The Dantzig-Wolfe reformulation approach has shown to be very efficient on applications well suited to decomposition. It is to be placed in the broader context of reformulation techniques. A recent didactic effort was done where we reviewed the set of methods in Lagrangian approaches, present the method as a generic algorithm and reviewed possible approaches to branching.

We have finalized our work on developing a branching scheme that is compatible with the column generation procedure and that implies no structural modifications to the pricing problem. Our generic branching scheme proceeds by recursively partitioning the sub-problem solution set. Branching constraints are enforced in the pricing problem instead of being dualized in a Lagrangian way. The sub-problem problem is solved by a limited number of calls to the provided solver. The scheme avoids the enumeration of symmetric solutions. Its computational efficiency was demonstrated, solving problem to integrality without modifying the subproblem or expanding its variable space, which was a first .

In the past decade, significant progress has been achieved in developing generic primal heuristics that made their way into commercial mixed integer programming (MIP) solver. Extensions to the context of a column generation solution approach are not straightforward. The Dantzig-Wolfe decomposition principle can indeed be exploited in greedy, local search, rounding or truncated exact methods. The price coordination mechanism can bring a global view that may be lacking in some “myopic” approaches based on a compact formulation. However, the dynamic generation of variables requires specific adaptation of heuristic paradigms.

Based on our application specific experience with these techniques , , , , and on a review of generic classes of column generation based primal heuristics, in we focus on a so-called “diving” method in which we introduce diversification based on Limited Discrepancy Search. In , we moreover consider its combination with sub-mipping and relaxation induced neighborhood search. These add-ons can be interpreted as local-search or diversification mechanisms. While being a general purpose approach, the implementation of the selected heuristics illustrates the technicalities specific to column generation. The methods are numerically tested on variants of the cutting stock and vehicle routing problems.

Working in an extended variable space allows one to develop tight reformulations for mixed integer programs. However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIP-solver. Then, one can use projection tools to derive valid inequalities for the original formulation and implement a cutting plane approach. Or, one can approximate the reformulation, using techniques such as variable aggregation or by reformulating a submodel only. Such approaches result in outer approximation of the intended extended formulation.

The alternative considered in  is an inner approximation obtained by generating dynamically the variables of the extended formulation. It assumes that the extended formulation stems from a decomposition principle: a sub-problem admits an extended formulation from which an extended formulation for the original problem can be derived. Then, one can implement column generation for the extended formulation of the original problem by transposing the equivalent procedure for the Dantzig-Wolfe reformulation. Pricing sub-problem solutions are expressed in the variables of the extended formulation and added to the current restricted version of the extended formulation along with the sub-problem constraints that are active for the sub-problem solution.

Our paper reviews the applications of the literature of such “column-and-row generation” procedure and analyses this approach's potential benefits compared to a standard column generation approach. Numerical experiments highlight a key observation: lifting pricing problem solutions in the space of the extended formulation permits their recombination into new sub-problem solutions and results in faster convergence.

In the follow-up of , we developed the combination of Dantzig-Wolfe and Bender's decomposition: Bender's Master is solved by column generation .

Our team is involved in the effort to synthesize advances and inspire new ideas in order to transform MINLP into an area in which researchers and practitioners can access robust tools and methods capable of solving a wide range of decision support problems. In , we exploited the information generated by solvers as they make branching decisions to define a structured family of disjunctive cuts. Even for MIPs, these ideas seem capable of significantly reducing the size of branch-and-bound trees ; moreover, the ideas themselves are directly applicable to MINLPs, and we are currently investigating how best to apply them.

In , we study convex hulls of sets defined by multi linear functions (involving a product of variables) in which the variables have lower and upper bounds, and in which the product itself is also bounded. Multi linear functions appear in many global optimization problems, including blending and electricity transmission, among many others. Since global branch-and-bound solvers for such problems use polyhedral relaxations of such sets to compute bounds, having tight relaxations can improve performance. For two variables, the well-known McCormick inequalities define the convex hull for an unbounded product. Our research defines an infinite set of linear inequalities that defines the convex hull for a boundedproduct of two variables. For a bounded product or more than two variables, we define valid linear inequalities that support as many points of the convex hull as possible. Though uncountably infinite in number, these inequalities can be separated for exactly in polynomial time.

We have extended some of these results to products of more than two variables, considering both convex hull descriptions and separation. One important result (first presented during an invited presentation at the 2010 Mixed Integer Programming Workshop, http:// www2. isye. gatech. edu/ mip2010/ ) is that the set of inequalities that we have defined suffice to describe the convex hull of the set defined by bounded product of nvariables, regardless of the size of n.

