Truncated Markov bases and Gr"obner bases for Integer Programming (original) (raw)

Integer Programming, Lattices, and Results in Fixed Dimension

Handbooks in Operations Research and Management Science, 2005

We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixed.

Lattice Reduction for Modular Knapsack

Lecture Notes in Computer Science, 2013

In this paper, we present a new methodology to adapt any kind of lattice reduction algorithms to deal with the modular knapsack problem. In general, the modular knapsack problem can be solved using a lattice reduction algorithm, when its density is low. The complexity of lattice reduction algorithms to solve those problems is upper-bounded in the function of the lattice dimension and the maximum norm of the input basis. In the case of a low density modular knapsack-type basis, the weight of maximum norm is mainly from its first column. Therefore, by distributing the weight into multiple columns, we are able to reduce the maximum norm of the input basis. Consequently, the upper bound of the time complexity is reduced. To show the advantage of our methodology, we apply our idea over the floating-point LLL (L 2 ) algorithm. We bring the complexity from O(d 3+ε β 2 + d 4+ε β) to O(d 2+ε β 2 + d 4+ε β) for ε < 1 for the low density knapsack problem, assuming a uniform distribution, where d is the dimension of the lattice, β is the bit length of the maximum norm of knapsack-type basis. We also provide some techniques when dealing with a principal ideal lattice basis, which can be seen as a special case of a low density modular knapsack-type basis.

Computing generating sets of lattice ideals and Markov bases of lattices

Journal of Symbolic Computation, 2009

In this article, we present an algorithm for computing generating sets of lattice ideals or equivalently for computing Markov bases of lattices. Generating sets of lattice ideals and Markov bases of lattices are essentially equivalent concepts. In contrast to other existing methods, the algorithm in this article computes with projections of lattices. This algorithm clearly outperforms other algorithms in our computational experience. Two areas of application for generating sets of lattice ideals and Markov bases lattices are algebraic statistics and integer programming.

Non-standard approaches to integer programming

Discrete Applied Mathematics, 2002

In this survey we address three of the principal algebraic approaches to integer programming. After introducing lattices and basis reduction, we ÿrst survey their use in integer programming, presenting among others Lenstra's algorithm that is polynomial in ÿxed dimension, and the solution of diophanine equations using basis reduction. The second topic concerns augmentation algorithms and test sets, including the role played by Hilbert and Gr obner bases in the development of a primal approach to solve a family of problems for all right-hand sides. Thirdly we survey the group approach of Gomory, showing the importance of subadditivity in integer programming and the generation of valid inequalities, as well the relation to the parametric problem cited above of solving for all right-hand sides.

Hard Equality Constrained Integer Knapsacks

Mathematics of Operations Research, 2004

We consider the following integer feasibility problem: Given positive integer numbers a 0 a 1 a n , with gcd a 1 a n = 1 and a = a 1 a n , does there exist a vector x ∈ n ≥0 satisfying a x = a 0 ? We prove that if the coefficients a 1 a n have a certain decomposable structure, then the Frobenius number associated with a 1 a n , i.e., the largest value of a 0 for which a x = a 0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we take a 0 to be the Frobenius number. Furthermore, we show that the decomposable structure of a 1 a n makes the solution of a lattice reformulation of our problem almost trivial, since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small. For branch-and-bound such instances are difficult to solve, since they are infeasible and have large values of a 0 /a i 1 ≤ i ≤ n. We illustrate our results by some computational examples.

Computing generating sets of lattice ideals

In this article, we present a new algorithm for computing generating sets and Gröbner bases of lattice ideals. In contrast to other existing methods, our algorithm starts computing in projected subspaces and then iteratively lifts the results back into higher dimensions, by using a completion procedure, until the original dimension is reached. We give a completely geometric presentation of our Projectand-Lift algorithm and describe also the two other existing main algorithms in this geometric framework. We then give more details on an efficient implementation of this algorithm, in particular on critical-pair criteria specific to lattice ideal computations. Finally, we conclude the paper with a computational comparison of our implementation of the Project-and-Lift algorithm in 4ti2 with algorithms for lattice ideal computations implemented in CoCoA and Singular. Our algorithm outperforms the other algorithms in every single instance we have tried.

A note on reducing the number of variables in integer programming problems

Computational Optimization and Applications, 1997

A necessary and sufficient condition for identification of dominated columns, which correspond to one type of redundant integer variables, in the matrix of a general Integer Programming problem, is derived. The given condition extends our recent work on eliminating dominated integer variables in Knapsack problems, and revises a recently published procedure for reducing the number of variables in general Integer Programming problems given in the literature. A report on computational experiments for one class of large scale Knapsack problems, illustrating the function of this approach, is included.

Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition

INFORMS Journal on Computing, 2000

This paper is about modeling and solving mixed integer programming (MIP) problems. In the last decade, the use of mixed integer programming models has increased dramatically. Fifteen years ago, mainframe computers were required to solve problems with a hundred integer variables. Now it is possible to solve problems with thousands of integer variables on a personal computer and obtain provably good approximate solutions to problems such as set partitioning with millions of binary variables. These advances have been made possible by developments in modeling, algorithms, software, and hardware. This paper focuses on effective modeling, preprocessing, and the methodologies of branch-and-cut and branch-and-price, which are the techniques that make it possible to treat problems with either a very large number of constraints or a very large number of variables. We show how these techniques are useful in important application areas such as network design and crew scheduling. Finally, we discuss the relatively new research areas of parallel integer programming and stochastic integer programming.

Partial Gröbner Bases for Multiobjective Integer Linear Optimization

SIAM Journal on Discrete Mathematics, 2009

This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gröbner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gröbner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “á la Buchberger” is developed to compute partial Gröbner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.