The Wheels of the Orthogonal Latin Squares Polytope: Classification and Valid Inequalities

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A wheel in a graph G(V,E) is an induced subgraph consisting of an odd hole and an additional node connected to all nodes of the hole. In this paper, we study the wheels of the intersection graph of the Orthogonal Latin Squares polytope (PI). Our work builds on structural properties of wheels which are used to categorise them into a number of collectively exhaustive classes. These classes give rise to families of inequalities that are valid for PI and facet-defining for its set-packing relaxation. The classification introduced allows us to establish the cardinality of the whole wheel class and determine the range of the coefficients of any variable included in a lifted wheel inequality. Finally, based on this classification, a constant-time recognition algorithm for wheel-inducing circulant matrices, is introduced.

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... The current work adds to the (knowledge of this) description for the OLS polytope by identifying families of facet-defining inequalities based on graphs called wheels (see next section for definitions). Our work builds on the results presented in [1], where all the wheels associated with the OLS polytope are described and categorised into a number of collectively exhaustive classes. For all classes of wheels of size 5, we introduce the corresponding families of maximally lifted wheel (MLW) inequalities and show that they are facet-defining for P I . ...
... In the current work, we study the families of MLW inequalities induced by wheels of size 5 (p = 2). In [1], it is shown that each such wheel belongs to one of six mutually exclusive classes and that the associated MLW inequalities include variables multiplied by 1, with the exception of the variable indexed by the hub which is multiplied by 2 (see [1, Proposition 6.3] for a more general result). Hence, the general form of the inequalities considered hereafter is ...
... For all these wheels, c = (n, n, n, n). In the first column, the class identification number, as defined in [1], is displayed. In the second and third column, the nodes (tuples) of H (c) and V (W (c)) are illustrated. ...
Orthogonal Latin squares (OLS) are fundamental combinatorial objects with important theoretical properties and interesting applications. OLS can be represented by integer points satisfying a certain system of equalities. The convex hull of these points is the OLS polytope. This paper adds to the description of the OLS polytope by providing non-trivial facets arising from wheels. Specifically, for each wheel of size five, we identify the variables that can be added to the induced inequality, thus obtaining all distinct families of maximally lifted wheel inequalities. Each of these families induces facets of the OLS polytope which can be efficiently separated in polynomial time.
... Again, while we do not discuss those here, the work put forth provides a solid foundation for further reading into the usage of this work. A great deal of further discussion is provided by Appa, Magos, Mourtos and Janssen [3], who give a general description of many properties of the polytope, and by Appa, Magos, and Mourtos [1], who exploit further structures within the intersection graph. ...
A significant problem in finite optimization is the assignment problem. In essence, the assignment problem asks for a minimum weight assignment of elements of distinct sets to one another. Assignment problems have many applications. We give descriptions of several of the most com-mon assignment problems. We relate the planar assignment problems to Latin squares and orthogonal Latin squares and discuss many known results on the existence of Latin squares and orthogonal Latin squares as the feasible sets for these assignment problems. Finally, we give a mathematical formulation for planar assignment problems and look in some depth at the intersection graph induced by the four index planar assignment problem. This intersection graph has application to the structure of the solution polytope; however, we do not directly explore this application.
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A graph G with a two-node cutset decomposes into two pieces. A technique to describe the stable set polytope for G based on stable set polytopes associated with the pieces is studied. This gives a way to characterize this polytope for classes of graphs that can be recursively decomposed. This also gives a procedure to describe new facets of this polytope. A compact system for the stable set problem in series-parallel graphs is derived. This technique is also applied to characterize facet- defining inequalities for graphs with no K 5 ∖e minor. The stable set problem is polynomially solvable for this class of graphs. Compositions of h-perfect graphs are also studied.
