436 research outputs found

    A Unifying Theory for Graph Transformation

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    The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Space-Query Tradeoffs in Range Subgraph Counting and Listing

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    This paper initializes the study of range subgraph counting and range subgraph listing, both of which are motivated by the significant demands in practice to perform graph analytics on subgraphs pertinent to only selected, as opposed to all, vertices. In the first problem, there is an undirected graph G where each vertex carries a real-valued attribute. Given an interval q and a pattern Q, a query counts the number of occurrences of Q in the subgraph of G induced by the vertices whose attributes fall in q. The second problem has the same setup except that a query needs to enumerate (rather than count) those occurrences with a small delay. In both problems, our goal is to understand the tradeoff between space usage and query cost, or more specifically: (i) given a target on query efficiency, how much pre-computed information about G must we store? (ii) Or conversely, given a budget on space usage, what is the best query time we can hope for? We establish a suite of upper- and lower-bound results on such tradeoffs for various query patterns

    Enumerating Subgraphs of Constant Sizes in External Memory

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    We present an indivisible I/O-efficient algorithm for subgraph enumeration, where the objective is to list all the subgraphs of a massive graph G : = (V, E) that are isomorphic to a pattern graph Q having k = O(1) vertices. Our algorithm performs O((|E|^{k/2})/(M^{{k/2}-1} B) log_{M/B}(|E|/B) + (|E|^?)/(M^{?-1} B) I/Os with high probability, where ? is the fractional edge covering number of Q (it always holds ? ? k/2, regardless of Q), M is the number of words in (internal) memory, and B is the number of words in a disk block. Our solution is optimal in the class of indivisible algorithms for all pattern graphs with ? > k/2. When ? = k/2, our algorithm is still optimal as long as M/B ? (|E|/B)^? for any constant ? > 0

    Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

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    Two graphs GG and HH are homomorphism indistinguishable over a class of graphs F\mathcal{F} if for all graphs F∈FF \in \mathcal{F} the number of homomorphisms from FF to GG is equal to the number of homomorphisms from FF to HH. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various question raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.Comment: 26 pages, 1 figure, 1 tabl

    Improving Expressivity of Graph Neural Networks using Localization

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    In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of k−k-WL for any kk. We analyze the power of Local k−k-WL and prove that it is more expressive than k−k-WL and at most as expressive as (k+1)−(k+1)-WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local k−k-WL equivalent. We also introduce two variants of k−k-WL: Layer k−k-WL and recursive k−k-WL. These methods are more time and space efficient than applying k−k-WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just 1−1-WL. The same idea can be extended further for larger patterns using k>1k>1. We also compare the expressive power of Local k−k-WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a]

    Constant-Depth Circuits vs. Monotone Circuits

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    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

    The Weisfeiler-Leman Dimension of Existential Conjunctive Queries

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    The Weisfeiler-Leman (WL) dimension of a graph parameter ff is the minimum kk such that, if G1G_1 and G2G_2 are indistinguishable by the kk-dimensional WL-algorithm then f(G1)=f(G2)f(G_1)=f(G_2). The WL-dimension of ff is ∞\infty if no such kk exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ\varphi, we quantify the WL-dimension of the function that maps every graph GG to the number of answers of φ\varphi in GG. The works of Dvor\'ak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ\varphi, the WL-dimension is equal to the treewidth of the Gaifman graph of φ\varphi. In this work, we give a characterisation that applies to all conjunctive qureies. Given any conjunctive query φ\varphi, we prove that its WL-dimension is equal to the semantic extension width sew(φ)\mathsf{sew}(\varphi), a novel width measure that can be thought of as a combination of the treewidth of φ\varphi and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ\varphi are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ\varphi cannot be computed by GNNs of order smaller than sew(φ)\mathsf{sew}(\varphi).Comment: 36 pages, 4 figures, abstract shortened due to ArXiv requirement
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