We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NP-hard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing their results to optimum solutions.

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Scatter search is a population-based method that has recently been shown to yield promising outcomes for solving combinatorial and nonlinear optimization problems. Based on formulations originally proposed in the 1960s for combining decision rules and problem constraints, such as the surrogate constraint method, scatter search uses strategies for combining solution vectors that have proved effective in a variety of problem settings. In this paper, we present a scatter search implementation designed to find high quality solutions for the NP-hard linear ordering problem, which has a significant number of applications in practice. The LOP, for example, is equivalent to the so-called triangulation problem for input-output tables in economics. Our implementation goes beyond a simple exercise on applying scatter search, by incorporating innovative mechanisms to combine solutions and to create a balance between quality and diversification in the reference set. We also use a tracking process that generates solution statistics disclosing the nature of combinations and the ranks of antecedent solutions that produced the best final solutions. Our extensive computational experiments with more than 300 instances establishes the effectiveness of our procedure in relation to those previously identified to be best.

The first computer implementation of the Dantzig-Fulkerson-Johnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique

This paper introduces a new hybrid method for efficiently integrating Pseudo-Boolean (PB) constraints into generic SAT solvers in order to solve PB satisfiability and optimization problems. To achieve this, we adopt the cutting-plane technique to draw inferences among PB constraints and combine it with generic implication graph analysis for conflictinduced learning. Novel features of our approach include a light-weight and efficient hybrid learning and backjumping strategy for analyzing PB constraints and CNF clauses in order to simultaneously learn both a CNF clause and a PB constraint with minimum overhead and use both to determine the backtrack level. Several techniques for handling the original and learned PB constraints are introduced. Overall, our method benefits significantly from the pruning power of the learned PB constraints, while keeping the overhead of adding them into the problem low. In this paper, we also address two other methods for solving PB problems, namely Integer Linear Programming (ILP) and pre-processing to CNF SAT, and present a thorough comparison between them and our hybrid method. Experimental comparison of our method against other hybrid approaches is also demonstrated. Additionally, we provide details of the MiniSAT-based implementation of our solver Pueblo to enable the reader to construct a similar one.

Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.

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Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primal-dual interior point method to solve the linear programming relaxations. A point which isagoodwarm start for a simplex-based cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some real-world problems; the algorithm appears to be competitive with a simplex-based cutting plane algorithm.

. The state equation is a verification technique that has been applied - not always under this name -- to numerous systems modelled as Petri nets or communicating automata. Given a safety property P, the state equation is used to derive a necessary condition for P to hold which can be mechanically checked. The necessary conditions derived from the state equation are known to be of little use for systems communicating by means of shared variables, in the sense that many of these systems satisfying the property but not the conditions. In this paper, we use traps, a well-known notion of net theory, to obtain stronger conditions that can still be efficiently checked. We show that the new conditions significantly extend the range of verifiable systems. Keywords: State equation, traps, approximation techniques, linear programming 1. Introduction The application of linear algebra and integer programming techniques to verification problems has been the subject of a large number of papers [3,...