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A wheel in a graph G(V, E) is an induced subgraph consisting of an odd hole and an additional node connected to all nodes of the hole. In this paper, we study the wheels of the column intersection graph of the OLS polytope (P I). These structures induce valid inequalities for this polytope, which are facet defining for its set packing relaxation. Our work builds on simple structural properties of wheels which are used to categorise them into a number of collectively exhaustive classes. Each such class gives rise to a set of valid inequalities for P I . Moreover, this classification allows us to estimate the cardinality of the whole wheel class as well as to derive a recognition algorithm for the circulant matrices corresponding to wheels of a particular type. In a forthcoming paper, we show for some of the wheel classes presented here that they give rise to facet-defining inequalities for P I .
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Since 1782, when Euler addressed the question of existence of a pair of Orthogonal Latin Squares (OLS) by stating his famous conjecture ([8, 9, 13]), these structures have remained an active area of research due to their theoretical properties as well as their applications in a variety of fields. In the current work we consider the polyhedral aspects of OLS. In particular we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. For two of these classes we show that the related inequalities have Chvátal rank two and both are facet defining. For each such class, we give a separation algorithm of the lowest possible complexity, i.e. linear in the number of variables.
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. A stable set in a graph G is a set of pairwise nonadjacent vertices. The problem of finding a maximum weight stable set is one of the most basic ℕℙ-hard problems. An important approach to this problem is to formulate it as the problem of optimizing a linear function over the convex hull STAB(G) of incidence vectors of stable sets. Since it is impossible (unless ℕℙ=coℕℙ) to obtain a “concise” characterization of STAB(G) as the solution set of a system of linear inequalities, it is a more realistic goal to find large classes of valid inequalities with the property that the corresponding separation problem (given a point x *, find, if possible, an inequality in the class that x * violates) is efficiently solvable.�Some known large classes of separable inequalities are the trivial, edge, cycle and wheel inequalities. In this paper, we give a polynomial time separation algorithm for the (t)-antiweb inequalities of Trotter. We then introduce an even larger class (in fact, a sequence of classes) of valid inequalities, called (t)-antiweb-s-wheel inequalities. This class is a common generalization of the (t)-antiweb inequalities and the wheel inequalities. We also give efficient separation algorithms for them.
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We introduce new classes of valid inequalities, called wheel inequalities, for the stable set polytope PG of a graph G. Each "wheel configuration" gives rise to two such inequalities. The simplest wheel configuration is an "odd" subdivision W of a wheel, and for these we give necessary and sufficient conditions for the wheel inequality to be facet-inducing for PW . Generalizations arise by allowing subdivision paths to intersect, and by replacing the "hub" of the wheel by a clique. The separation problem for these inequalities can be solved in polynomial time. 1 Introduction Let G = (V; E) be a simple connected graph with jV j = n 2 and jEj = m. A subset of V is called a stable set if it does not contain adjacent vertices of G. Let N be a stable set. The incidence vector of N is x 2 f0; 1g V such that x v = 1 if and only if v 2 N . The stable set polytope of G, denoted by PG , is the convex hull of incidence vectors of stable sets of G. Some well-known valid inequalities for PG ...
In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form . This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.
By identifying all latin squares of order n with certain n2-element subsets of an n3-element ground set En a clutter Bn is obtained, which induces an independence system (En, In) in a natural way. Starting from Ryser's conditions for the completion of latin rectangles (cf. Mirsky [15]) we present special classes of circuits of (En, In) and extend Ryser's conditions slightly.Latin squares of order n correspond to the solutions of the planar 3-dimensional assignment problem and, in view of its solution via linear programming techniques, we present some first classes of facet-defining inequalities for P(In) resp. P(Bn), the convex hull of all those 0–1 vectors, which correspond to members of In resp. Bn.
Geometric algorithms and combinatorial optimization / Martin Grötschel ; László Lovász ; Alexander Schrijver. - Berlin u. a. : Springer, 1988. - XII, 362 S. - (Algorithms and combinatorics ; 2).
Wheel inequalities for stable set polytopes Antiweb-wheel inequalities and their separation problems over the stable set polytopes On Latin squares and the facial structure of related polytopes On the facial structure of the set packing polyhedra
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  • A R Mahjoub
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  • R Cunningham
  • R E Euler
  • R Burkard
  • C F Grommes
  • G L Laywine
  • Mullen
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