In a previous work, we proposed a new integer programming formulation for the graph coloring problem which, to a certain extent, avoids symmetry. We studied the facet structure of the 0/1-polytope associated with it. Based on these theoretical results, we present now a Branch-and-Cut algorithm for the graph coloring problem. Our computational experiences compare favorably with the well-known exact graph coloring algorithm DSATUR.

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.

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a b s t r a c tThe 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.

Abstract-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 article, 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 characterize its complexity. We prove that the problem 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 us how to adopt a two-phase approach (Produceand-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.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 a large real-world database of user-generated restaurant reviews from Yahoo! Local, exploring their performance under a variety of settings. 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.

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Reward-based scheduling of real-time systems of periodic, preemptable, and independent tasks with mandatory and optional parts in homogeneous multiprocessors with energy considerations is a problem that has not been analyzed before. The problem is NP-hard. In this paper, a restricted migration schedule is adopted in which different jobs of the same task may execute in different processors and at different power modes but no migration is allowed after the job has started its execution. An objective function to maximize the performance of the system considering the execution of optional parts, the benefits of slowing down the processor, and a penalty for changing the operation frequency is introduced together with a set of constraints that guarantee the real-time performance of the system. Different algorithms are proposed to find a feasible schedule maximizing the objective function and are compared using synthetic systems of tasks generated following guidelines proposed in previous papers.

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