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12
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 122 (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.
Orbital branching
 in Proceedings of IPCO XII
, 2007
"... Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an ar ..."
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Cited by 21 (3 self)
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Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint. Many important families of integer programs are structured such that small instances from the family are embedded in larger instances. This structural information can be exploited to define a group of strong constraints on which to base the orbital branching disjunction. The symmetric nature of the problems is further exploited by enumerating nonisomorphic solutions to instances of the small family and using these solutions to create a collection of typically easytosolve integer programs. The solution of each integer program in the collection is equivalent to solving the original large instance. The effectiveness of this methodology is demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.
Fundamental Domains for Integer Programs with Symmetries
"... We define a fundamental domain for a linear programming relaxation of a combinatorial integer program which is symmetric under a group action. We then describe a straightforward way to construct fundamental domains defined by the maximization of a linear function. The computation of this fundament ..."
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Cited by 11 (0 self)
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We define a fundamental domain for a linear programming relaxation of a combinatorial integer program which is symmetric under a group action. We then describe a straightforward way to construct fundamental domains defined by the maximization of a linear function. The computation of this fundamental domain is at worst polynomial in the size of the group; however, for the symmetric group, which has exponential size, we show how to compute separation in polynomial time (in the size of the integer program). Fundamental domains are a simple and flexible approach to reducing the computation difficulties that often arise in integer programs with symmetries. Their construction is closely related to the constructions of orbitopes (by Kaibel and Pfetsch), but more general and easier to analyze, although the computations required may be somewhat more complex.
Hard multidimensional multiple choice knapsack problems, an empirical study
 COMPUTERS OPERATIONS RESEARCH
, 2009
"... Recent advances in algorithms for the multidimensional multiple choice knapsack problems have enabled us to solve rather large problem instances. However, these algorithms are evaluated with very limited benchmark instances. In this study, we propose new methods to systematically generate comprehen ..."
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Cited by 6 (0 self)
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Recent advances in algorithms for the multidimensional multiple choice knapsack problems have enabled us to solve rather large problem instances. However, these algorithms are evaluated with very limited benchmark instances. In this study, we propose new methods to systematically generate comprehensive benchmark instances. Some instances with special correlation properties between parameters are found to be several orders of magnitude harder than those currently used for benchmarking the algorithms. Experiments on an existing exact algorithm and two generic solvers show that instances whose weights are uncorrelated with the profits are easier compared with weakly or strongly correlated cases. Instances with classes containing similar set of profits for items and with weights strongly correlated to the profits are the hardest among all instance groups investigated. These hard instances deserve further study and understanding their properties may shed light to better algorithms.
A supernodal formulation of vertex colouring with applications in course timetabling
, 2009
"... For many problems in Scheduling and Timetabling the choice of an mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new formulation using ..."
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Cited by 6 (2 self)
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For many problems in Scheduling and Timetabling the choice of an mathematical programming formulation is determined by the formulation of the graph colouring component. This paper briefly surveys seven known integer programming formulations of vertex colouring and introduces a new formulation using “supernodes”. In the definition of George and McIntyre [SIAM J. Numer. Anal. 15 (1978), no. 1, 90–112], “supernode” is a complete subgraph, where each two vertices have the same neighbourhood outside of the subgraph. Seen another way, the algorithm for obtaining the best possible partition of an arbitrary graph into supernodes, which we give and show to be polynomialtime, makes it possible to use any formulation of vertex multicolouring to encode vertex colouring. The power of this approach is shown on the benchmark problem of Udine Course Timetabling. Results from empirical tests on DIMACS colouring instances, in addition to instances from other timetabling applications, are also provided and discussed.
A BranchandCut Algorithm based on Semidefinite Programming for the Minimum kPartition Problem
, 2007
"... The minimum kpartition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branchandcut ..."
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Cited by 4 (1 self)
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The minimum kpartition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branchandcut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming (SDP) with polyhedral results; and the iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branchandbound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing the best exact approach to date for k ≥ 3.
A Local Dominance Procedure for MixedInteger Linear Programming
, 2007
"... Among the hardest MixedInteger Linear Programming (MILP) problems, the ones that exhibit a symmetric nature are particularly important in practice, as they arise in both theoretical and practical combinatorial optimization problems. A theoretical concept that generalizes the notion of symmetry is t ..."
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Among the hardest MixedInteger Linear Programming (MILP) problems, the ones that exhibit a symmetric nature are particularly important in practice, as they arise in both theoretical and practical combinatorial optimization problems. A theoretical concept that generalizes the notion of symmetry is that of dominance. This concept, although known since a long time, is typically not used in generalpurpose MILP codes, due to the intrinsic difficulties arising when using the classical definitions in a completely general context. In this paper we study a generalpurpose dominance procedure proposed in the 80’s by Fischetti and Toth, that overcomes some of the main drawbacks of the classical dominance criteria. Both theoretical and practical issues concerning this procedure are considered, and important improvements are proposed. Computational results on a testbed made of hard (single and multiple) knapsack problems are reported, showing that the proposed method can lead to considerable speedup when embedded in a generalpurpose MILP solver.
Branchcutandpropagate for the maximum kcolorable subgraph problem with symmetry
, 2011
"... Given an undirected graph and a positive integer k, the maximum kcolorable subgraph problem consists of selecting a kcolorable induced subgraph of maximum cardinality. The natural integer programming formulation for this problem exhibits two kinds of symmetry: arbitrarily permuting the color clas ..."
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Cited by 1 (1 self)
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Given an undirected graph and a positive integer k, the maximum kcolorable subgraph problem consists of selecting a kcolorable induced subgraph of maximum cardinality. The natural integer programming formulation for this problem exhibits two kinds of symmetry: arbitrarily permuting the color classes and/or applying a nontrivial graph automorphism gives equivalent solutions. It is well known that such symmetries have negative effects on the performance of constraint/integer programming solvers. We investigate the integration of a branchandcut algorithm for solving the maximum kcolorable subgraph problem with constraint propagation techniques to handle the symmetry arising from the graph. The latter symmetry is handled by (nonlinear) lexicographic ordering constraints and linearizations thereof. In experiments, we evaluate the influence of several components of our algorithm on the performance, including the different symmetry handling methods. We show that several components are crucial for an efficient algorithm; in particular, the handling of graph symmetries yields a significant performance speedup.
Mathematisches Forschungsinstitut Oberwolfach Moduli Spaces in Algebraic Geometry
"... Abstract. Invariants of topological spaces of dimension three play a major role in many areas, in particular . . . Mathematics Subject Classification ..."
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Abstract. Invariants of topological spaces of dimension three play a major role in many areas, in particular . . . Mathematics Subject Classification
Symmetry in Scheduling Problems
, 2010
"... The presence of symmetry is common in certain types of scheduling problems. Symmetry can occur when one is scheduling a collection of jobs on multiple identical machines, or if one is determining production schedules for identical machines. General symmetrybreaking methods can be strengthened by ta ..."
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The presence of symmetry is common in certain types of scheduling problems. Symmetry can occur when one is scheduling a collection of jobs on multiple identical machines, or if one is determining production schedules for identical machines. General symmetrybreaking methods can be strengthened by taking advantage of the special structure of the symmetry group in scheduling problems. In this paper, we examine the effectiveness of symmetrybreaking methods for scheduling problems. We also present a modified version of orbital branching, a powerful symmetrybreaking procedure, and discuss when it should and should not be used in practice. Using operating room and power generator scheduling problems as sample problems, we will provide computational results comparing different methods of symmetry breaking.