1,682 research outputs found
Single-edge monotonic sequences of graphs and linear-time algorithms for minimal completions and deletions
AbstractWe study graph properties that admit an increasing, or equivalently decreasing, sequence of graphs on the same vertex set such that for any two consecutive graphs in the sequence their difference is a single edge. This is useful for characterizing and computing minimal completions and deletions of arbitrary graphs into having these properties. We prove that threshold graphs and chain graphs admit such sequences. Based on this characterization and other structural properties, we present linear-time algorithms both for computing minimal completions and deletions into threshold, chain, and bipartite graphs, and for extracting a minimal completion or deletion from a given completion or deletion. Minimum completions and deletions into these classes are NP-hard to compute
Exploring Subexponential Parameterized Complexity of Completion Problems
Let be a family of graphs. In the -Completion problem,
we are given a graph and an integer as input, and asked whether at most
edges can be added to so that the resulting graph does not contain a
graph from as an induced subgraph. It appeared recently that special
cases of -Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of , and the problem of completing into a split graph,
i.e., the case of , are solvable in parameterized
subexponential time . The exploration of this
phenomenon is the main motivation for our research on -Completion.
In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized
subexponential time , that is -Completion for , a cycle and a path on four
vertices.
- The problems known in the literature as Pseudosplit Completion, the case
where , and Threshold Completion, where , are also solvable in time .
We complement our algorithms for -Completion with the following
lower bounds:
- For , , , and
, -Completion cannot be solved in time
unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of -Completion problems for .Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in
the proceedings of STACS'1
Multiplicative Lidskii's inequalities and optimal perturbations of frames
In this paper we study two design problems in frame theory: on the one hand,
given a fixed finite frame \cF for \hil\cong\C^d we compute those dual
frames \cG of \cF that are optimal perturbations of the canonical dual
frame for \cF under certain restrictions on the norms of the elements of
\cG. On the other hand, for a fixed finite frame \cF=\{f_j\}_{j\in\In} for
\hil we compute those invertible operators such that is a
perturbation of the identity and such that the frame V\cdot
\cF=\{V\,f_j\}_{j\in\In} - which is equivalent to \cF - is optimal among
such perturbations of \cF. In both cases, optimality is measured with respect
to submajorization of the eigenvalues of the frame operators. Hence, our
optimal designs are minimizers of a family of convex potentials that include
the frame potential and the mean squared error. The key tool for these results
is a multiplicative analogue of Lidskii's inequality in terms of
log-majorization and a characterization of the case of equality.Comment: 22 page
Effective description of brane terms in extra dimensions
We study how theories defined in (extra-dimensional) spaces with localized
defects can be described perturbatively by effective field theories in which
the width of the defects vanishes. These effective theories must incorporate a
``classical'' renormalization, and we propose a renormalization prescription a
la dimensional regularization for codimension 1, which can be easily used in
phenomenological applications. As a check of the validity of this setting, we
compare some general predictions of the renormalized effective theory with
those obtained in a particular ultraviolet completion based on deconstruction.Comment: 28 page
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