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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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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...
Low-Dimensional Linear Ordering Polytopes
- SOLVING LINEAR ORDERING PROBLEMS 15
, 1997
"... In this paper we discuss some new results on the structure of linear ordering polytopes. We give a linear description of the linear ordering polytope for n = 8 which we believe to be complete. Furthermore, we address the question of comparing facet-defining inequalities with respect to various crite ..."
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In this paper we discuss some new results on the structure of linear ordering polytopes. We give a linear description of the linear ordering polytope for n = 8 which we believe to be complete. Furthermore, we address the question of comparing facet-defining inequalities with respect to various criteria which may have impact on the development of branch-and-cut algorithms for the linear ordering problem. 1 Introduction Let D n = (V n ; A n ) be the complete digraph on n nodes. A tournament T is a subset of arcs of D n such that for every pair i; j 2 V n either ij 2 T or ji 2 T but not both. Having arc weights c a , a 2 A n , associated with every arc, the linear ordering problem consists of determining an acyclic tournament in D n of maximum weight. It is easy to see that an acyclic tournament corresponds to a linear ordering of the nodes of D n and vice versa. The linear ordering problem has various applications. For a survey as well as reports on experience with branch-and-cut algori...
Facets of the Linear Ordering Polytope: a unification for the fence family through weighted graphs
, 2005
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How to recycle your facets
- DISCRETE OPTIMIZATION
"... We show how to transform any inequality defining a facet of some 0/1-polytope into an inequality defining a facet of the acyclic subgraph polytope. While this facet-recycling procedure can potentially be used to construct ‘nasty’ facets, it can also be used to better understand and extend the polyh ..."
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We show how to transform any inequality defining a facet of some 0/1-polytope into an inequality defining a facet of the acyclic subgraph polytope. While this facet-recycling procedure can potentially be used to construct ‘nasty’ facets, it can also be used to better understand and extend the polyhedral theory of the acyclic subgraph and linear ordering problems.
{0, 1/2}-CUTS AND THE LINEAR ORDERING PROBLEM: SURFACES THAT DEFINE FACETS
"... We find new facet-defining inequalities for the linear ordering polytope generalizing the well-known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1/2}-cuts produces triangulations of the Möbius band and of the corr ..."
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We find new facet-defining inequalities for the linear ordering polytope generalizing the well-known Möbius ladder inequalities. Our starting point is to observe that the natural derivation of the Möbius ladder inequalities as {0, 1/2}-cuts produces triangulations of the Möbius band and of the corresponding (closed) surface, the projective plane. In that sense, Möbius ladder inequalities have the same ‘shape’ as the projective plane. Inspired by the classification of surfaces, a classic result in topology, we prove that a surface has facet-defining {0, 1/2}-cuts of the same ‘shape ’ if and only if it is nonorientable.
LAGRANGIAN RELAXATION AND PEGGING TEST FOR LINEAR ORDERING PROBLEMS
, 2011
"... We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian ..."
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We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian relaxation of a binary integer linear programming formulation of the problem. Since the number of the constraints is proportional to the third power of the number of items and grows rapidly, we propose a modified subgradient method that temporarily ignores a large part of the constraints and gradually adds constraints whose Lagrangian multipliers are likely to be positive at an optimal multiplier vector. We also propose an improvement on the ordinary pegging test by using the problem structure.
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.
Solution of the Linear Ordering Problem (NP=P)
, 2003
"... We consider the following problem n ∑ n∑ max i=1,i=j j=1 s. t. 0 � xij � 1, cijxij xij + xji = 1, 0 � xij + xjk − xik � 1, i = j, i = k, j = k, i, j, k = 1,..., n. We denote the corresponding polytope by Bn. The polytope Bn has integer vertices corresponding to feasible solutions of the linear or ..."
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We consider the following problem n ∑ n∑ max i=1,i=j j=1 s. t. 0 � xij � 1, cijxij xij + xji = 1, 0 � xij + xjk − xik � 1, i = j, i = k, j = k, i, j, k = 1,..., n. We denote the corresponding polytope by Bn. The polytope Bn has integer vertices corresponding to feasible solutions of the linear ordering problem as well as non-integer vertices. We denote the polytope of integer vertices as Pn. Let us give an example of non-integer vertex in Bn and describe an exact facet cut. In what follows we will interested only in generating exact facet cuts. Fig. 1 shows a graph interpretation of a non-integer vertex [1], i1 i2 im j1 j2
