94 research outputs found
Treewidth versus clique number. II. Tree-independence number
In 2020, we initiated a systematic study of graph classes in which the
treewidth can only be large due to the presence of a large clique, which we
call -bounded. While -bounded graph
classes are known to enjoy some good algorithmic properties related to clique
and coloring problems, it is an interesting open problem whether
-boundedness also has useful algorithmic implications for
problems related to independent sets.
We provide a partial answer to this question by means of a new min-max graph
invariant related to tree decompositions. We define the independence number of
a tree decomposition of a graph as the maximum independence
number over all subgraphs of induced by some bag of . The
tree-independence number of a graph is then defined as the minimum
independence number over all tree decompositions of . Generalizing a result
on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is
given together with a tree decomposition with bounded independence number, then
the Maximum Weight Independent Packing problem can be solved in polynomial
time.
Applications of our general algorithmic result to specific graph classes will
be given in the third paper of the series [Dallard, Milani\v{c}, and
\v{S}torgel, Treewidth versus clique number. III. Tree-independence number of
graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A
previous version of this arXiv post has been reorganized into two parts; this
is the first of the two parts (the second one is arXiv:2206.15092
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
What's wrong with the growth of simple closed geodesics on nonorientable hyperbolic surfaces
A celebrated result of Mirzakhani states that, if is a finite area
\emph{orientable} hyperbolic surface, then the number of simple closed
geodesics of length less than on is asymptotically equivalent to a
positive constant times , where
denotes the space of measured laminations on . We observed on some explicit
examples that this result does not hold for \emph{nonorientable} hyperbolic
surfaces. The aim of this article is to explain this surprising phenomenon. Let
be a finite area \emph{nonorientable} hyperbolic surface. We show that
the set of measured laminations with a closed one--sided leaf has a peculiar
structure. As a consequence, the action of the mapping class group on the
projective space of measured laminations is not minimal. We determine a partial
classification of its orbit closures, and we deduce that the number of simple
closed geodesics of length less than on is negligible compared to
. We extend this result to general multicurves. Then
we focus on the geometry of the moduli space. We prove that its Teichm\"uller
volume is infinite, and that the Teichm\"uller flow is not ergodic. We also
consider a volume form introduced by Norbury. We show that it is the right
generalization of the Weil--Petersson volume form. The volume of the moduli
space with respect to this volume form is again infinite (as shown by Norbury),
but the subset of hyperbolic surfaces whose one--sided geodesics have length at
least has finite volume. These results suggest that the moduli
space of a nonorientable surface looks like an infinite volume geometrically
finite orbifold. We discuss this analogy and formulate some conjectures
Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks
We develop a global twistor correspondence for pseudo-Riemannian conformal
structures of signature (++--) with self-dual Weyl curvature. Near the
conformal class of the standard indefinite product metric on S^2 x S^2, there
is an infinite-dimensional moduli space of such conformal structures, and each
of these has the surprising global property that its null geodesics are all
periodic. Each such conformal structure arises from a family of holomorphic
disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3.
An interesting sub-class of these conformal structures are represented by
scalar-flat indefinite K\"ahler metrics, and our methods give particularly
sharp results in this more restrictive setting.Comment: 56 pages, LaTeX2
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