2,546 research outputs found
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Polynomial Norms
In this paper, we study polynomial norms, i.e. norms that are the
root of a degree- homogeneous polynomial . We first show
that a necessary and sufficient condition for to be a norm is for
to be strictly convex, or equivalently, convex and positive definite. Though
not all norms come from roots of polynomials, we prove that any
norm can be approximated arbitrarily well by a polynomial norm. We then
investigate the computational problem of testing whether a form gives a
polynomial norm. We show that this problem is strongly NP-hard already when the
degree of the form is 4, but can always be answered by testing feasibility of a
semidefinite program (of possibly large size). We further study the problem of
optimizing over the set of polynomial norms using semidefinite programming. To
do this, we introduce the notion of r-sos-convexity and extend a result of
Reznick on sum of squares representation of positive definite forms to positive
definite biforms. We conclude with some applications of polynomial norms to
statistics and dynamical systems
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