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58
Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems
"... A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞ ..."
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Cited by 61 (1 self)
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A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of nonzero coefficients in a row of U, and provide a polynomialtime approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first nontrivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomialtime approximation algorithms to match this bound. This improves on the best previous mapproximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentiallymany) inequalities added. Our contribution here is to describe an appropriate
SCIP: solving constraint integer programs
, 2009
"... Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), wh ..."
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Cited by 53 (0 self)
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Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and noncommercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly nonlinear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current stateoftheart techniques for proving the validity of properties on circuits containing arithmetic.
Implementing the DantzigFulkersonJohnson Algorithm for Large Traveling Salesman Problems
, 2003
"... Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et ..."
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Cited by 36 (6 self)
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Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cuttingplane method in largescale applications.
The 01 Knapsack Problem With A Single Continuous Variable
 MATHEMATICAL PROGRAMMING
, 1997
"... Constraints arising in practice often contain many 01 variables and one or a small number of continuous variables. Existing knapsack separation routines cannot be used on such constraints. Here we study such constraint sets, and derive valid inequalities that can be used as cuts for such sets, as w ..."
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Cited by 34 (7 self)
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Constraints arising in practice often contain many 01 variables and one or a small number of continuous variables. Existing knapsack separation routines cannot be used on such constraints. Here we study such constraint sets, and derive valid inequalities that can be used as cuts for such sets, as well for more general mixed 01 constraints. Specifically we investigate the polyhedral structure of the knapsack problem with a single continuous variable, called the continuous 01 knapsack problem. First di#erent classes of facetdefining inequalities are derived based on projection and lifting. The order of lifting, particularly of the continuous variable, plays an important role. Secondly we show that the flow cover inequalities derived for the single node flow set, consisting of arc flows into and out of a single node with binary variable lower and upper bounds on each arc, can be obtained from valid inequalities for the continuous 01 knapsack problem. Thus the separation heuristic we derive for continuous knapsack sets can also be used to derive cuts for more general mixed 01 constraints. Initial computational results on a variety of problems are presented.
Lifted Cover Inequalities for 01 Integer Programs: Computation
 INFORMS Journal on Computing
, 1998
"... We investigate the algorithmic and implementation issues related to the e ective and e cient use of lifted cover inequalities and lifted GUB cover inequalities in a branchandcut algorithm for 01 integer programming. We have tried various strategies on several test problems and we identify the bes ..."
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Cited by 33 (5 self)
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We investigate the algorithmic and implementation issues related to the e ective and e cient use of lifted cover inequalities and lifted GUB cover inequalities in a branchandcut algorithm for 01 integer programming. We have tried various strategies on several test problems and we identify the best ones for use in practice.
On the facets of the mixed–integer knapsack polyhedron
 MATH. PROGRAM., SER. B 98: 145–175 (2003)
, 2003
"... We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities cont ..."
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Cited by 28 (12 self)
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We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.
TSP cuts which do not conform to the template paradigm
 IN COMPUTATIONAL COMBINATORIAL OPTIMIZATION
, 2001
"... The first computer implementation of the DantzigFulkersonJohnson cuttingplane 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 c ..."
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Cited by 25 (1 self)
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The first computer implementation of the DantzigFulkersonJohnson cuttingplane 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
Sequence Independent Lifting in Mixed Integer Programming
 Journal of Combinatorial Optimization
, 1998
"... We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive ..."
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Cited by 25 (2 self)
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We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coefficient for each variable. To relieve this computational burden, we study sequence independent lifting, which only involves the solution of one optimization problem. We show that if a certain lifting function is superadditive, then the lifting coefficients are independent of the lifting sequence. We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting. We apply these results to strengthen Balas' lifting theorem for cover inequalities and to produce lifted flow cover inequalities for a single node flow problem. December 1995 Revised July 1997 Revised Janua...
Integerprogramming software systems
 Annals of Operations Research
, 1995
"... Abstract. Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when ..."
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Cited by 24 (0 self)
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Abstract. Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer–programming software systems and a perspective on features we may expect to see in these systems in the near future. 1.
SCIP  a framework to integrate constraint and mixed integer programming
, 2004
"... Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem int ..."
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Cited by 24 (1 self)
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Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today’s stateoftheart algorithms of both fields have several strategies in common, in particular the branchandbound process to recursively divide the problem into smaller subproblems. On the other hand, the main techniques to process each subproblem are different, and it was observed that they have complementary strengths. We present the programming framework Scip that integrates techniques from both fields in order to exploit the strengths of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, Scip does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, indepth control and high performance.