1,083 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Structured Semidefinite Programming for Recovering Structured Preconditioners

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    We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite KRd×d\mathbf{K} \in \mathbb{R}^{d \times d} with nnz(K)\mathrm{nnz}(\mathbf{K}) nonzero entries, computes an ϵ\epsilon-optimal diagonal preconditioner in time O~(nnz(K)poly(κ,ϵ1))\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1})), where κ\kappa^\star is the optimal condition number of the rescaled matrix. We give an algorithm which, given MRd×d\mathbf{M} \in \mathbb{R}^{d \times d} that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in M\mathbf{M} in O~(d2)\widetilde{O}(d^2) time. Our diagonal preconditioning results improve state-of-the-art runtimes of Ω(d3.5)\Omega(d^{3.5}) attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of Ω(dω)\Omega(d^{\omega}) where ω>2.3\omega > 2.3 is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.Comment: Merge of arXiv:1812.06295 and arXiv:2008.0172

    Local computation algorithms for hypergraph coloring – following Beck’s approach

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    We investigate local computation algorithms (LCA) for two-coloring of k-uniform hypergraphs. We focus on hypergraph instances that satisfy strengthened assumption of the Lovász Local Lemma of the form 21αk(+1)e<121−αk (∆ + 1)e < 1, where ∆ is the bound on the maximum edge degree. The main question which arises here is for how large α there exists an LCA that is able to properly color such hypergraphs in polylogarithmic time per query. We describe briefly how upgrading the classical sequential procedure of Beck from 1991 with Moser and Tardos’ Resample yields polylogarithmic LCA that works for α up to 1/4. Then, we present an improved procedure that solves wider range of instances by allowing α up to 1/3

    On the graph theory of majority illusions

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    The popularity of an opinion in one's direct circles is not necessarily a good indicator of its popularity in one's entire community. For instance, when confronted with a majority of opposing opinions in one's circles, one might get the impression that they belong to a minority. From this perspective, network structure makes local information about global properties of the group potentially inaccurate. However, the way a social network is wired also determines what kind of information distortion can actually occur. In this paper, we discuss which classes of networks allow for a majority of agents to have the wrong impression about what the majority opinion is, that is, to be in a 'majority illusion'

    Summer 2023 Full Issue

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    Unsolved Problems in Spectral Graph Theory

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    Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of 2020 topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction

    Quantum Codes on Graphs

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    We consider some questions related to codes constructed using various graphs, in particular focusing on graphs which are not lattices in two or three dimensions. We begin by considering Floquet codes which can be constructed using ``emergent fermions". Here, we are considering codes that in some sense generalize the honeycomb code[1] to more general, non-planar graphs. We then consider a class of these codes that is related to (generalized) toric codes on 22-complexes. For (generalized) toric codes on 22-complexes, the following question arises: can the distance of these codes grow faster than square-root? We answer the question negatively, and remark on recent systolic inequalities[2]. We then turn to the case that of planar codes with vacancies, or ``dead qubits", and consider the statistical mechanics of decoding in this setting. Although we do not prove a threshold, our results should be asymptotically correct for low error probability and high degree decoding graphs (high degree taken before low error probability). In an appendix, we discuss a toy model of vacancies in planar quantum codes, giving a phenomenological discussion of how errors occur when ``super-stabilizers" are not measured, and in a separate appendix we discuss a relation between Floquet codes and chain maps.Comment: 25 pages, 1 figur

    The long-range Falicov-Kimball model and the amorphous Kitaev model: Quantum many-body systems I have known and loved

