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18
ALL LINEAR AND INTEGER PROGRAMS ARE SLIM 3WAY TRANSPORTATION PROGRAMS
, 2006
"... We show that any rational convex polytope is polynomialtime representable as a 3way linesum transportation polytope of “slim” (r, c, 3) format. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. We provide a po ..."
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Cited by 20 (4 self)
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We show that any rational convex polytope is polynomialtime representable as a 3way linesum transportation polytope of “slim” (r, c, 3) format. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. We provide a polynomialtime embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs. Our construction resolves several standing problems on 3way transportation polytopes. For example, it demonstrates that, unlike the case of 2way contingency tables, the range of values an entry can attain in any slim 3way contingency table with specified 2margins can contain arbitrary gaps. Our smallest such example has format (6, 4, 3). Our construction provides a powerful automatic tool for studying concrete questions about transportation polytopes and contingency tables. For example, it automatically provides new proofs for some classical results, including a wellknown “realfeasible but integerinfeasible” (6, 4, 3)transportation polytope of M. Vlach, and bitransportation programs where any feasible bitransportation must have an arbitrarily large prescribed denominator.
Graphs of Transportation Polytopes
, 2007
"... This paper discusses properties of the graphs of 2way and 3way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3way transportation polytopes and a catalogue of nondegenerate transp ..."
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Cited by 12 (5 self)
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This paper discusses properties of the graphs of 2way and 3way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3way transportation polytopes and a catalogue of nondegenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed to discover some new results. For example, we prove that the number of vertices of an m × n transportation polytope is a multiple of the greatest common divisor of m and n.
Convex Integer Maximization via Graver Bases
, 2008
"... We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. ..."
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Cited by 12 (7 self)
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We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w1x,..., wdx are linear forms on R n, max {c(w1x,..., wdx) : Ax = b, x ∈ N n}. This method works for arbitrary input data A, b, d, w1,..., wd, c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multiway transportation problems, packing problems, and partitioning problems in variable dimension.
Nonlinear Bipartite Matching
 DISC. OPTIM
, 2008
"... We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization ..."
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Cited by 8 (3 self)
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We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization and maximization, and a randomized algorithm for optimizing arbitrary objectives.
The convex dimension of a graph
 Discrete Applied Mathematics
"... The convex dimension of a graph G = (V, E) is the smallest dimension d for which G admits an injective map f: V − → R d of its vertices into dspace, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admit ..."
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Cited by 6 (0 self)
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The convex dimension of a graph G = (V, E) is the smallest dimension d for which G admits an injective map f: V − → R d of its vertices into dspace, such that the barycenters of the images of the edges of G are in convex position. The strong convex dimension of G is the smallest d for which G admits a map as above such that the images of the vertices of G are also in convex position. In this paper we study the convex and strong convex dimensions of graphs. 1
Nonlinear optimization for matroid intersection and extensions
, 2008
"... We address optimization of nonlinear functions of the form f(Wx) , where f: R d → R is a nonlinear function, W is a d × n matrix, and feasible x are in some large finite set F of integer points in R n. Generally, such problems are intractable, so we obtain positive algorithmic results by looking a ..."
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Cited by 6 (1 self)
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We address optimization of nonlinear functions of the form f(Wx) , where f: R d → R is a nonlinear function, W is a d × n matrix, and feasible x are in some large finite set F of integer points in R n. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes of f, W and F. One of our main motivations is multiobjective discrete optimization, where f trades off the linear functions given by the rows of W. Another motivation is that we want to extend as much as possible the known results about polynomialtime linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures. We assume that the convex hull of F is welldescribed by linear inequalities (i.e., we have an efficient separation oracle). For example, the set of characteristic vectors of common bases of a pair of matroids on a common ground set satisfies this property for F. In this setting, the problem is already known to be intractable (even for a single matroid), for general f (given by a comparison oracle), for (i) d = 1