查看更多>>摘要：? 2019 Elsevier B.V.We present well known properties related to the topology of Steiner minimal trees and to the geometric position of Steiner points, and investigate their application in the exact algorithms that have been proposed for the Euclidean Steiner problem. We discuss the difficulty in the application of properties that were very successfully applied to solve the problem in the plane, when the dimension of the space increases, and point out that the application of some geometric conditions for Steiner points is hindered when the well known implicit enumeration scheme proposed by Smith in 1992 is considered. Finally, we mention mathematical-optimization methods as a direction to explore in the search for good formulations of inequalities that would allow the application of these restrictive geometric conditions.
查看更多>>摘要：? 2020 Elsevier B.V.In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. We present a complete description of this polytope. Apart from linearization constraints one needs appropriately strengthened rank inequalities. The separation problem for these inequalities reduces to a submodular function minimization problem. These polyhedral results give rise to a new hierarchy for the solution of matroid optimization problems with polynomial objectives. This hierarchy allows to strengthen the relaxations of arbitrary linearized combinatorial optimization problems with polynomial objective functions and matroidal substructures. Finally, we give suggestions for future work.
查看更多>>摘要：? 2019 Elsevier B.V.Speakman and Lee (2017) gave a formula for the volume of the convex hull of the graph of a trilinear monomial, y=x1x2x3, over a box in the nonnegative orthant, in terms of the upper and lower bounds on the variables. This was done in the context of using volume as a measure for comparing alternative convexifications to guide the implementation of spatial branch-and-bound for mixed integer nonlinear optimization problems. Here, we introduce an alternative method for computing this volume, making use of the rich theory of mixed volumes. This new method may lead to a natural approach for considering extensions of the problem.
查看更多>>摘要：? 2019 The AuthorsWe consider the maximum vertex-weighted matching problem (MVM) for non-bipartite graphs in which non-negative weights are assigned to the vertices of a graph and a matching that maximizes the sum of the weights of the matched vertices is desired. In earlier work we have described a 2/3-approximation algorithm for the MVM on bipartite graphs (Florin Dobrian et al., 2019). Here we show that a 2/3-approximation algorithm for MVM on non-bipartite graphs can be obtained by restricting the length of augmenting paths to at most three. The algorithm has time complexity O(mlogΔ+nlogn), where n is the number of vertices, m is the number of edges, and Δ is the maximum degree of a vertex. The approximation ratio of the algorithm is obtained by considering failed vertices, i.e., vertices that the approximation algorithm fails to match but the exact algorithm does. We show that there are two distinct heavier matched vertices that we can charge each failed vertex to. Our proof techniques characterize the structure of augmenting paths in a novel way. We have implemented the 2/3-approximation algorithm and show that it runs in under a minute on graphs with tens of millions of vertices and hundreds of millions of edges. We compare its performance with five other algorithms: an exact algorithm for MVM, an exact algorithm for the maximum edge-weighted matching (MEM) problem, as well as three approximation algorithms. The approximation algorithms include a 1/2-approximation algorithm for MVM, and (2/3?ε)- and (1?ε)-approximation algorithms for the MEM. In our test set of nineteen problems, there are graphs on which the exact algorithms fail to terminate in 100 hours. In addition, the new 2/3-approximation algorithm for MVM outperforms the other approximation algorithms by either being faster (often by orders of magnitude) or obtaining better weights.
查看更多>>摘要：? 2019 Elsevier B.V.Circuits play a fundamental role in polyhedral theory and linear programming. For instance, circuits are used as step directions in various augmentation schemes for solving linear programs or to leave degenerate vertices while running the simplex method. However, there are significant challenges when implementing these approaches: The set of circuits of a polyhedron may be of exponential size and is highly sensitive to the representation of the polyhedron. In this paper, we provide a universal framework for enumerating the set of circuits and optimizing over sets of circuits of a polyhedron in any representation—we propose a polyhedral model in which the circuits of the original polyhedron are encoded as extreme rays or vertices. Many methods in the literature and software assume that a polyhedron is in standard form; our framework is a direct generalization. We demonstrate its value by showing that the conversion of a general representation to standard form may introduce exponentially many new circuits. We then discuss the main advantages of the generalized polyhedral model. It enables the direct enumeration of useful subsets of circuits such as strictly feasible circuits or sign-compatible circuits, as well as optimization over these sets. In particular, this leads to the efficient computation of a steepest-descent circuit, which can be used in an augmentation scheme for solving linear programs or the construction of sign-compatible circuit walks with additional properties.
查看更多>>摘要：? 2019 Elsevier B.V.Within the framework of the superadditive duality theory of integer programming, we study two types of dual-feasible functions of a single real variable (Alves et al., 2016). We introduce software that automates testing piecewise linear functions for maximality and extremality, enabling a computer-based search. We build a connection to cut-generating functions in the Gomory–Johnson and related models, complete the characterization of maximal functions, and prove analogues of the Gomory–Johnson 2-slope theorem and the Basu–Hildebrand–Molinaro approximation theorem.
查看更多>>摘要：? 2020 Elsevier B.V.We prove a new PPA parity theorem: Given a bipartite graph G with bipartition (A,B) where B is a set of even-degree vertices, and given a tree T? of G containing all of A, such that any vertex of B in T? has degree 2 in T? and such that each vertex of A which is not a leaf of T? is met by an odd number of edges not in T?, then there is an even number of trees of G containing all of A, with degree 0 or 2 at each vertex of B and with the same degree as T? at each vertex of A. This theorem generalizes Berman's generalization of Thomason's generalization of Smith's Theorem.
查看更多>>摘要：? 2019 Elsevier B.V.A graph G is a B0-VPG graph if one can associate a horizontal or vertical path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect in at least one grid-point. A graph G is a contact B0-VPG graph if it is a B0-VPG graph admitting a representation with no one-point paths, no two paths crossing, and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact B0-VPG graphs within four special graph classes: chordal graphs, tree-cographs, P4-tidy graphs and P5-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact B0-VPG graphs.
查看更多>>摘要：? 2020 Elsevier B.V.We investigate efficient randomized methods for approximating the number of perfect matchings in bipartite graphs and general undirected graphs. Our approach is based on assigning probabilities to edges, randomly selecting an edge to be in a perfect matching, and discarding edges that cannot be put in a perfect matching. The probabilities are set according to the entries in the doubly stochastically scaled version of the adjacency matrix of the given graph. The experimental analysis on random and real-life graphs shows improvements in the approximation over previous and similar methods from the literature.
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