We present an approximation algorithm for solving large 0-1 integer programming problems where A is 0-1 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems ...

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We propose a new branch-and-bound algorithm to solve hard instances of set covering problems arising from Steiner triple systems. 1 Introduction. The set covering probem (SC) is the problem of finding the minimum number of elements of a ground set E intersecting each member of a given family of subsets of E. (SC) is known to be NP-hard ([14]). Several algorithms have been studied and implemented in order to solve (SC), exactly or approximately, see [3, 4, 7, 8, 10, 11, 12, 13, 16]. In 1974 Fulkerson, Nemhauser and Trotter [13] described a class of computationally difficult set covering problems arising from a class of set systems known as Steiner triple systems (STS), and they suggested that these problems could be used as benchmarks to test the quality of different algorithms for the set covering problem. In particular, they introduced four special instances of (STS) with 9, 15, 27 and 45 elements; we denote them by STS 9 , STS 15 , STS 27 and STS 45 . They were able to solve STS 9...

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We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to `real-life' problems show that the algorithm is competitive with the fastest algorithms known so far. 1 Introduction We denote by G = (V; E) an undirected graph. V is the set of nodes and E the set of edges. A stable set is a subset of V such that no two nodes of the subset are pairwise adjacent. The cardinality of a maximum stable set of G will be denoted by ff(G). A clique is a subset of V with the property that all the nodes are pairwise adjacent. A clique covering is a set of disjoint cliques whose union is equal to V ; the cardinality of a minimum clique covering is denoted by `(G), and since at most one nod...

. Edge projection is a specialization of Lov'asz and Plummer's clique projection when restricted to edges. We discuss some properties of the edge projection which are then exploited to develop a new upper bound procedure for the Maximum Cardinality Stable Set Problem (MSS Problem). The upper bound computed by our heuristic, incorporated in a branch-and-bound scheme in conjunction with Balas and Yu branching rule, seems to be very effective for sparse graphs, which are typically hard instances of the MSS Problem. 1. Introduction We denote by G = (V; E) an undirected graph. V is the set of nodes and E the set of edges. A stable set is a subset of V such that no two nodes of the subset are pairwise adjacent. The cardinality of a maximum stable set of G will be denoted by ff(G). When G is the empty graph, ff(G) = 0. A clique is a subset of V with the property that all the nodes are pairwise adjacent. A clique partitioning of G is a family of cliques such that each node of G is contained ...

Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branch-and-bound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint. Many important families of integer programs are structured such that small instances from the family are embedded in larger instances. This structural information can be exploited to define a group of strong constraints on which to base the orbital branching disjunction. The symmetric nature of the problems is further exploited by enumerating non-isomorphic solutions to instances of the small family and using these solutions to create a collection of typically easy-to-solve integer programs. The solution of each integer program in the collection is equivalent to solving the original large instance. The effectiveness of this methodology is demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.

Performance aspects of a Lagrangian relaxation based heuristic for solving large 0-1 integer linear programs are discussed. In particular, we look at its application to airline and railway crew scheduling problems. We present a scalable parallelization of the original algorithm used in production at Carmen Systems AB, Göteborg, Sweden, based on distributing the variables. A lazy variant of this approach which decouples communication and computation is even useful on networks of workstations. Furthermore,

We describe the design of an optimizer for 0-1 integer programming aimed at solving large problems. The algorithm is based on very simple operations giving it a low complexity, and we show that for large set covering problems it can produce very good solutions compared to other methods. This is the only optimizer in the Carmen system for airline crew scheduling, used by several major European airlines, and we discuss how it has affected the overall design and success of this system. Keywords: 0-1 integer linear programming, efficient algorithms, set covering, airline crew scheduling. 1 Introduction In this paper I will describe the design of an optimizer for solving large 0-1 integer programs, and its application in the Carmen system for airline crew scheduling. The design of the optimizer itself has mainly been an academic research project aimed at discrete optimization in general. However, the specific application in mind has been the usually very large optimization problems that a...

Abstract. We present a biased random-key genetic algorithm (BRKGA) for finding small covers of computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Using a parallel implementation of the BRKGA, we compute improved covers for the two largest instances in a standard set of test problems used to evaluate solution procedures for this problem. The new covers for instances A405 and A729 have sizes 335 and 617, respectively. On all other smaller instances our algorithm consistently produces covers of optimal size. 1.