94 research outputs found

    Treewidth versus clique number. II. Tree-independence number

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    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 (tw,ω)(\mathrm{tw},\omega)-bounded. While (tw,ω)(\mathrm{tw},\omega)-bounded graph classes are known to enjoy some good algorithmic properties related to clique and coloring problems, it is an interesting open problem whether (tw,ω)(\mathrm{tw},\omega)-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 T\mathcal{T} of a graph as the maximum independence number over all subgraphs of GG induced by some bag of T\mathcal{T}. The tree-independence number of a graph GG is then defined as the minimum independence number over all tree decompositions of GG. 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

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    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 D>2D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension DD. 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

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    A celebrated result of Mirzakhani states that, if (S,m)(S,m) is a finite area \emph{orientable} hyperbolic surface, then the number of simple closed geodesics of length less than LL on (S,m)(S,m) is asymptotically equivalent to a positive constant times LdimML(S)L^{\dim\mathcal{ML}(S)}, where ML(S)\mathcal{ML}(S) denotes the space of measured laminations on SS. 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 (S,m)(S,m) 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 LL on (S,m)(S,m) is negligible compared to LdimML(S)L^{\dim\mathcal{ML}(S)}. 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 ε>0\varepsilon>0 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

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    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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