8,367 research outputs found
Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope
We study a new geometric graph parameter \egd(G), defined as the smallest
integer for which any partial symmetric matrix which is completable to
a correlation matrix and whose entries are specified at the positions of the
edges of , can be completed to a matrix in the convex hull of correlation
matrices of \rank at most . This graph parameter is motivated by its
relevance to the problem of finding low rank solutions to semidefinite programs
over the elliptope, and also by its relevance to the bounded rank Grothendieck
constant. Indeed, \egd(G)\le r if and only if the rank- Grothendieck
constant of is equal to 1. We show that the parameter \egd(G) is minor
monotone, we identify several classes of forbidden minors for \egd(G)\le r
and we give the full characterization for the case . We also show an upper
bound for \egd(G) in terms of a new tree-width-like parameter \sla(G),
defined as the smallest for which is a minor of the strong product of a
tree and . We show that, for any 2-connected graph on at
least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been
modified to accommodate the suggestions of the referees. Furthermore, the
title has been changed since we feel that the new title reflects more
accurately the content and the main results of the pape
Recognising Multidimensional Euclidean Preferences
Euclidean preferences are a widely studied preference model, in which
decision makers and alternatives are embedded in d-dimensional Euclidean space.
Decision makers prefer those alternatives closer to them. This model, also
known as multidimensional unfolding, has applications in economics,
psychometrics, marketing, and many other fields. We study the problem of
deciding whether a given preference profile is d-Euclidean. For the
one-dimensional case, polynomial-time algorithms are known. We show that, in
contrast, for every other fixed dimension d > 1, the recognition problem is
equivalent to the existential theory of the reals (ETR), and so in particular
NP-hard. We further show that some Euclidean preference profiles require
exponentially many bits in order to specify any Euclidean embedding, and prove
that the domain of d-Euclidean preferences does not admit a finite forbidden
minor characterisation for any d > 1. We also study dichotomous preferencesand
the behaviour of other metrics, and survey a variety of related work.Comment: 17 page
On the graph condition regarding the -inverse cover problem
In their paper titled "On -inverse covers of inverse monoids", Auinger and
Szendrei have shown that every finite inverse monoid has an -inverse cover
if and only if each finite graph admits a locally finite group variety with a
certain property. We study this property and prove that the class of graphs for
which a given group variety has the required property is closed downwards in
the minor ordering, and can therefore be described by forbidden minors. We find
these forbidden minors for all varieties of Abelian groups, thus describing the
graphs for which such a group variety satisfies the above mentioned condition
Simple PTAS's for families of graphs excluding a minor
We show that very simple algorithms based on local search are polynomial-time
approximation schemes for Maximum Independent Set, Minimum Vertex Cover and
Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups
We examine two criteria for balance of a gain graph, one based on binary
cycles and one on circles. The graphs for which each criterion is valid depend
on the set of allowed gain groups. The binary cycle test is invalid, except for
forests, if any possible gain group has an element of odd order. Assuming all
groups are allowed, or all abelian groups, or merely the cyclic group of order
3, we characterize, both constructively and by forbidden minors, the graphs for
which the circle test is valid. It turns out that these three classes of groups
have the same set of forbidden minors. The exact reason for the importance of
the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in
Journal of Graph Theor
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
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