Graph Modification Research Articles

Bipartite field theories: from D-brane probes to scattering amplitudes

Abstract We introduce and initiate the investigation of a general class of 4d, $\mathcal{N}=1$ quiver gauge theories whose Lagrangian is defined by a bipartite graph on a Riemann surface, with or without boundaries. We refer to such class of theories as Bipartite Field Theories (BFTs). BFTs underlie a wide spectrum of interesting physical systems, including: D3-branes probing toric Calabi-Yau 3-folds, their mirror configurations of D6-branes, cluster integrable systems in (0 + 1) dimensions and leading singularities in scattering amplitudes for $\mathcal{N}=4$ SYM. While our discussion is fully general, we focus on models that are relevant for scattering amplitudes. We investigate the BFT perspective on graph modifications, the emergence of Calabi-Yau manifolds (which arise as the master and moduli spaces of BFTs), the translation between square moves in the graph and Seiberg duality and the identification of dual theories by means of the underlying Calabi-Yaus, the phenomenon of loop reduction and the interpretation of the boundary operator for cells in the positive Grassmannian as higgsing in the BFT. We develop a technique based on generalized Kasteleyn matrices that permits an efficient determination of the Calabi-Yau geometries associated to arbitrary graphs. Our techniques allow us to go beyond the planar limit by both increasing the number of boundaries of the graphs and the genus of the underlying Riemann surface. Our investigation suggests a central role for Calabi-Yau manifolds in the context of leading singularities, whose full scope is yet to be uncovered.

A fundamental approach to model versioning based on graph modifications: from theory to implementation

In model-driven engineering, models are primary artifacts that can evolve heavily during their life cycle. Therefore, versioning of models is a key technique to be offered by integrated development environments for model-driven engineering. In contrast to text-based versioning systems, we present an approach that takes model structures and their changes over time into account. Considering model structures as graphs, we define a fundamental approach where model revisions are considered as graph modifications consisting of delete and insert actions. Two different kinds of conflict detection are presented: (1) the check for operation-based conflicts between different graph modifications, and (2) the check for state-based conflicts on merged graph modifications. For the merging of graph modifications, a two-phase approach is proposed: First, operational conflicts are temporarily resolved by always giving insertion priority over deletion to keep as much information as possible. Thereafter, this tentative merge result is the basis for manual conflict resolution as well as for the application of repair actions that resolve state-based conflicts. If preferred by the user, giving deletion priority over insertion might be one solution. The fundamental concepts are illustrated by versioning scenarios for simplified statecharts. Furthermore, we show an implementation of this fundamental approach to model versioning based on the Eclipse Modeling Framework as technical space.

BOUNDED SEARCH TREE ALGORITHMS FOR PARAMETRIZED COGRAPH DELETION: EFFICIENT BRANCHING RULES BY EXPLOITING STRUCTURES OF SPECIAL GRAPH CLASSES

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (2003) 89–102] and computer-aided branching rules [J. Gramm, J. Guo, F. Hüffner and R. Niedermeier, Automated generation of search tree algorithms for hard graph modification problems, Algorithmica39(4) (2004) 321–347].

On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property ź consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph $H=(V,E\vartriangle F)$ satisfies the property ź. In the źedge-completion problem, the set F is constrained to be disjoint from E; in the źedge-deletion problem, F is a subset of E; no constraint is imposed on F in the źedge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171---176, 1996; Fellows et al. in FCT, pp. 312---321, 2007; Heggernes et al. in STOC, pp. 374---381, 2007). When parameterized by the size k of the set F, it has been proved that if ź is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three ź edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171---176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (Gź,kź) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively (Kratsch and Wahlstrom in IWPEC, pp. 264---275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free edge-deletion and the Cl-free edge-deletion problems for lź7 and lź4 respectively. Indeed, if they exist, then NP⊆coNP/poly.

Dependence of Confounding on the Target Population: A Modification of Causal Graphs to Account for Co-Action

Directed acyclic graphs (DAGs) are useful tools for assessing confounding. However, the basic approach to detecting confounding in DAGs does not distinguish among various forms of interaction between the exposure and covariates and between different target populations for which effects are estimated. We propose a simple modification of DAG rules to overcome this limitation. Using fundamental concepts and DAGs, we show that the basic approach can suggest confounding even when absent if the covariate has no effect in the reference population being used as the comparator for the target population, as occurs when the target population is the exposed and the covariate acts only when the exposure is present. We present three examples that illustrate this scenario and propose a simple revision to the basic approach to detecting confounding in DAGs that makes confounding in the presence of causal interaction easier to evaluate visually. This revision extends to other scenarios involving other target populations and variables with multiple levels. A simple modification of the basic approach to identifying confounding in DAGs allows more frequent exclusion of confounding when it is absent.

