Isabel Méndez-Díaz

A Branch-and-Cut algorithm for graph coloring

In this paper a Branch-and-Cut algorithm, based on a formulation previously introduced by us, is proposed for the Graph Coloring Problem. Since colors are indistinguishable in graph coloring, there may typically exist many different symmetrical colorings associated with a same number of colors. If solutions to an integer programming model of the problem exhibit that property, the Branch-and-Cut method tends to behave poorly even for small size graph coloring instances. Our model avoids, to certain extent, that bottleneck. Computational experience indicates that the results we obtain improve, in most cases, on those given by the well-known exact solution graph coloring algorithm Dsatur.

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.

A new formulation for the Traveling Deliveryman Problem

The Traveling Deliveryman Problem is a generalization of the Minimum Cost Hamiltonian Path Problem where the starting vertex of the path, i.e. a depot vertex, is fixed in advance and the cost associated with a Hamiltonian path equals the sum of the costs for the layers of paths (along the Hamiltonian path) going from the depot vertex to each of the remaining vertices. In this paper, we propose a new Integer Programming formulation for the problem and computationally evaluate the strength of its Linear Programming relaxation. Computational results are also presented for a cutting plane algorithm that uses a number of valid inequalities associated with the proposed formulation. Some of these inequalities are shown to be facet defining for the convex hull of feasible solutions to that formulation. These inequalities proved very effective when used to reinforce Linear Programming relaxation bounds, at the nodes of a Branch and Bound enumeration tree.

A branch-and-cut algorithm for the latent-class logit assortment problem

We study the product assortment problem of a retail operation that faces a stream of customers who are heterogeneous with respect to preferences. Each customer belongs to a market segment characterized by a consideration set that includes the alternatives viewed as options, and by the preference weights that the segment assigns to each of those alternatives. Upon arrival, he checks the offer set displayed by the firm, and either chooses one of those products or quits without purchasing according to a multinomial-logit (MNL) criterion. The firms goal is to maximize the expected revenue extracted during a fixed time horizon. This problem also arises in the growing area of choice-based, network revenue management, where computational speed is a critical factor for the practical viability of a solution approach. This so-called latent-class, logit assortment problem is known to be NP-Hard. In this paper, we analyze unconstrained and constrained (i.e., with a limited number of products to display) versions of it, and propose a branch-and-cut algorithm that is computationally fast and leads to (nearly) optimal solutions.

Composite Retrieval of Diverse and Complementary Bundles

Users are often faced with the problem of finding complementary items that together achieve a single common goal (e.g., a starter kit for a novice astronomer, a collection of question/answers related to low-carb nutrition, a set of places to visit on holidays). In this paper, we argue that for some application scenarios returning item bundles is more appropriate than ranked lists. Thus we define composite retrieval as the problem of finding k bundles of complementary items. Beyond complementarity of items, the bundles must be valid w.r.t. a given budget, and the answer set of k bundles must exhibit diversity. We formally define the problem and show that in its general form is NP-hard and that also the special cases in which each bundle is formed by only one item, or only one bundle is sought, are hard. Our characterization however suggests how to adopt a two-phase approach (Produce-and-Choose, or PAC) in which we first produce many valid bundles, and then we choose k among them. For the first phase we devise two ad-hoc clustering algorithms, while for the second phase we adapt heuristics with approximation guarantees for a related problem. We also devise another approach which is based on first finding a k-clustering and then selecting a valid bundle from each of the produced clusters (Cluster-and-Pick, or CAP). We compare experimentally the proposed methods on two real-world data sets: the first data set is given by a sample of touristic attractions in 10 large European cities, while the second is a large database of user-generated restaurant reviews from Yahoo! Local. Our experiments show that when diversity is highly important, CAP is the best option, while when diversity is less important, a PAC approach constructing bundles around randomly chosen pivots, is better.

A polyhedral approach for the equitable coloring problem

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm.

An integer programming approach for the time-dependent TSP

The Time-Dependent Travelling Salesman Problem (TDTSP) is a generalization of the traditional TSP where the travel cost between two cities depends on the moment of the day the arc is travelled. In this paper, we focus on the case where the travel time between two cities depends not only on the distance between them, but also on the position of the arc in the tour. We consider the formulations proposed in Picard and Queryanne [J.C. Picard and M. Queyranne. The time-dependent traveling salesman problem and its application to the tardiness problem in one-machine scheduling. Operations Res., 26(1):86–110, 1978] and Vander Wiel and Sahinidis [R.J. Vander Wiel and N.V. Sahinidis. An exact solution approach for the time-dependent traveling-salesman problem. Naval Res. Logist., 43(6):797–820, 1996], analyze the relationship between them and derive some valid inequalities and facets. Computational results are also presented for a Branch and Cut algorithm (B&C) that uses these inequalities, which showed to be very effective.

Complexity and algorithms for composite retrieval

Online search has become a daily activity and a source of a variety of valuable information, from the finest granularity such as finding the address of a specific restaurant, to more complex tasks like looking for accessories compatible with an iPhone or planning a trip. The latter typically involves running multiple search queries to gather information about different places, reading online reviews to find out about hotels, and checking geographic proximity of places to visit. We refer to this information seeking activity as composite retrieval and propose to organize results into item bundles that together constitute an improved exploratory experience over ranked lists. As a first step towards composite retrieval definition, we need to formalize intuitive desirable properties of item bundles. We distinguish between properties of each bundle in the answer and properties of the answer as a whole. Consider the case of a user selecting the restaurants to try during a visit to a new city. The user has a limited budget which might be either financial, or simply the number of nights spent in the city. The user prefers suggested restaurants to serve different cuisines. The validity of a bundle of restaurants is given by the budget constraint and the complementarity of the restaurants in the bundle w.r.t. the cuisine they serve. Other restaurant attributes could be used for defining valid bundles. For example, instead of cuisines, different dress codes could be required to every restaurant in a single bundle. Moreover, in order to provide meaningful bundles, restaurants forming each bundle must be compatible, e.g., close geographically, or liked by similar reviewers. The degree of compatibility of the items forming a bundle defines the quality of the bundle. Intuitively, in the case geographic distance is used, the closer restaurants are from each other, the higher the quality of the bundle they belong to. Similarly, when common reviewers are used as the

Solving a multicoloring problem with overlaps using integer programming

This paper presents a new generalization of the graph multicoloring problem. We propose a Branch-and-Cut algorithm based on a new integer programming formulation. The cuts used are valid inequalities that we could identify to the polytope associated with the model. The Branch-and-Cut system includes separation heuristics for the valid inequalities, specific initial and primal heuristics, branching and pruning rules. We report on computational experience with random instances.

A cluster-first route-second approach for the swap body vehicle routing problem

The swap body vehicle routing problem (SB-VRP) is a generalization of the classical vehicle routing problem where a particular structure as well as several operational aspects for the trucks composing the fleet are considered. This research has been motivated by the VeRoLog Solver Challenge 2014, organized together by VeRoLog and PTV group, aiming to motivate the study of real-world logistic problems. A truck can carry either only one swap body or, in addition, an extra trailer with an extra swap body. For the latter, special depots, called swap locations, can be used to drop and pickup the swap bodies. These operations may affect the feasibility and the cost of a route, and therefore the overall operational cost. In this paper, we propose a cluster-first route-second heuristic for the SB-VRP. Computational experiments are conducted over the benchmark instances proposed for the competition, simulating a practical environment by considering limited resources and execution time. The results obtained are of very good quality, where our approach ended as runner-up in the final set of instances and performs similarly to the other algorithms in the remaining cases, showing its potential to be applied in practice.