311 research outputs found

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

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    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of LovĂĄsz local lemma when Δ â‰Č qᔏ. In prior, however, fast approximate counting algorithms exist when Δ â‰Č qᔏ/Âł, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. ‱ When q, k ≄ 4 are evens and Δ ≄ 5·qᔏ/ÂČ, approximating the number of hypergraph colourings is NP-hard. ‱ When the input hypergraph is linear and Δ â‰Č qᔏ/ÂČ, a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Nonlocal games and their device-independent quantum applications

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    Device-independence is a property of certain protocols that allows one to ensure their proper execution given only classical interaction with devices and assuming the correctness of the laws of physics. This scenario describes the most general form of cryptographic security, in which no trust is placed in the hardware involved; indeed, one may even take it to have been prepared by an adversary. Many quantum tasks have been shown to admit device-independent protocols by augmentation with "nonlocal games". These are games in which noncommunicating parties jointly attempt to fulfil some conditions imposed by a referee. We introduce examples of such games and examine the optimal strategies of players who are allowed access to different possible shared resources, such as entangled quantum states. We then study their role in self-testing, private random number generation, and secure delegated quantum computation. Hardware imperfections are naturally incorporated in the device-independent scenario as adversarial, and we thus also perform noise robustness analysis where feasible. We first study a generalization of the Mermin–Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these "magic rectangle" games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for m×n magic rectangle games with dimensions m,n≄3, there are quantum strategies that win with certainty, while for dimensions 1×n quantum strategies do not outperform classical strategies. The final case of dimensions 2×n is richer, and we give upper and lower bounds that both outperform the classical strategies. As an initial usage scenario, we apply our findings to quantum certified randomness expansion to find noise tolerances and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probabilities of games with a distinguished input for which the devices give a deterministic outcome and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)]. Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. We use our 3×n magic rectangle games to obtain a self-test for n Bell states where one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes [constant for Alice and O(log n) bits for Bob] and is robust with robustness O(n⁔/ÂČ√Δ), where Δ is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test, we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, we generalize this strategy to the family of 3×n magic rectangle games, and we supplement these nonlocal games with extra check rounds (of single and pairs of observables). Finally, we introduce a device-independent two-prover scheme in which a classical verifier can use a simple untrusted quantum measurement device (the client device) to securely delegate a quantum computation to an untrusted quantum server. To do this, we construct a parallel self-testing protocol to perform device-independent remote state preparation of n qubits and compose this with the unconditionally secure universal verifiable blind quantum computation (VBQC) scheme of J. F. Fitzsimons and E. Kashefi [Phys. Rev. A 96, 012303 (2017)]. Our self-test achieves a multitude of desirable properties for the application we consider, giving rise to practical and fully device-independent VBQC. It certifies parallel measurements of all cardinal and intercardinal directions in the XY-plane as well as the computational basis, uses few input questions (of size logarithmic in n for the client and a constant number communicated to the server), and requires only single-qubit measurements to be performed by the client device

    Faster Deterministic Distributed MIS and Approximate Matching

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    \renewcommand{\tilde}{\widetilde} We present an O~(log⁥2n)\tilde{O}(\log^2 n) round deterministic distributed algorithm for the maximal independent set problem. By known reductions, this round complexity extends also to maximal matching, Δ+1\Delta+1 vertex coloring, and 2Δ−12\Delta-1 edge coloring. These four problems are among the most central problems in distributed graph algorithms and have been studied extensively for the past four decades. This improved round complexity comes closer to the Ω~(log⁥n)\tilde{\Omega}(\log n) lower bound of maximal independent set and maximal matching [Balliu et al. FOCS '19]. The previous best known deterministic complexity for all of these problems was Θ(log⁥3n)\Theta(\log^3 n). Via the shattering technique, the improvement permeates also to the corresponding randomized complexities, e.g., the new randomized complexity of Δ+1\Delta+1 vertex coloring is now O~(log⁥2log⁥n)\tilde{O}(\log^2\log n) rounds. Our approach is a novel combination of the previously known two methods for developing deterministic algorithms for these problems, namely global derandomization via network decomposition (see e.g., [Rozhon, Ghaffari STOC'20; Ghaffari, Grunau, Rozhon SODA'21; Ghaffari et al. SODA'23]) and local rounding of fractional solutions (see e.g., [Fischer DISC'17; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Ghaffari, Kuhn FOCS'21; Faour et al. SODA'23]). We consider a relaxation of the classic network decomposition concept, where instead of requiring the clusters in the same block to be non-adjacent, we allow each node to have a small number of neighboring clusters. We also show a deterministic algorithm that computes this relaxed decomposition faster than standard decompositions. We then use this relaxed decomposition to significantly improve the integrality of certain fractional solutions, before handing them to the local rounding procedure that now has to do fewer rounding steps

    A Theory of Link Prediction via Relational Weisfeiler-Leman on Knowledge Graphs

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    Graph neural networks are prominent models for representation learning over graph-structured data. While the capabilities and limitations of these models are well-understood for simple graphs, our understanding remains incomplete in the context of knowledge graphs. Our goal is to provide a systematic understanding of the landscape of graph neural networks for knowledge graphs pertaining to the prominent task of link prediction. Our analysis entails a unifying perspective on seemingly unrelated models and unlocks a series of other models. The expressive power of various models is characterized via a corresponding relational Weisfeiler-Leman algorithm. This analysis is extended to provide a precise logical characterization of the class of functions captured by a class of graph neural networks. The theoretical findings presented in this paper explain the benefits of some widely employed practical design choices, which are validated empirically.Comment: Proceedings of the Thirty-Seventh Annual Conference on Advances in Neural Information Processing Systems (NeurIPS 2023). Code available at: https://github.com/HxyScotthuang/CMPN

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Learning-powered computer-assisted counterexample search

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    Treballs Finals de Grau de MatemĂ tiques, Facultat de MatemĂ tiques, Universitat de Barcelona, Any: 2023, Director: Kolja Knauer[en] This thesis explores the great potential of computer-assisted proofs in the advancement of mathematical knowledge, with a special focus on using computers to refute conjectures by finding counterexamples, sometimes a humanly impossible task. In recent years, mathematicians have become more aware that machine learning techniques can be extremely helpful for finding counterexamples to conjectures in a more efficient way than by using exhaustive search methods. In this thesis we do not only present the theoretical background behind some of these methods but also implement them to try to refute some graph theory conjectures

    Asymptotics for Palette Sparsification

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    It is shown that the following holds for each Δ>0\varepsilon>0. For GG an nn-vertex graph of maximum degree DD and "lists" LvL_v (v∈V(G)v \in V(G)) chosen independently and uniformly from the ((1+Δ)ln⁥n(1+\varepsilon)\ln n)-subsets of {1,...,D+1}\{1, ..., D+1\}, G admits a proper coloring σ with σv∈Lv∀v G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v with probability tending to 1 as D→∞D \to \infty. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.Comment: 29 page

    On the structure of graphs with forbidden induced substructures

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    One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. ErdƑs and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every 22-edge-colouring of the complete graph on nn vertices there is a monochromatic clique on at least 12log⁡n\frac{1}{2}\log n vertices. The famous ErdƑs-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair (n,e)(n,e) is the family consisting of all graphs of order nn and size ee, i.e. on nn vertices with ee edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs (m,f)(m,f), i.e. for nn approaching infinity, the limit superior of the fraction of all possible sizes ee, such that the order-size pair (n,e)(n,e) does not avoid the pair (m,f)(m,f)

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the ChvĂĄtal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
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