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SOLVING LINEAR ORDERING PROBLEMS WITH A COMBINED INTERIOR POINT/SIMPLEX CUTTING PLANE ALGORITHM
"... We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primal-dual interior point method to solve the first few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm ..."
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Cited by 36 (11 self)
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We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primal-dual interior point method to solve the first few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm with one that uses only an interior point method and with one that uses only a simplex method. We solve integer programming problems with as many as 31125 binary variables. Computational results show that the combined approach can dramatically outperform the other two methods.
Determining the Automorphism Group of the Linear Ordering Polytope
- Discrete Applied Mathematics
, 1999
"... In this paper we explore the combinatorial automorphism group of the linear ordering polytope P n LO for each n ? 1. We establish that this group is isomorphic to Z 2 \Theta Sym(n + 1) if n ? 2 (and to Z 2 if n = 2). Doing so, we provide a simple and unified interpretation of all the automorphisms ..."
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Cited by 8 (4 self)
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In this paper we explore the combinatorial automorphism group of the linear ordering polytope P n LO for each n ? 1. We establish that this group is isomorphic to Z 2 \Theta Sym(n + 1) if n ? 2 (and to Z 2 if n = 2). Doing so, we provide a simple and unified interpretation of all the automorphisms. Key words: Linear ordering polytope, automorphism group, facets 1 Introduction The linear ordering polytope is a familiar object from polyhedral combinatorics. It is defined as the convex hull of the 0/1--vectors encoding linear orders (or total orders) on a given base set. Exploiting results on the facial structure of this family of polytopes and using advanced techniques in linear programming, efficient algorithms could be designed to solve real-world instances of some hard combinatorial optimization problems. For example, the triangulation problem for input-output tables can be formulated as a linear program on the linear ordering polytope. This problem asks, given a matrix of n \The...
Algorithmic Aspects of Using Small Instance Relaxations in Parallel Branch-and-Cut
- IEEE Trans. El. Dev
, 1998
"... Essential for the success of branch-and-cut algorithms for solving combinatorial optimization problems are the availability of reasonable tight relaxations and effective routines for solving the associated separation problems. In this paper we introduce the concept of small instance relaxations whic ..."
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Essential for the success of branch-and-cut algorithms for solving combinatorial optimization problems are the availability of reasonable tight relaxations and effective routines for solving the associated separation problems. In this paper we introduce the concept of small instance relaxations which can be particularly useful for problems with symmetric structure. Small instance relaxations base on the facets of polytopes associated with small instances of the combinatorial optimization problem to be solved and can be generated automatically by facet enumeration. For a certain class of symmetric problems, we describe a general approach to the separation problem. Algorithmic aspects of using small instance relaxations effectively (parallel separation, facet selection, cutting plane selection) are discussed in detail. Extensive computational results are presented for the linear ordering problem and a certain betweenness problem. 1 Introduction During the last years, branch-and-cut algo...
Revised GRASP with Path-Relinking for the Linear Ordering Problem
, 2009
"... The linear ordering problem (LOP) is an N P-hard combinatorial optimization problem with a wide range of applications in economics, archaeology, the social sciences, scheduling, and biology. It has, however, drawn little attention compared to other closely related problems such as the quadratic as ..."
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The linear ordering problem (LOP) is an N P-hard combinatorial optimization problem with a wide range of applications in economics, archaeology, the social sciences, scheduling, and biology. It has, however, drawn little attention compared to other closely related problems such as the quadratic assignment problem and the traveling salesman problem. Due to its computational complexity, it is essential in practice to develop solution approaches to rapidly search for solution of high-quality. In this paper we propose a new algorithm based on a greedy randomized adaptive search procedure (GRASP) to efficiently solve the LOP. The algorithm is integrated with a Path-Relinking (PR) procedure and a new local search scheme. We tested our implementation on the set of 49 real-world instances of input-output tables (LOLIB instances) proposed in Reinelt (Dec. 2002). In addition, we tested a set of 30 large randomly-generated instances proposed in Mitchell (1997). Most of the LOLIB instances were solved to optimality within 0.87 seconds on average. The average gap for the randomly-generated instances was 0.0173 % with an average running time of 21.98 seconds. The results indicate the efficiency and high-quality of the proposed heuristic procedure.
