve moderate-size strategic safety stock placement problems to optimality. The speed-up arises because the solver automatically generates strong flow cover cuts using the redundant constraints.     

A lexicographic pricer for the fractional bin packing problem

We propose an exact lexicographic dynamic programming pricing algorithm for solving the Fractional Bin Packing Problem with column generation. The new algorithm is designed for generating maximal columns of minimum reduced cost which maximize, lexicographically, one of the measures of maximality we investigate. Extensive computational experiments reveal that a column generation algorithm based on this pricing technique can achieve a substantial reduction in the number of columns and the computing time, also when combined with a classical smoothing technique from the literature.