Although well studied, important questions on the rank of the Gomory-Chàtal operator when restricting to polytopes contained in the n-dimensional 0/1 cube have not been answered yet. In particular, the question on the maximal rank of the Gomory-Chàtal-procedure for this class of polytopes is still open. So far, the best-known upper bound is O( n2log( n))and the best-known lower bound, which is based on a randomized construction of a family of polytopes, is (1 + ) n, both of which were established in Eisenbrand and Scultz . The main techniques to prove lower bounds were introduced in . In we revisit one of those techniques and we develop a simpler method to establish lower bounds. We show the power and applicability of this method on classical examples from the literature as well as provide new families of polytopes with high rank. Furthermore, we provide a deterministic family of polytopes achieving a Chvàtal-Gomory rank of at least (1 + 1/ e) n-1> nand we conclude the paper with showing how to obtain a lower bound on the rank from solely examining the integrality gap.

We built on our previous work on the stable set problem (selecting disjoint nodes) in claw-free graphs (graphs with no vertex having a stable set of size 3 in its neighborhood). This problem is a fundamental generalization of the matching problem that offers a nice playground for building the theory of polyhedral Combinatorics further. Indeed Minty gave the first polynomial time algorithm for solving this problem in 1980 but unfortunately he did not reveal the polyhedral counterpart. Describing the stable set polytope of claw-free graphs was thus ranked as one of the top ten open problems in combinatorial optimization by Groetschel, Lovasz and Schrijver in 1986. Our team has had significant contributions on this problem both on the polyhedral and algorithmic aspects.

After closing a 25-years old conjecture for an interesting subclass of claw-free graphs , giving a complete description of the rank facets of the stable set of fuzzy circular interval graphs, we provided new facets for claw-free graphs , we gave a characterization of strongly minimal facets for quasi-line graphs , an extended formulation and polytime separation procedure for the stable set polytope of claw-free graphs and a full characterization of the polytope for claw-free graphs with 4 .

In we propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O( n3)-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present paper. Despite being weaker than the well-known structure result for claw-free graphs given by Chudnovsky and Seymour , our decomposition theorem is, on the other hand, algorithmic, i.e. it is coupled with an O( n3)-time procedure that actually produces the decomposition. We also believe that our algorithmic decomposition result is interesting on its own and might be also useful to solve other kind of problems on claw-free graphs.

Interestingly the composition operation at the core of this decomposition theorem, which as we have just seen has some nice algorithmic consequences, appears to also have a nice polyhedral behavior for the stable set polytope that go much beyond claw-free graphs. Indeed, in we show how one can use the structure of the composition to describe the stable set polytope from the matching one and, more importantly, how one can use it to separate over this stable set polytope in polynomial time. We then apply those general results to the stable set in claw-free graphs, to show that the stable set polytope can be reduced to understanding the polytope in very basic structures (for most of which it is already known). In particular for a general claw-free graph G, we show two integral extended formulation for STAB( G)and a procedure to separate in polynomial time over STAB( G); moreover, we provide a complete characterization of STAB( G)when Gis any claw-free graph with stability number at least 4 having neither homogeneous pairs nor 1-joins.

In , we focus on the facets of the stable set polytope of quasi-line graphs (a subclass of claw-free graphs). While Ben Rebea Theoremprovides a complete linear description of this polytope (see Eisenbrand et al. ), no minimal description is available. In this paper, we shed some light on this question. We show that any facet of this polytope is such that the restriction of the inequality to the graph induced by the vertices with maximal coefficient yield a rank facet for this subgraph. We build upon this result and a result from Galluccio and Sassano for rank-minimal facets in claw-free graphs to provide a complete description of the strongly minimal facets for quasi-line graphs. Finally we show that our result supports two conjectures refining Ben Rebea Theorem for the stable set polytope of circulant graphs and fuzzy circular interval graphs due respectively to Pêcher and Wagler and Oriolo and Stauffer and that it offers a possible line of attack.

Another central contribution of our team concerns the chromatic number of a graph (the minimum number of independent stable sets needed to cover the graph). We investigated the circular-chromatic number. It is a well-studied refinement of the chromatic number of a graph (designed for problems with periodic solutions): the chromatic number of a graph is the integer ceiling of its circular-chromatic number. Xuding Zhu noticed in 2000 that circular cliques are the relevant circular counterpart of cliques, with respect to the circular chromatic number, thereby introducing circular-perfect graphs, a super-class of perfect graphs. It is unknown whether the circular-chromatic number of a circular-perfect graph is computable in polynomial time in general.

