We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.

We describe a fixed-point based approach to the theory of bipartite stable matchings. By this, we provide a common framework that links together seemingly distant results, like the stable marriage theorem of Gale and Shapley [11], the Menelsohn-Dulmage theorem [21], the Kundu-Lawler theorem [19], Tarski's fixed point theorem [32], the Cantor-Bernstein theorem, Pym's linking theorem [22, 23] or the monochromatic path theorem of Sands et al. [29]. In this framework, we formulate a matroid-generalization of the stable marriage theorem and study the lattice structure of generalized stable matchings. Based on the theory of lattice polyhedra and blocking polyhedra, we extend results of Vande Vate [33] and Rothblum [28] on the bipartite stable matching polytope.

We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem.

In recent years approximation algorithms based on primal-dual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primal-dual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design. The worst-case bound of the proposed algorithm is related to a fundamental relationship (called strength) between the set of valid inequalities and the set of minimal solutions to the covering problems. In this way, we can construct an approximation algorithm simply by constructing the required valid inequalities. We apply the proposed algorithm to several problems, such as covering problems related to totally balanced matrices, cyclic scheduling, vertex cover, general set covering, intersections of polymatroids, and several network design problems attaining (in most cases) the best worst-case bound known in the literature. In the last 20 years, two approaches to discrete optimization problems have emerged: polyhedral combinatorics and approximation algorithms. Under the first approach, researchers formulate problems as integer programs and solve their linear programming relaxations. By adding strong valid inequalities (preferably facets of the convex hull of solutions) to enhance the formulations, researchers

We address the problem of estimating the spectrum required in a wireless network for a given demand and interference pattern. This problem can be abstracted as a generalization of the graph coloring problem, which typically presents additional degree of hardness compared to the standard coloring problem. It is worthwhile to note that the question of estimating the spectrum requirement differs markedly from that of allocating channels. The main focus of this work is to obtain strong upper and lower bounds on the spectrum requirement, as opposed to the study of spectrum allocation/management. While the relation to graph coloring establishes the intractability of the spectrum estimation problem for arbitrary network topologies, useful bounds and algorithms are obtainable for specific topologies. In the first part of this work, we establish some new results regarding generalized coloring, which we use to derive tight bounds for specific families of graphs. The latter part is devoted to the...

doi 10.1287/opre.1040.0169

A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved e#ciently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first

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The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V,E), a set of terminals R ⊆ V, and non-negative costs ce for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices V\R are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J. Discrete Math, 2005); it achieves a performance guarantee of 1 + ln3 2 ≈ 1.55. The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math. Programming, 1993) and achieves an approximation ratio of 2 − 2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of G\R has at most b vertices. We show that Robins ’ and Zelikovsky’s algorithm has an approximation ratio better than 1 + ln3 2 for such instances, and we prove that the integrality gap of our LP is between 8 7

A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0; 1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron fx 0 : Ax 1g is integral. Examples of ideal binary clutters are st-paths, st-cuts, T -joins or T -cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour conjectured that a binary clutter is ideal if and only if it does not contain L F 7 , OK 5 , or b(O K 5 ) as a minor. In this paper, we show that a binary clutter is ideal if it does not contain five specified minors, namely the three above minors plus two others. This generalizes Guenin's characterization of weakly bipartite graphs, as well as the theorem of Edmonds and Johnson on T -joins and T -cuts. 1.