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    Large systems of interacting objects can give rise to a rich array of emergent behaviours. Make those objects quantum and the possibilities only expand. Interacting quantum many-body systems, as such systems are called, include essentially all physical systems. Luckily, we don't usually need to consider this full quantum many-body description. The world at the human scale is essentially classical (not quantum), while at the microscopic scale of condensed matter physics we can often get by without interactions. Strongly correlated materials, however, do require the full description. Some of the most exciting topics in modern condensed matter fall under this umbrella: the spin liquids, the fractional quantum Hall effect, high temperature superconductivity and much more. Unfortunately, strongly correlated materials are notoriously difficult to study, defying many of the established theoretical techniques within the field. Enter exactly solvable models, these are interacting quantum many-body systems with extensively many local symmetries. The symmetries give rise to conserved charges. These charges break the model up into many non-interacting quantum systems which are more amenable to standard theoretical techniques. This thesis will focus on two such exactly solvable models. The first, the Falicov-Kimball (FK) model is an exactly solvable limit of the famous Hubbard model which describes itinerant fermions interacting with a classical Ising background field. Originally introduced to explain metal-insulator transitions, it has a rich set of ground state and thermodynamic phases. Disorder or interactions can turn metals into insulators and the FK model features both transitions. We will define a generalised FK model in 1D with long-range interactions. This model shows a similarly rich phase diagram to its higher dimensional cousins. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. This allows us to look at how the move to 1D affects the physics of the model. We show that the model can be understood by comparison to a simpler model of fermions coupled to binary disorder. The second, the Kitaev Honeycomb (KH) model, was the one of the first solvable 2D models with a Quantum Spin Liquid (QSL) ground state. QSLs are generally expected to arise from Mott insulators, when frustration prevents magnetic ordering all the way to zero temperature. The QSL state defies the traditional Landau-Ginzburg-Wilson paradigm of phases being defined by local order parameters. It is instead a topologically ordered phase. Recent work generalising non-interacting topological insulator phases to amorphous lattices raises the question of whether interacting phases like the QSLs can be similarly generalised. We extend the KH model to random lattices with fixed coordination number three generated by Voronoi partitions of the plane. We show that this model remains solvable and hosts a chiral amorphous QSL ground state. The presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian QSL phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a phase transition to a conducting thermal metal state and discuss possible experimental realisations.Open Acces

    On the choosability of HH-minor-free graphs

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    Given a graph HH, let us denote by fχ(H)f_\chi(H) and f(H)f_\ell(H), respectively, the maximum chromatic number and the maximum list chromatic number of HH-minor-free graphs. Hadwiger's famous coloring conjecture from 1943 states that fχ(Kt)=t1f_\chi(K_t)=t-1 for every t2t \ge 2. In contrast, for list coloring it is known that 2to(t)f(Kt)O(t(loglogt)6)2t-o(t) \le f_\ell(K_t) \le O(t (\log \log t)^6) and thus, f(Kt)f_\ell(K_t) is bounded away from the conjectured value t1t-1 for fχ(Kt)f_\chi(K_t) by at least a constant factor. The so-called HH-Hadwiger's conjecture, proposed by Seymour, asks to prove that fχ(H)=v(H)1f_\chi(H)=\textsf{v}(H)-1 for a given graph HH (which would be implied by Hadwiger's conjecture). In this paper, we prove several new lower bounds on f(H)f_\ell(H), thus exploring the limits of a list coloring extension of HH-Hadwiger's conjecture. Our main results are: For every ε>0\varepsilon>0 and all sufficiently large graphs HH we have f(H)(1ε)(v(H)+κ(H))f_\ell(H)\ge (1-\varepsilon)(\textsf{v}(H)+\kappa(H)), where κ(H)\kappa(H) denotes the vertex-connectivity of HH. For every ε>0\varepsilon>0 there exists C=C(ε)>0C=C(\varepsilon)>0 such that asymptotically almost every nn-vertex graph HH with Cnlogn\left\lceil C n\log n\right\rceil edges satisfies f(H)(2ε)nf_\ell(H)\ge (2-\varepsilon)n. The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of HH-minor-free graphs is separated from the natural lower bound (v(H)1)(\textsf{v}(H)-1) by a constant factor for all large graphs HH of linear connectivity. The second result tells us that even when HH is a very sparse graph (with an average degree just logarithmic in its order), f(H)f_\ell(H) can still be separated from (v(H)1)(\textsf{v}(H)-1) by a constant factor arbitrarily close to 22. Conceptually these results indicate that the graphs HH for which f(H)f_\ell(H) is close to (v(H)1)(\textsf{v}(H)-1) are typically rather sparse.Comment: 14 page

    Constant-Depth Circuits vs. Monotone Circuits

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