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17
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 99 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
Compressed polytopes and statistical disclosure limitation. arXiv:math.CO/0412535. 19 Akimichi Takemura and Satoshi Aoki. Some characterizations of minimal Markov basis for sampling from discrete conditional distributions
 Ann. Inst. Statist. Math
"... We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1polytope. As an application, we characterize those graphs whose cut polytopes are compressed and discuss ..."
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Cited by 17 (1 self)
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We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1polytope. As an application, we characterize those graphs whose cut polytopes are compressed and discuss consequences for studying linear programming relaxations in statistical disclosure limitation. 1
On the Embeddability of Weighted Graphs in Euclidean Spaces
, 1998
"... Given an incomplete edgeweighted graph, G = (V; E; !), G is said to be embeddable in ! r , or rembeddable, if the vertices of G can be mapped to points in ! r such that every two adjacent vertices v i , v j of G are mapped to points x i , x j 2 ! r whose Euclidean distance is equal to t ..."
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Cited by 12 (1 self)
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Given an incomplete edgeweighted graph, G = (V; E; !), G is said to be embeddable in ! r , or rembeddable, if the vertices of G can be mapped to points in ! r such that every two adjacent vertices v i , v j of G are mapped to points x i , x j 2 ! r whose Euclidean distance is equal to the weight of the edge (v i ; v j ). Barvinok [3] proved that if G is rembeddable for some r, then it is r embeddable where r = b( p 8jEj + 1 \Gamma 1)=2c. In this paper we provide a constructive proof of this result by presenting an algorithm to construct such an r embedding. 1 Introduction Let G = (V; E;!) be an incomplete undirected edgeweighted graph with vertex set V = fv 1 ; v 2 ; : : : ; v n g, edge set E ae V \Theta V and a nonnegative weight Email aalfakih@orion.math.uwaterloo.ca y Research supported by Natural Sciences Engineering Research Council Canada. Email henry@orion.math.uwaterloo.ca. ! ij for each (v i ; v j ) 2 E. G is said to be rembeddable if th...
Compression functions of uniform embeddings of groups into Hilbert and Banach spaces
 J. Reine Angew. Math
"... We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the Ecompression of these groups coincides with their Hilbert space compression. Moreover, the groups that we c ..."
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Cited by 11 (0 self)
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We construct finitely generated groups with arbitrary prescribed Hilbert space compression α ∈ [0, 1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the Ecompression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. 1
Tighter linear and semidefinite relaxations for maxcut based on the LovászSchrijver liftandproject procedure
 SIAM Journal on Optimization
"... Abstract. We study how the liftandproject method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5minor. ..."
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Cited by 7 (4 self)
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Abstract. We study how the liftandproject method introduced by Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] applies to the cut polytope. We show that the cut polytope of a graph can be found in k iterations if there exist k edges whose contraction produces a graph with no K5minor. Therefore, for a graph G with n ≥ 4 nodes with stability number α(G), n − 4 iterations suffice instead of the m (number of edges) iterations required in general and, under some assumption, n − α(G) − 3 iterations suffice. The exact number of needed iterations is determined for small n ≤ 7 by a detailed analysis of the new relaxations. If positive semidefiniteness is added to the construction, then one finds in one iteration a relaxation of the cut polytope which is tighter than its basic semidefinite relaxation and than another one introduced recently by Anjos and Wolkowicz [Discrete Appl. Math., to appear]. We also show how the Lovász–Schrijver relaxations for the stable set polytope of G can be strengthened using the corresponding relaxations for the cut polytope of the graph G ∇ obtained from G by adding a node adjacent to all nodes of G.
On Equicut Graphs
 QUARTERLY JOURNAL OF MATHEMATICS OXFORD
, 2000
"... The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equic ..."
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Cited by 7 (3 self)
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The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equicut graph, that is, admits an `1 embedding f that for any 1 i n f satis es x2V f(x) i 2 fdv=2e; bv=2cg. Basic properties of equicut graphs are investigated. A construction of equicut graphs from `1graphs via a natural doubling construction is given. It generalizes several wellknown constructions of polytopes and distanceregular graphs. Finally, large families of examples, mostly related to polytopes and distanceregular graphs, are presented.
On the Solitaire Cone and Its Relationship to MultiCommodity Flows
 PREPRINT CAMS 142 ECOLE DES HAUTES ETUDES EN SCIENCES SOCIALES
, 2001
"... The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities ov ..."
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Cited by 7 (3 self)
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The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NPcompleteness result; 3. a method of generating large classes of facets; 4. a complete characterization of 01 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards; 7. a complete characterization of the adjacency of extreme rays, diameter, number of 2faces and edge connectivity for rectangular toric boards.
Solitaire Cones
 Discrete Applied Mathematics
, 1996
"... The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities o ..."
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Cited by 5 (0 self)
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The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give: 1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone; 2. a related NPcompleteness result; 3. a method of generating large classes of facets; 4. a complete characterization of 01 facets; 5. exponential upper and lower bounds (in the dimension) on the number of facets; 6. results on the number of facets, incidence, adjacency and ...
On the Face Lattice of the Metric Polytope
"... In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailormade algorithms using their rich combinatorial features can exhib ..."
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Cited by 4 (1 self)
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In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, tailormade algorithms using their rich combinatorial features can exhibit strong performances. The main engine of these combinatorial algorithms is the use of the large symmetry group of combinatorial polytopes. Specifically we consider a polytope with applications to the wellknown maxcut and multicommodity flow problems: the metric polytope mn on n nodes. We prove that for n 9 the faces of codimension 3 of the metric polytope are partitioned into 15 orbits of its symmetry group. For n 8, we describe additional upper layers of the face lattice of mn . In particular, using the list of orbits of high dimensional faces of m8 , we prove that the description of m8 given in [9] is complete with 1 550 825 000 vertices and that the LaurentPoljak conjecture [14] holds for n 8. Many vertices of m9 are computed and additional results on the structure of the metric polytope are presented...
HIERARCHICAL MODELS, MARGINAL POLYTOPES, AND LINEAR CODES
"... Abstract. In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with ..."
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Cited by 3 (3 self)
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Abstract. In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them. 1.