Combinatorics and Optimization

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Browsing Combinatorics and Optimization by Author "Edmonds, Jack"

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    FACES OF MATCHING POLYHEDRA
    (University of Waterloo, 2016-09-30) Pulleyblank, William R.; Edmonds, Jack
    Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of the over the edges j of G incident with i is no greater than bi. The matching polyhedron P(G, b) is the convex hull of the set of feasible matchings. In Chapter 3 we describe a version of Edmonds' blossom algorithm which solves the problem of maximizing C • X over P (G, b) where c =. (c.: j e: E) J is an arbitrary real vector. This algorithm proves a theorem of Edmonds which gives a set of linear inequalities sufficient to define P(G, b). In Chapter 4 we prescribe the unique subset of these inequalities which are necessary to define P(G, b), that is, we characterize the facets of P(G, b). We also characterize the vertices of P(G, b), thus describing the structure possessed by the members of the minimal set X of feasible matchings of G such that for any real vector c = (c.: j e: E), c • x is maximized over P(G, b) J member of X. by a In Chapter 5 we present a generalization of the blossom algorithm which solves the problem: maximize c • x over a face F of P(G, b) for any real vector c = (c.: j e: E). J In other words, we find a feasible matching x of G which satisfies the constraints obtained by replacing an arbitrary subset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this restriction. We also describe an application of this algorithm to matching problems having a hierarchy of objective functions, so called ''multi-optimization'' problems. In Chapter 6 we show how the blossom algorithm can be combined with relatively simple initialization algorithms to give an algorithm which solves the following postoptimality problem. Given that we know a matching 0 x £ P(G, b) maximizes c · x over P(G, b), we wish to utilize 0 X which to find a feasible matching x' £ P(G, b') which maximizes c • x over P(G, b'), where b' = (b!: i £ V) ]_ vector of positive integers and arbitrary real vector. c=(c.:j£E) J is a is an In Chapter 7 we describe a computer implementation of the blossom algorithm described herein.
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    SUBMODULAR FUNCTIONS, GRAPHS AND INTEGER POLYHEDRA
    (University of Waterloo, 2016-09-12) Giles, Frederick Richard; Edmonds, Jack
    This thesis is a study of the faces of certain combinatorially ­defined polyhedra. In particular, we examine the vertices and facets of these polyhedra. Chapter 2 contains the essential mathematical background in polyhedral theory, linear programming and graph theory. We also discuss the existence of an integer-valued optimum solution to a linear program. This is essential for determining that the vertices of certain polyhedra are integer-valued, and for establishing related combinatorial min-max relations. Chapter 3 is the a application of the results of Chapter 4 to the polymatroid aspects of matroid theory. We characterize the vertices and the facets of the intersection of two matroid polyhedra. We use the characterization of the facets of this intersection to derive a graph theoretic description of the facets of the polyhedron associated with branchings in a graph. In Chapter 5 we discuss certain polyhedra which can be associated with strong k-covers and strong k-matchings of an acyclic graph. By proving the existence of integer¥valued optimum solutions to particular primal-dual pairs of linear programs we are able to demon­strate certain combinatorial min-max relations. Chapter 6 is a unification of the polyhedra described in Chapters 4 and 5, We give a combinatorial definition of a class of polyhedra which includes polymatroid intersection and the polyhedra associated with strong k-covers and strong k-matchings of an acyclic graph. We establish the existence of integer-valued optimum solutions to certain dual linear programs and thereby draw conclusions concerning the integrality of the vertices of particular polyhedra within this class. The applications include establishing the integrality of the vertices of the intersection of two integral polymatroids and the integrality of the vertices of strong k-cover and strong k-matching polyhedra. Chapter 7 is a discussion of the facets of polyhedra defined in Chapter 6 and we obtain a description of the facets of a subclass of these polyhedra which includes a description of the facets of the intersection of two polymatroids and the facets of strong k-cover and strong k-matching polyhedra.