In , we design a polynomial time algorithm that computes this circular chromatic number when the circular-perfect graphs is claw-free. We also proved that the chromatic number of circular-perfect graphs is computable in polynomial time , thereby extending Grötschel, Lovász and Schrijver's result to the whole family of circular-perfect graphs. Last but not least, we managed recently to give closed formulas for the Lovász Theta number of circular-cliques (previously, closed formulas were known for circular-cliques with clique number at most 3 only), which implies that the circular-chromatic number of densecircular-perfect graphs is computable in polynomial time .

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In , we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation.

To accommodate the increase of traffic in telecommunication networks, today's optical networks use grooming and wavelength division multiplexing technologies. Packing multiple requests together in the same optical stream requires to convert the signal in the electrical domain at each aggregation of disaggregation of traffic at an origin, a destination or a bifurcation node. Traffic grooming and routing decisions along with wavelength assignments must be optimized to reduce opto-electronic system installation cost. In collaboration with B. Jaumard from Concordia University in Quebec, we developed and compared several decomposition approaches to deal with backbone optical network with relatively few nodes (around 20) but thousands of requests for which traditional multi-commodity network flow approaches are completely overwhelmed. We also studied the impact of imposing a restriction on the number of optical hops in any request route.

In collaboration with Teresa Godinho from the Polytechnic Institute of Beja, Portugal, and Luis Gouveia from the University of Lisbon, Portugal, Pierre Pesneau has studied the time-dependent travelling salesman problem. This problem is a generalization of the common Asymmetric Travelling Salesman Problem where the cost of a link depends on its position in the tour. Observing that the main feature of the well-known Picard and Queyranne formulation for the problem is the use, as a sub-problem, of the exact description of a circuit on nnodes (that may repeat arcs and nodes), they proposed a new formulation by strengthening this sub-problem. For a given node k, the new sub-problem describes exactly a circuit on nnodes (as in the previous model) but going through node kexactly once. An even stronger formulation is obtained by duplicating this sub-problem for each possible node k. To obtain such formulation, it has been necessary to consider additional sets of variables and the new formulations, even compact, are quite large. However, the projections of the linear relaxation of these formulations on the space of the original Picard and Queyranne variables are described and give potential issues to handle such formulations. The results given by the computational experiments on the extended formulation show very good dual bounds issued from the linear relaxation of the formulation. The integrality gap is even closed to zero for several instances of the literature. These results have led to a paper submitted for publication in Discrete Applied Mathematics.

Along with Laurent Alfandari, Sylvie Borne and Lucas Létocart from the University Paris 13, Pierre Pesneau is studying integer quadratic or integer linear programming formulations for some variants of the Asymmetric Travelling Salesman Problem. They study the case where there is a lower bound on the distance (in number of links) between cities of some subset. Such case appears, for instance, in the design the shortest circuit for a travelling salesman who has to visit a minimum number of clients before having a break (or a night). Another close application can be found in where the authors consider lower bounds on the capacity for a vehicle routing problem. In previous work they made several proposals of reformulations and solution approaches based on branch-and-price-and-cut. Now they experimented with one of these reformulations, developing a branch-and-price-and-cut under our software platform BaPCod. (In the process, BaPCod was developed further to encompass delayed generation of constraints needed for the validity of the formulation. The preliminary results are encouraging and it is planed to develop this application in more details and to further complete the cut generator framework within BaPCod.

Pierre Pesneau has started a new collaboration with Pierre Fouilhoux from the University Paris 6, France on a variant of the maximum multi-cut problem. Given a graph and an integer k, a multi-cut is a set of edges when deleted, they are disconnecting the graph in kcomponents. The maximum multi-cut problem finds such a cut of maximum weight. Note that when kis equal to 2, the problem is simply the well-known maximum cut problem. In the considered case, one reduces the search on cuts such that, when deleted, it leaves exactly k connectedcomponents. This problem has various applications. We can cite for instance an application coming from the Cemagref (French research institute in environmental sciences and technologies) where the goal is to decompose a region in several connected areas while maximizing the dissimilarities between these areas. Other applications could be found for domain decomposition in numerical calculus. This study has already led to several reformulations of the problem, in particular a quadratic formulation that has been linearised and a formulation suited for a branch-and-price solution. The perspectives are to study the polyhedron associated with this problem and develop solution algorithms. To this aim, concerning the branch-and-price based formulation, it will be necessary to study the sub-problem induced by the formulation that consist in finding an induced connected sub-graph of minimum weight, where the weights are unrestricted and are carried by the edges and the nodes. It seems this problem is itself a challenge.

