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.

On the reduction method for integer linear programs, II

Discrete Applied Mathematics, 1985

The problem of the sequential reduction of linear equations in non-negative discrete variables has been treated by several authors since its introduction by Elmaghraby and Wig in 1969. This paper provides tighter conditions on the multipliers, especially for the case of bivalent (0, 1) variables in homogeneous equations (zero right side). An earlier version of this paper appeared in 1980 under the same title (OR Report No. 125, N.C. State University). The current paper extends the previous results, and presents a comparative analysis with other approaches. In particular, we give conditions under which our multipliers are 'better' than others, where 'better' is precisely defined.

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.

On compact formulations for integer programs solved by column generation

2005

Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a socalled compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.

50 Years of Integer Programming 1958-2008

2010

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.

AN ALGORITHM FOR SOLVING INTEGER LINEAR PROGRAMMING PROBLEMS

The paper describes a method to solve an ILP by describing whether an approximated integer solution to the RLP is an optimal solution to the ILP. If the approximated solution fails to satisfy the optimality condition, then a search will be conducted on the optimal hyperplane to obtain an optimal integer solution using a modified form of Branch and Bound Algorithm.

Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix

Mathematical Programming, 2017

We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the affine TUdimension of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls.

An integer programming approach for linear programs with probabilistic constraints

Mathematical Programming, 2010

Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.

A Polynomial-Time Algorithm for Optimizing over N-Fold 4-Block Decomposable Integer Programs

Lecture Notes in Computer Science, 2010

In this paper we generalize N -fold integer programs and two-stage integer programs with N scenarios to N -fold 4-block decomposable integer programs. We show that for fixed blocks but variable N , these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions.

Decomposition Methods for Integer Linear Programming

2010

Within the field of mathematical programming, discrete optimization has become the focus of a vast body of research and development due to the increasing number of industries now employing it to model the decision analysis for their most complex systems. Mixed integer linear programming problems involve minimizing (or maximizing) the value of some linear function over a polyhedral feasible region subject to integrality restrictions on some of the variables.

A note on the shortest lattice vector problem

1999

Is it easier to decide instances of NP-hard problems when they are given with the additional promise that the associated search problem has exactly zero or one solution? Over a decade ago, Valiant and Vazirani [VV86] proved a beautiful result that shows that this is not the case. More formally, they gave a probabilistic many-one reduction from the NP-complete Boolean formula satisfiability problem to the problem of deciding whether a Boolean formula is satisfiable under the promise that it has either zero or one satisfying assignment.