2,672 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

    Covering and Separation for Permutations and Graphs

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    This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph

    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 Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, 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·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², 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

    From coordinate subspaces over finite fields to ideal multipartite uniform clutters

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    Take a prime power qq, an integer n2n\geq 2, and a coordinate subspace SGF(q)nS\subseteq GF(q)^n over the Galois field GF(q)GF(q). One can associate with SS an nn-partite nn-uniform clutter C\mathcal{C}, where every part has size qq and there is a bijection between the vectors in SS and the members of C\mathcal{C}. In this paper, we determine when the clutter C\mathcal{C} is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether qq is 2,42,4, a higher power of 22, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of C\mathcal{C} depends solely on the underlying matroid of SS. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and τ=2\tau=2 Conjectures for this class of clutters.Comment: 32 pages, 6 figure

    A BIM - GIS Integrated Information Model Using Semantic Web and RDF Graph Databases

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    In recent years, 3D virtual indoor and outdoor urban modelling has become an essential geospatial information framework for civil and engineering applications such as emergency response, evacuation planning, and facility management. Building multi-sourced and multi-scale 3D urban models are in high demand among architects, engineers, and construction professionals to achieve these tasks and provide relevant information to decision support systems. Spatial modelling technologies such as Building Information Modelling (BIM) and Geographical Information Systems (GIS) are frequently used to meet such high demands. However, sharing data and information between these two domains is still challenging. At the same time, the semantic or syntactic strategies for inter-communication between BIM and GIS do not fully provide rich semantic and geometric information exchange of BIM into GIS or vice-versa. This research study proposes a novel approach for integrating BIM and GIS using semantic web technologies and Resources Description Framework (RDF) graph databases. The suggested solution's originality and novelty come from combining the advantages of integrating BIM and GIS models into a semantically unified data model using a semantic framework and ontology engineering approaches. The new model will be named Integrated Geospatial Information Model (IGIM). It is constructed through three stages. The first stage requires BIMRDF and GISRDF graphs generation from BIM and GIS datasets. Then graph integration from BIM and GIS semantic models creates IGIMRDF. Lastly, the information from IGIMRDF unified graph is filtered using a graph query language and graph data analytics tools. The linkage between BIMRDF and GISRDF is completed through SPARQL endpoints defined by queries using elements and entity classes with similar or complementary information from properties, relationships, and geometries from an ontology-matching process during model construction. The resulting model (or sub-model) can be managed in a graph database system and used in the backend as a data-tier serving web services feeding a front-tier domain-oriented application. A case study was designed, developed, and tested using the semantic integrated information model for validating the newly proposed solution, architecture, and performance

    Parameterized Complexity of Binary CSP: Vertex Cover, Treedepth, and Related Parameters

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    We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: - Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d+1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. - We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. - We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ? S[t] ? A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems

    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

    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

    Mining Butterflies in Streaming Graphs

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    This thesis introduces two main-memory systems sGrapp and sGradd for performing the fundamental analytic tasks of biclique counting and concept drift detection over a streaming graph. A data-driven heuristic is used to architect the systems. To this end, initially, the growth patterns of bipartite streaming graphs are mined and the emergence principles of streaming motifs are discovered. Next, the discovered principles are (a) explained by a graph generator called sGrow; and (b) utilized to establish the requirements for efficient, effective, explainable, and interpretable management and processing of streams. sGrow is used to benchmark stream analytics, particularly in the case of concept drift detection. sGrow displays robust realization of streaming growth patterns independent of initial conditions, scale and temporal characteristics, and model configurations. Extensive evaluations confirm the simultaneous effectiveness and efficiency of sGrapp and sGradd. sGrapp achieves mean absolute percentage error up to 0.05/0.14 for the cumulative butterfly count in streaming graphs with uniform/non-uniform temporal distribution and a processing throughput of 1.5 million data records per second. The throughput and estimation error of sGrapp are 160x higher and 0.02x lower than baselines. sGradd demonstrates an improving performance over time, achieves zero false detection rates when there is not any drift and when drift is already detected, and detects sequential drifts in zero to a few seconds after their occurrence regardless of drift intervals
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