The bin-packing problem raise the question of the minimum number of bin of fixed size we need to pack a set of items of different sizes. We studied a generalization of this problem where items can be in conflicts and thus cannot be put together in the same bin. We show in that the instances of the literature with 120 to 1000 items can be solved to optimality with a generic Branch-and-Price algorithm, such as our prototype BaPCod, within competitive computing time. Moreover, we solved to optimality all the 37 open instances. The approach involves generic primal heuristics, generic branching, but a specific pricing procedure.

The knapsack variant encountered in our bin packing problem resolution considers conflicts between items. This problem is quite difficult to solve compared to the usual knapsack problem. The latter is already NP-hard, but can be usually efficiently solved by dynamic programming. We have shown that when the conflict graph (the graph defining the conflicts between the items) is an interval graph, this generalization of the knapsack can also be solved quite efficiently by dynamic programming with the same complexity than the one to solve the common knapsack problem. For the case, when the conflict graph is arbitrary, we proposed a very efficient enumeration algorithm which outperforms the approaches used in the literature.

Another research concerns scheduling parallel jobs i.e. which can be executed on more than one processor at the same time. With the emergence of new production, communication and parallel computing system, the usual scheduling requirement that a job is executed only on one processor has become, in many cases, obsolete and unfounded. Therefore, parallel jobs scheduling is becoming more and more widespread. In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed. We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution  .

We have also studied in a scheduling problem that takes place at cross docking terminals. In such places, products from incoming trucks are sorted according to there destinations and transferred to outgoing trucks using a temporary storage. Such terminals allow companies to reduce storage and transportation costs in supply chain. We focus on the operational activities at cross docking terminals. In , we consider the trucks scheduling problem with the objective to minimise the storage usage during the product transfer. We show that a simplification of this NP-hard problem in which the arrival sequences of incoming and outgoing trucks are fixed and outgoing tracks can take products of only one type is polynomially solvable by proposing a dynamic programming algorithm for it. In  , this result has been extended to the case in which outgoing trucks can take products of several types. This work also presents the results of numerical tests of the algorithm on randomly generated instances are presented.

In , we developed rigorous computational methods to find high quality production plans for big bucket lot-sizing problems of realistic size. By big bucketwe mean problems in which multiple product categories compete for the same capacities (of machines, labor, etc.) In , we have compared various methods for finding performance guarantees (lower bounds) for realistically sized instances of such problems. These methods include both those previously proposed in the literature and those we have developed ourselves. Our methods of comparison are both theoretical and computational; one of the primary contributions of this research is to identify and highlight those aspects of these problems that prevent us from solving them more effectively. This identification could be crucial in improving our ability to solve such models.

In research described in , we build on these results to identify and analyze a minimal submodel that captures those aspects of these problems, mentioned above, that make them currently intractable. We are in the process of developing polyhedral approaches to these problems based on our analyses of these submodels, and of compiling extensive computational results for multi-level, capacitated big bucket instances of realistic size. The evidence suggests that the strengthened formulations generated by our procedure are tighter than those produced by these and other authors in previous research.

The following theoretical research was directly inspired by production planning problems. In , we discuss a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. These results directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in O( nlogn). Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.

We are currently working on a project aiming to plan the energy production and the maintenance breaks for a set of power plants generating electricity. This problem has two different levels of decisions. The first one consist in determining, for a certain time horizon, when the different power plants will have to stop in order to perform a refueling and to decide the amount of this refueling. Given a set of scenarios defining variable levels of energy consumption, the second decision level aims to decide the quantity of power each plant will have to produce.

As the number of periods composing the time horizon, and the number of scenario are quite large, the size of any MIP formulation for such problem will forbid an exact resolution of the problem in an acceptable time. However, our objective is to show that exact methods can be used on a simplified problem (implementing an hierarchical optimization) and combined to design heuristics for the solution of large scale problem.

In , we deal with the inventory control of one single product in a distribution network made of one warehouse and nretailers where the retailers face continuous demands with constant rates. We develop two new heuristics for this problem based on the application of the classical EOQ formula at each locations and we show that those heuristics are respectively - and 1.275-optimal. We then conduct computational experiments and show that our second heuristic surprisingly shows excellent average performance : it deviates from Roundy's celebrated algorithm by less than 5%. This result shows evidences (in a very simple setting) that slightly adapting single echelon strategies might be a reasonable approach to manage efficiently the inventory in simple distribution networks, confirming observations from practitioners.

