On multi-index assignment polytopes

Appa, Gautam; Magos, D.; Mourtos, Ioannis

doi:10.1016/j.laa.2005.11.009

Cited by 16 publications

(35 citation statements)

References 25 publications

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“…We will now describe two families of facet-defining inequalities that were originally described in [5]. See Appa et al [2] for the kindex case. The column intersection graph of A n , namely G(V, E), has a node for each column of A n and an edge for every pair of columns that have a +1 entry in the same row.…”

Section: The 3-index Assignment Polytopementioning

confidence: 99%

On the Complexity of Separation: The Three-Index Assignment Problem

2012

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A fundamental step in any cutting plane algorithm is separation: deciding whether a violated inequality exists within a certain class of inequalities. It is customary to express the complexity of a separation algorithm in n, the number of variables. Here, we argue that the input to a separation algorithm can be expressed in |sup(x)|, where sup(x) denotes the vector containing the positive components of x. This input measure allows one to take sparsity into account. We apply this idea to two known classes of valid inequalities for the three-index assignment problem, and we find separation algorithms with a better complexity than the ones known in literature. We also show empirically the performance of our separation algorithms. MotivationCutting plane algorithms constitute a fundamental way of solving combinatorial optimization problems. Typically, in such an approach, a specific combinatorial optimization problem is formulated as an Integer Linear Program (ILP)This paper is an improved version of an extended abstract that appeared as "Fast separation algorithms for three index assignment problems" in the proceedings of ISCO

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Section: The 3-index Assignment Polytopementioning

confidence: 99%

On the Complexity of Separation: The Three-Index Assignment Problem

2012

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“…Counting Latin rectangles is a topic broadly studied in combinatorics; some examples listed in chronological order are [16], [4], [7], [13] and [18]. Definition 1 Two m-row Latin rectangles of order n, with m < n, form an orthogonal pair (OLR) if and only if when superimposed each of the n 2 ordered pairs of values (1, 1), (1,2), ..., (n, n) appears at most once.…”

Section: Introductionmentioning

confidence: 99%

“…Also note that for m = n we have the case of orthogonal Latin squares (OLS) where each of the n 2 ordered pairs of values (1,1), (1,2), ..., (n, n) appears exactly once when the two squares are superimposed. The definition for OLR naturally extends to a set T of m-row Latin rectangles of order n, which are called mutually orthogonal Latin rectangles (MOLR), if and only if all Latin rectangles are pairwise orthogonal.…”

Section: Introductionmentioning

confidence: 99%

“…After reviewing the results of [6], along with counting the circuits associated with the completion of 2-row (single) Latin rectangles in Section 3, we present our main contribution in Section 4. In Section 5, we discuss the implications of our work regarding the polytope associated with orthogonal Latin squares and thus planar multi-dimensional assignment [1]. We conclude in Section 6 with ideas for future work.…”

Section: Introductionmentioning

confidence: 99%

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On the completability of incomplete orthogonal Latin rectangles

Appa

Euler

Kouvela

et al. 2016

Discrete Mathematics

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We address the problem of completability for 2-row orthogonal Latin rectangles (OLR2). Our approach is to identify all pairs of incomplete 2-row Latin rectangles that are not completable to an OLR2 and are minimal with respect to this property; i.e., we characterize all circuits of the independence system associated with OLR2. Since there can be no polytime algorithm generating the clutter of circuits of an arbitrary independence system, our work adds to the few independence systems for which that clutter is fully described. The result has a direct polyhedral implication; it gives rise to inequalities that are valid for the polytope associated with orthogonal Latin squares and thus planar multi-dimensional assignment. A complexity result is also at hand: completing a set of (n − 1) incomplete M OLR2 is N P-complete.

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“…The k-index assignment polytopeThe ideas presented in the previous section are also applicable to the axial assignment problem comprising more than 3 sets, see Appa et al[2]. The k-index axial assignment problem can be defined using k disjoint n-sets M 1 , .…”

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confidence: 99%

Fast separation for the three-index assignment problem

2016

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A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x, denoted as supp(x), which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x, we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms This paper is an improved version of an extended abstract that appeared as "Fast separation algorithms for three index assignment

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On multi-index assignment polytopes

Cited by 16 publications

References 25 publications

On the Complexity of Separation: The Three-Index Assignment Problem

On the Complexity of Separation: The Three-Index Assignment Problem

On the completability of incomplete orthogonal Latin rectangles

Fast separation for the three-index assignment problem

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