242 research outputs found
Polyhredral techniques in combinatorial optimization I: theory
Combinatorial optimization problems appear in many disciplines ranging
from management and logistics to mathematics, physics, and chemistry. These
problems are usually relatively easy to formulate mathematically, but most
of them are computationally hard due to the restriction that a subset of the
variables have to take integral values. During the last two decades there has
been a remarkable progress in techniques based on the polyhedral description
of combinatorial problems, leading to a large increase in the size of several
problem types that can be solved. The basic idea behind polyhedral techniques
is to derive a good linear formulation of the set of solutions by identifying
linear inequalities that can be proved to be necessary in the description of the
convex hull of feasible solutions. Ideally we can then solve the problem as
a linear programming problem, which can be done eciently. The purpose of
this manuscript is to give an overview of the developments in polyhedral theory,
starting with the pioneering work by Dantzig, Fulkerson and Johnson on the
traveling salesman problem, and by Gomory on integer programming. We also
present some modern applications, and computational experience
Integer programming, lattices, and results in fixed dimension
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 fixe
Lattice based extended formulations for integer linear equality systems
We study different extended formulations for the set
X^+ = X\cap Z^n_+(X^+)Aaaa$ can be obtained that will provide a useful extended formulation.
Our theoretical results are accompanied by a small computational study
Polyhedral techniques in combinatorial optimization II: applications and computations
The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems
Polyhedral Techniques in Combinatorial Optimization
Combinatorial optimization problems arise in several areas ranging from management to mathematics and graph theory. Most combinatorial optimization problems are compu- tationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable development in polyhedral techniques leading to an increase in the size of several problem types that can be solved by a factor hundred. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. The purpose of this article is to give anoverview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also discuss several computational aspects and implementation issues related to the use of polyhedral methods.
Hard equality constrained integer knapsacks
We consider the following integer feasibility problem: Given positive integer numbers a0, a1,&mellip;,an with gcd(a1,&mellip;,an) = 1 and a = (a1,&mellip;,an), does there exist a vector x Ε &Zdbl;≥0n satisfying a x = a0? We prove that if the coefficients a1,&mellip;,an have a certain decomposable structure, then the Frobenius number associated with a1,&mellip;,an, i.e., the largest value of a0 for which a x = a0 does not have a nonnegative integer solution, is close to a known upper bound. In the instances we consider, we take a0 to be the Frobenius number. Furthermore, we show that the decomposable structure of a1,&mellip;,an 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 a0/ai, 1 &le i &le n. We illustrate our results by some computational examples. © 2004 INFORMS
Erratum: Hard equality constrained integer knapsacks (Mathematics of Operations Research (2004) 29:3 (724-738))
No abstract availabl
Improved approximation algorithm for k-level UFL with penalties, a simplistic view on randomizing the scaling parameter
The state of the art in approximation algorithms for facility location
problems are complicated combinations of various techniques. In particular, the
currently best 1.488-approximation algorithm for the uncapacitated facility
location (UFL) problem by Shi Li is presented as a result of a non-trivial
randomization of a certain scaling parameter in the LP-rounding algorithm by
Chudak and Shmoys combined with a primal-dual algorithm of Jain et al. In this
paper we first give a simple interpretation of this randomization process in
terms of solving an aux- iliary (factor revealing) LP. Then, armed with this
simple view point, Abstract. we exercise the randomization on a more
complicated algorithm for the k-level version of the problem with penalties in
which the planner has the option to pay a penalty instead of connecting chosen
clients, which results in an improved approximation algorithm
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