In we consider a well-known NP-hard deterministic inventory control problem: the One-Warehouse Multi-Retailer (OWMR) problem. We present a simple combinatorial algorithm to recombine the optimal solutions of the natural single-echelon inventory sub-problems into a feasible solution of the OWMR problem. This approach yields a 3-approximation. We then show how this algorithm can be improved to a 2-approximation by halving the demands at the warehouse and at the retailers in the sub-problems. Both algorithms are purely combinatorial and can be implemented to run in linear time for traditional linear holding costs and quadratic time for more general holding cost structures. We finally show that our technique can be extended to the Joint Replenishment Problem (JRP) with back-orders and to the OWMR problem with non-linear holding costs.

The allocation of surgeries to operating rooms (ORs) is a challenging combinatorial optimization problem. There is moreover significant uncertainty in the duration of surgical procedures, which further complicates assignment decisions. In , we present stochastic optimization models for the assignment of surgeries to ORs on a given day of surgery. The objective includes a fixed cost of opening ORs and a variable cost of overtime relative to a fixed length-of-day. We describe two types of models. The first is a two-stage stochastic linear program with binary decisions in the first-stage and simple recourse in the second stage. The second is its robust counterpart, in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations. We describe the mathematical models, bounds on the optimal solution, and solution methodologies, including an easy-to-implement heuristic. Numerical experiments based on real data from a large health care provider are used to contrast the results for the two models, and illustrate the potential for impact in practice. Based on our numerical experimentation we find that a fast and easy-to-implement heuristic works fairly well on average across many instances. We also find that the robust method performs approximately as well as the heuristic, is much faster than solving the stochastic recourse model, and has the benefit of limiting the worst-case outcome of the recourse problem.

With C. Joncour (PhD student) and P. Valicov (PhD student), we investigate the orthogonal knapsack problem, with the help of graph theory  . Fekete and Schepers managed a recent breakthrough in solving multi-dimensional orthogonal placement problems by using an efficient representation of all geometrically symmetric solutions by a so called packing classinvolving one interval graph(whose complement admits a transitive orientation: each such orientation of the edges corresponds to a specific placement of the forms) for each dimension. Though Fekete & Schepers' framework is very efficient, we have however identified several weaknesses in their algorithms: the most obvious one is that they do not take advantage of the different possibilities to represent interval graphs.

In , , and , we give two new algorithms: the first one is based upon matrices with consecutive ones on each row as data structures and the second one uses so-called MPQ-trees. These two new algorithms are very efficient, as they outperform Fekete and Schepers' on most standard benchmarks.

The development of the prototype software platform has made good progress this year thanks to our junior engineer, F. Labat, who has been hired for one year on an ADT: the developments focus on the redesign of the version manager along with continuous integration tools and automatic bug reports; new compilation environment using cmake tools; code transfer to several platforms; a revised interface with MIP solvers; code profiling to identify the bottlenecks; performance improvements (by a factor 10 on large scale applications). These developments of the environment and progress in software re-engineering were done in parallel to the implementation of new methodologies (such generic primal heuristics and the prototyping of simultaneous column-and-row generation approach for extended formulations).

We have also launched a new collaboration with EURODECISION (Versailles): the company is testing our prototype on industrial applications in the aim of fostering future exchanges on further methodological developments and efficient implementation of core modules.

The prototype also enables us to be very responsive in our industrial contact. In particular, it was used in our approach to the powerplant planning optimization challenge proposed by EDF.

A research proposal on the generic methodologies underlying the Branch-and-Price approach has been submitted. The purpose is to install a collaboration with M. Poggi and E. Uchoa (from Rio) and the company GAPSO (a Brazilian spin-up launched by these academics). In this context, the software platform BaPCod shall serve as a proof-of-concept code and it will benefit from the transfer of knowledge between the parties.

2D-KNAP is a software available on LaForge INRIA to check whether a 2D orthogonal packing admits a feasible solution. Fekete and Schepers introduced a tuple of interval graphs as data structures to store a feasible packing, and gave a very efficient algorithm. In 2D-KNAP, feasibility checks are based on an alternative graph theory characterization of interval graphs: Fulkerson and Gross's decomposition into maximal cliques . The algorithm uses consecutive one matrices as data structures.