299 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index

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    Asymptotic separation index is a parameter that measures how easily a Borel graph can be approximated by its subgraphs with finite components. In contrast to the more classical notion of hyperfiniteness, asymptotic separation index is well-suited for combinatorial applications in the Borel setting. The main result of this paper is a Borel version of the Lov\'asz Local Lemma -- a powerful general-purpose tool in probabilistic combinatorics -- under a finite asymptotic separation index assumption. As a consequence, we show that locally checkable labeling problems that are solvable by efficient randomized distributed algorithms admit Borel solutions on bounded degree Borel graphs with finite asymptotic separation index. From this we derive a number of corollaries, for example a Borel version of Brooks's theorem for graphs with finite asymptotic separation index

    Computing a partition function of a generalized pattern-based energy over a semiring

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    Valued constraint satisfaction problems with ordered variables (VCSPO) are a special case of Valued CSPs in which variables are totally ordered and soft constraints are imposed on tuples of variables that do not violate the order. We study a restriction of VCSPO, in which soft constraints are imposed on a segment of adjacent variables and a constraint language Γ\Gamma consists of {0,1}\{0,1\}-valued characteristic functions of predicates. This kind of potentials generalizes the so-called pattern-based potentials, which were applied in many tasks of structured prediction. For a constraint language Γ\Gamma we introduce a closure operator, ΓΓ \overline{\Gamma^{\cap}}\supseteq \Gamma, and give examples of constraint languages for which Γ|\overline{\Gamma^{\cap}}| is small. If all predicates in Γ\Gamma are cartesian products, we show that the minimization of a generalized pattern-based potential (or, the computation of its partition function) can be made in O(VD2Γ2){\mathcal O}(|V|\cdot |D|^2 \cdot |\overline{\Gamma^{\cap}}|^2 ) time, where VV is a set of variables, DD is a domain set. If, additionally, only non-positive weights of constraints are allowed, the complexity of the minimization task drops to O(VΓDmaxρΓρ2){\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}| \cdot |D| \cdot \max_{\rho\in \Gamma}\|\rho\|^2 ) where ρ\|\rho\| is the arity of ρΓ\rho\in \Gamma. For a general language Γ\Gamma and non-positive weights, the minimization task can be carried out in O(VΓ2){\mathcal O}(|V|\cdot |\overline{\Gamma^{\cap}}|^2) time. We argue that in many natural cases Γ\overline{\Gamma^{\cap}} is of moderate size, though in the worst case Γ|\overline{\Gamma^{\cap}}| can blow up and depend exponentially on maxρΓρ\max_{\rho\in \Gamma}\|\rho\|

    Parameterized Graph Modification Beyond the Natural Parameter

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    Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices

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    This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the  Lambda-polytope. The complexity of classical simulation with respect to the polytope   is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the  Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of  Lambda. We conjecture that this class contains even more vertices of  Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n   24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Divide and Conquer Dynamic Programming: An Almost Linear Time Change Point Detection Methodology in High Dimensions

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    We develop a novel, general and computationally efficient framework, called Divide and Conquer Dynamic Programming (DCDP), for localizing change points in time series data with high-dimensional features. DCDP deploys a class of greedy algorithms that are applicable to a broad variety of high-dimensional statistical models and can enjoy almost linear computational complexity. We investigate the performance of DCDP in three commonly studied change point settings in high dimensions: the mean model, the Gaussian graphical model, and the linear regression model. In all three cases, we derive non-asymptotic bounds for the accuracy of the DCDP change point estimators. We demonstrate that the DCDP procedures consistently estimate the change points with sharp, and in some cases, optimal rates while incurring significantly smaller computational costs than the best available algorithms. Our findings are supported by extensive numerical experiments on both synthetic and real data.Comment: 84 pages, 4 figures, 6 table

    The Complexity of Some Geometric Proof Systems

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    In this Thesis we investigate proof systems based on Integer Linear Programming. These methods inspect the solution space of an unsatisfiable propositional formula and prove that this space contains no integral points. We begin by proving some size and depth lower bounds for a recent proof system, Stabbing Planes, and along the way introduce some novel methods for doing so. We then turn to the complexity of propositional contradictions generated uniformly from first order sentences, in Stabbing Planes and Sum-Of-Squares. We finish by investigating the complexity-theoretic impact of the choice of method of generating these propositional contradictions in Sherali-Adams

    Parameterized Graph Modification Beyond the Natural Parameter

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    Lower Bounds for (Batch) PIR with Private Preprocessing

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    In this paper, we study (batch) private information retrieval with private preprocessing. Private information retrieval (PIR) is the problem where one or more servers hold a database of nn bits and a client wishes to retrieve the ii-th bit in the database from the server(s). In PIR with private preprocessing (also known as offline-online PIR), the client is able to compute a private rr-bit hint in an offline stage that may be leveraged to perform retrievals accessing at most tt entries. For privacy, the client wishes to hide index ii from an adversary that has compromised some of the servers. In the batch PIR setting, the client performs queries to retrieve the contents of multiple entries simultaneously. We present a tight characterization for the trade-offs between hint size rr and number of accessed entries tt during queries. For any PIR scheme that enables clients to perform batch retrievals of kk entries, we prove a lower bound of tr=Ω(nk)tr = \Omega(nk) when rkr \ge k. When r<kr < k, we prove that t=Ω(n)t = \Omega(n). Our lower bounds hold when the scheme errs with probability at most 1/151/15 and against PPT adversaries that only compromise one out of \ell servers for any =O(1)\ell = O(1). Our work also closes the multiplicative logarithmic gap for the single query setting (k=1)(k = 1) as our lower bound matches known constructions. Our lower bounds hold in the model where each database entry is stored without modification but each entry may be replicated arbitrarily. Finally, we show connections between PIR and the online matrix-vector (OMV) conjecture from fine-grained complexity. We present barriers for proving lower bounds for two-server PIR schemes in general computational models as they would immediately imply the OMV conjecture
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