Daniel E. Steffy

MIPLIB 2010 - Mixed Integer Programming Library version 5

This paper reports on the fifth version of the Mixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.

Improving the accuracy of linear programming solvers with iterative refinement

We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, right-hand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, while numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We demonstrate that this algorithm is effective in practice for computing extended precision solutions and that this leads to direct improvement of the best known methods for solving LPs exactly over the rational numbers.

Solving Very Sparse Rational Systems of Equations

Efficient methods for solving linear-programming problems in exact precision rely on the solution of sparse systems of linear equations over the rational numbers. We consider a test set of instances arising from exact-precision linear programming and use this test set to compare the performance of several techniques designed for symbolic sparse linear-system solving. We compare a direct exact solver based on LU factorization, Wiedemann’s method for black-box linear algebra, Dixon’s p-adic-lifting algorithm, and the use of iterative numerical methods and rational reconstruction as developed by Wan.

Strong matching preclusion for augmented cubes

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. The concept was introduced by Park and Son. In this paper, we study the strong matching preclusion problem for the augmented cube graphs. As a result, we find smp(AQn) and classify all optimal solutions.

Exact solutions to linear systems of equations using output sensitive lifting

Many methods have been developed to symbolically solve systems of linear equations over the rational numbers. A common approach is to use p-adic lifting or iterative refinement to build a modular or approximate solution, then apply rational number reconstruction. An upper bound can be computed on the number of iterations these algorithms must perform before applying rational reconstruction. In practice such bounds can be conservative. Output sensitive lifting is the technique of performing rational reconstruction at intermediate steps of the algorithm and verifying correctness which allows the possibility of early termination when the solution size is small. In this paper we show how using an appropriate output sensitive lifting strategy can improve several algorithms. We show this procedure to be computationally effective and introduce a variant of the iterative-refinement method that incorporates warm starts into the rational reconstruction procedure.

Strong local diagnosability of (n,k)-star graphs and Cayley graphs generated by 2-trees with missing edges

Abstract In this paper, we study the local diagnosability and strong local diagnosability properties for ( n , k ) -star graphs and Cayley graphs generated by 2-trees. Moreover, we also consider the corresponding problem with missing edges.

Iterative Refinement for Linear Programming

We describe an iterative refinement procedure for computing extended-precision or exact solutions to linear programming (LP) problems. Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited-precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, right-hand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, and numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We show that this algorithm is effective in practice for computing extended-precision solutions and that it leads to direct improvement of the best known methods for solving LPs exactly over the rational numbers. Our implementation is publically available as an extensio...

Maximal vertex-connectivity of

The class of star graphs is a popular topology for interconnection networks. However, it has certain deficiencies. A class of generalization of star graphs called (n, k)-star graphs was introduced by Chiang and Chen to address these issues. In this article we will consider the vertex-connectivity of the directed (n, k)-star graph, , given by Cheng and Lipman, 8, and show that it is maximally connected.

Valid Linear Programming Bounds for Exact Mixed-Integer Programming

Fast computation of valid linear programming LP bounds serves as an important subroutine for solving mixed-integer programming problems exactly. We introduce a new method for computing valid LP bounds designed for this application. The algorithm corrects approximate LP dual solutions to be exactly feasible, giving a valid bound. Solutions are repaired by performing a projection and a shift to ensure all constraints are satisfied; bound computations are accelerated by reusing structural information through the branch-and-bound tree. We demonstrate this method to be widely applicable and faster than solving a sequence of exact LPs. Several variations of the algorithm are described and computationally evaluated in an exact branch-and-bound algorithm within the mixed-integer programming framework SCIP Solving Constraint Integer Programming.

On sublinear inequalities for mixed integer conic programs

This paper studies