134 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On the use of senders for minimal Ramsey theory

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    This thesis investigates problems related to extremal and probabilistic graph theory. Our focus lies on the highly dynamic field of Ramsey theory. The foundational result of this field was proved in 1930 by Franck P. Ramsey. It implies that for every integer t and every sufficiently large complete graph Kn, every colouring of the edges of Kn with colours red and blue contains a red copy or a blue copy of Kt. Let q ⩾ 2 represent a number of colours, and let H1,..., Hq be graphs. A graph G is said to be q-Ramsey for the tuple (H1,...,Hq) if, for every colouring of the edges of G with colours {1, . . . , q}, there exists a colour i and a monochromatic copy of Hi in colour i. As we often want to understand the structural properties of the collection of graphs that are q-Ramsey for (H1,..., Hq), we restrict our attention to the graphs that are minimal for this property, with respect to subgraph inclusion. Such graphs are said to be q-Ramsey-minimal for (H1,..., Hq). In 1976, Burr, Erdős, and Lovász determined, for every s, t ⩾ 3, the smallest minimum degree of a graph G that is 2-Ramsey-minimal for (Ks, Kt). Significant efforts have been dedicated to generalising this result to a higher number of colours, q⩾3, starting with the ‘symmetric’ q-tuple (Kt,..., Kt). In this thesis, we improve on the best known bounds for this parameter, providing state-of-the-art bounds in different (q, t)-regimes. These improvements rely on constructions based on finite geometry, which are then used to prove the existence of extremal graphs with certain key properties. Another crucial ingredient in these proofs is the existence of gadget graphs, called signal senders, that were initially developed by Burr, Erdős, and Lovász in 1976 for pairs of complete graphs. Until now, these senders have been shown to exist only in the two-colour setting, when q = 2, or in the symmetric multicolour setting, when H1,..., Hq are pairwise isomorphic. In this thesis, we then construct similar gadgets for all tuples of complete graphs, providing the first known constructions of these tools in the multicolour asymmetric setting. We use these new senders to prove far-reaching generalisations of several classical results in the area

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Neural function approximation on graphs: shape modelling, graph discrimination & compression

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    Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Fundamentals

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    Volume 1 establishes the foundations of this new field. It goes through all the steps from data collection, their summary and clustering, to different aspects of resource-aware learning, i.e., hardware, memory, energy, and communication awareness. Machine learning methods are inspected with respect to resource requirements and how to enhance scalability on diverse computing architectures ranging from embedded systems to large computing clusters

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    The Lov\'asz-Cherkassky theorem in infinite graphs

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    Infinite generalizations of theorems in finite combinatorics were initiated by Erd\H{o}s due to his famous Erd\H{o}s-Menger conjecture (now known as the Aharoni-Berger theorem) that extends Menger's theorem to infinite graphs in a structural way. We prove a generalization of this manner of the classical result about packing edge-disjoint T T -paths in an ``inner Eulerian'' setting obtained by Lov\'asz and Cherkassky independently in the '70s

    Towards the Erd\H{o}s-Gallai Cycle Decomposition Conjecture

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    In the 1960's, Erd\H{o}s and Gallai conjectured that the edges of any nn-vertex graph can be decomposed into O(n)O(n) cycles and edges. We improve upon the previous best bound of O(nloglogn)O(n\log\log n) cycles and edges due to Conlon, Fox and Sudakov, by showing an nn-vertex graph can always be decomposed into O(nlogn)O(n\log^{*}n) cycles and edges, where logn\log^{*}n is the iterated logarithm function.Comment: Final version, accepted for publicatio

    Side-Channel Analysis and Cryptography Engineering : Getting OpenSSL Closer to Constant-Time

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    As side-channel attacks reached general purpose PCs and started to be more practical for attackers to exploit, OpenSSL adopted in 2005 a flagging mechanism to protect against SCA. The opt-in mechanism allows to flag secret values, such as keys, with the BN_FLG_CONSTTIME flag. Whenever a flag is checked and detected, the library changes its execution flow to SCA-secure functions that are slower but safer, protecting these secret values from being leaked. This mechanism favors performance over security, it is error-prone, and is obscure for most library developers, increasing the potential for side-channel vulnerabilities. This dissertation presents an extensive side-channel analysis of OpenSSL and criticizes its fragile flagging mechanism. This analysis reveals several flaws affecting the library resulting in multiple side-channel attacks, improved cache-timing attack techniques, and a new side channel vector. The first part of this dissertation introduces the main topic and the necessary related work, including the microarchitecture, the cache hierarchy, and attack techniques; then it presents a brief troubled history of side-channel attacks and defenses in OpenSSL, setting the stage for the related publications. This dissertation includes seven original publications contributing to the area of side-channel analysis, microarchitecture timing attacks, and applied cryptography. From an SCA perspective, the results identify several vulnerabilities and flaws enabling protocol-level attacks on RSA, DSA, and ECDSA, in addition to full SCA of the SM2 cryptosystem. With respect to microarchitecture timing attacks, the dissertation presents a new side-channel vector due to port contention in the CPU execution units. And finally, on the applied cryptography front, OpenSSL now enjoys a revamped code base securing several cryptosystems against SCA, favoring a secure-by-default protection against side-channel attacks, instead of the insecure opt-in flagging mechanism provided by the fragile BN_FLG_CONSTTIME flag
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