52,164 research outputs found
QCSP on partially reflexive forests
We study the (non-uniform) quantified constraint satisfaction problem QCSP(H)
as H ranges over partially reflexive forests. We obtain a complexity-theoretic
dichotomy: QCSP(H) is either in NL or is NP-hard. The separating condition is
related firstly to connectivity, and thereafter to accessibility from all
vertices of H to connected reflexive subgraphs. In the case of partially
reflexive paths, we give a refinement of our dichotomy: QCSP(H) is either in NL
or is Pspace-complete
Shape Optimization Problems with Internal Constraint
We consider shape optimization problems with internal inclusion constraints,
of the form \min\big\{J(\Omega)\ :\ \Dr\subset\Omega\subset\R^d,\
|\Omega|=m\big\}, where the set \Dr is fixed, possibly unbounded, and
depends on via the spectrum of the Dirichlet Laplacian. We analyze the
existence of a solution and its qualitative properties, and rise some open
questions.Comment: 18 pages, 0 figure
Locally compact, -compact spaces
An -compact space is a space in which every closed discrete
subspace is countable. We give various general conditions under which a locally
compact, -compact space is -countably compact, i.e., the
union of countably many countably compact spaces. These conditions involve very
elementary properties.Comment: 21 pages, submitted, comments are welcom
Cancelation norm and the geometry of biinvariant word metrics
We study biinvariant word metrics on groups. We provide an efficient
algorithm for computing the biinvariant word norm on a finitely generated free
group and we construct an isometric embedding of a locally compact tree into
the biinvariant Cayley graph of a nonabelian free group. We investigate the
geometry of cyclic subgroups. We observe that in many classes of groups cyclic
subgroups are either bounded or detected by homogeneous quasimorphisms. We call
this property the bq-dichotomy and we prove it for many classes of groups of
geometric origin.Comment: 32 pages, to appear in Glasgow Journal of Mathematic
On the Assouad dimension of self-similar sets with overlaps
It is known that, unlike the Hausdorff dimension, the Assouad dimension of a
self-similar set can exceed the similarity dimension if there are overlaps in
the construction. Our main result is the following precise dichotomy for
self-similar sets in the line: either the \emph{weak separation property} is
satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the
\emph{weak separation property} is not satisfied, in which case the Assouad
dimension is maximal (equal to one).
In the first case we prove that the self-similar set is Ahlfors regular, and
in the second case we use the fact that if the \emph{weak separation property}
is not satisfied, one can approximate the identity arbitrarily well in the
group generated by the similarity mappings, and this allows us to build a
\emph{weak tangent} that contains an interval. We also obtain results in higher
dimensions and provide illustrative examples showing that the
`equality/maximal' dichotomy does not extend to this setting.Comment: 24 pages, 2 figure
The Complexity of the List Partition Problem for Graphs
The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete
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