FACES OF MATCHING POLYHEDRA
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Date
2016-09-30
Authors
Pulleyblank, William R.
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
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.
Description
Keywords
set theory, graph theory, polyhedral theory, first facet characterization, second facet characterization, vertices of polyhedra, Blossom algorithm, alternating forests, Hungarian forests, Post-Optimality algorithm