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For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by , is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group of an LP. This method computes when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints. However, when the feasible set of the LP satisfies equality constraints this method finds only a subgroup of and can miss symmetries. We develop a method for finding the symmetry group of a feasible LP whose feasible set satisfies equality constraints. We apply this method to find and exploit the previously unexploited symmetries of an orthogonal array defining integer linear program (ILP) within the branch-and-bound (B&B) with isomorphism pruning algorithm (Margot, 2007). Our method reduced the running time for finding all OD-equivalence classes of OA and OA by factors of and compared to the fastest known method (Bulutoglu and Ryan, 2018). These were the two bottleneck cases that could not have been solved until the B&B with isomorphism pruning algorithm was applied. Another key finding of this paper is that converting inequalities to equalities by introducing slack variables and exploiting the symmetry group of the resulting ILP’s LP relaxation within the B&B with isomorphism pruning algorithm can reduce the computation time by several orders of magnitude when enumerating a set of all non-isomorphic solutions of an ILP. 相似文献
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Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that does not intersect a deterministic hull. For general with , no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that does not intersect this random hull for any . As a corollary, we will give a new proof of Werner's result on conformal restriction measures. 相似文献
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Moosa Gabeleh 《Indagationes Mathematicae》2019,30(1):227-239
Given and two nonempty subsets of a metric space, a mapping is noncyclic provided that and . A point is called a best proximity pair for the noncyclic mapping if and . In this article, we survey the convergence of Picard’s iteration to a best proximity pair for noncyclic contractions using a projection algorithm in uniformly convex Banach spaces, where the initial point is in the proximal set of . We also provide some sufficient conditions to ensure the existence of a common best proximity pairs for a pair of noncyclic mappings. 相似文献
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《Indagationes Mathematicae》2022,33(2):421-439
We prove the irreducibility of integer polynomials whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae and , with ratio of the distances to these points depending on the canonical decomposition of and . In particular, we obtain irreducibility criteria for the case where and have few prime factors, and is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting. 相似文献
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In the b-coloring problem, we aim to assign colors from a set to the vertices of a graph in such a way that adjacent vertices do not receive the same color, and for every we have a -colored vertex in such that every color in is assigned to at least one of ’s neighbors. It has been shown that b-coloring is NP-complete, so we propose in this article an approach for the problem under integer programming techniques. To this end, we give an integer programming formulation and study the associated polytope. We provide several families of valid inequalities, and analyze facetness conditions for them. Finally, we show computational evidence suggesting that the given inequalities may be useful in a branch-and-cut environment. 相似文献
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Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension of is at most 36, although the example with largest projective dimension he constructed has . Based on computational evidence, it had been conjectured that . In the present paper we prove this conjectured sharp bound. 相似文献
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《Discrete Mathematics》2020,343(1):111640
For any graph with , a shortest path reconfiguration graph can be formed with respect to and ; we denote such a graph as . The vertex set of is the set of all shortest paths from to in while two vertices in are adjacent if and only if the vertex sets of the paths that represent and differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles. 相似文献
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A -list assignment of a graph is a mapping that assigns to each vertex a list of at least colors satisfying for each edge . A graph is -choosable if there exists an -coloring of for every -list assignment . This concept is also known as choosability with separation. In this paper, we prove that any planar graph is -choosable if any -cycle is not adjacent to a -cycle, where and . 相似文献
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Hemanshu Kaul Jeffrey A. Mudrock Michael J. Pelsmajer Benjamin Reiniger 《Discrete Mathematics》2019,342(8):2371-2383
In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A -assignment for a graph specifies a list of available colors for each vertex of . An -coloring assigns a color to each vertex from its list . For each color , let be the number of vertices whose list contains . A proportional-coloring of is a proper -coloring in which each color is used or times. A graph is proportionally-choosable if a proportional -coloring of exists whenever is a -assignment for . We show that if a graph is proportionally -choosable, then every subgraph of is also proportionally -choosable and also is proportionally -choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph is proportionally -choosable whenever , and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques. 相似文献
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We view an undirected graph as a symmetric digraph, where each edge is replaced by two opposite arcs and . Assume is an inverse closed subset of permutations of positive integers. We say is --colourable if for any mapping with , there is a mapping such that for each arc . The concept of --colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets such that every triangle-free planar graph is -3-colourable. Such a set is called TFP-good. Grötzsch’s theorem is equivalent to say that is TFP-good. We prove that for any inverse closed subset of which is not isomorphic to , is TFP-good if and only if either or there exists such that for each , . It remains an open question to determine whether or not is TFP-good. 相似文献