Approximation and Tidying—A Problem Kernel for s-Plex Cluster Vertex Deletion

We introduce the s-Plex Cluster Vertex Deletion problem. Like the Cluster Vertex Deletion problem, it is NP-hard and motivated by graph-based data clustering. While the task in Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become cliques, the task in s-Plex Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become s-plexes. An s-plex is a graph in which every vertex is nonadjacent to at most s-1 other vertices; a clique is an 1-plex. In contrast to Cluster Vertex Deletion, s-Plex Cluster Vertex Deletion allows to balance the number of vertex deletions against the sizes and the density of the resulting clusters, which are s-plexes instead of cliques. The focus of this work is the development of provably efficient and effective data reduction rules for s-Plex Cluster Vertex Deletion. In terms of fixed-parameter algorithmics, these yield a so-called problem kernel. A similar problem, s-Plex Editing, where the task is the insertion or the deletion of edges so that the connected components of a graph become s-plexes, has also been studied in terms of fixed-parameter algorithmics. Using the number of allowed graph modifications as parameter, we expect typical parameter values for s-Plex Cluster Vertex Deletion to be significantly lower than for s-Plex Editing, because one vertex deletion can lead to a high number of edge deletions. This holds out the prospect for faster fixed-parameter algorithms for s-Plex Cluster Vertex Deletion.

Nauty in Macaulay2

We introduce a new Macaulay2 package, Nauty, which gives access to powerful methods on graphs provided by the software nauty by Brendan McKay. The primary motivation for accessing nauty is to determine if two graphs are isomorphic. We also implement methods to generate families of graphs restricted in various ways using tools provided with the software nauty. INTRODUCTION. Let G and H be two finite, simple, undirected graphs on the common vertex set V with edge sets E(G) and E(H), respectively. We say that G and H are isomorphic if there is a bijection j from V to itself which preserves edges, that is,fu; vg2 E(G) if and only iffj(u);j(v)g2 E(H). Determining whether two given graphs are isomorphic is known as the Graph Isomorphism problem. When Garey and Johnson wrote their classic book (GJ) on the complexity of algorithms, they specified twelve problems of ambiguous complexity, one of which was the Graph Isomorphism problem. Unfortunately, it is still unknown if the Graph Isomorphism problem is P or NP-complete. Moreover, the problem is of such notoriety that some have even begun referring to a new complexity class, GI, of problems which reduce in polynomial time to the Graph Isomorphism problem (J). Despite this, there exists computer software which is capable of determining whether two graphs are isomorphic in reasonable time. One such piece of software is nauty (N) by McKay. The nauty software is written in highly portable C and is designed to, above all else, compute whether two graphs are isomorphic. It also includes an extensive family of tools, collectively called gtools, to generate systematic modifications of graphs, to generate specific families of graphs, to generate random graphs, to filter a set of graphs for given properties, and to canonically relabel graphs. Most of these features would be beneficial to any computer software that handles graphs. The package EdgeIdeals (FHT) by Francisco, Hoefel, and Van Tuyl implements structures and methods for manipulating graphs (and hypergraphs) within Macaulay2 (M2), a software system by Grayson and Stillman designed to aid in research of commutative algebra and algebraic geometry. We introduce a new package, Nauty, for Macaulay2, which provides an interface with nauty. 1 Most of the aforementioned tools in gtools are accessible through Nauty. In particular, the methods of perhaps the greatest interest are areIsomorphic, filterGraphs, generateGraphs, and generateRandomGraphs. The remainder of this note is broken in to two sections: the first describes briefly the theoretical underpinnings of nauty and the second gives an example session of using Nauty along with a few useful caveats. CANONICAL LABELLINGS. In (M), McKay describes the improved algorithms which he developed to canonically label a graph; these algorithms are the heart of nauty and are summarised in (N). We recall briefly the theoretical ideas which make such algorithms useful.

Publishing anonymous survey rating data

We study the challenges of protecting privacy of individuals in the large public survey rating data in this paper. Recent study shows that personal information in supposedly anonymous movie rating records are de-identified. The survey rating data usually contains both ratings of sensitive and non-sensitive issues. The ratings of sensitive issues involve personal privacy. Even though the survey participants do not reveal any of their ratings, their survey records are potentially identifiable by using information from other public sources. None of the existing anonymisation principles (e.g., k-anonymity, l-diversity, etc.) can effectively prevent such breaches in large survey rating data sets. We tackle the problem by defining a principle called $${(k,\epsilon)}$$ -anonymity model to protect privacy. Intuitively, the principle requires that, for each transaction t in the given survey rating data T, at least (k ? 1) other transactions in T must have ratings similar to t, where the similarity is controlled by $${\epsilon}$$ . The $${(k,\epsilon)}$$ -anonymity model is formulated by its graphical representation and a specific graph-anonymisation problem is studied by adopting graph modification with graph theory. Various cases are analyzed and methods are developed to make the updated graph meet $${(k,\epsilon)}$$ requirements. The methods are applied to two real-life data sets to demonstrate their efficiency and practical utility.

Graph-based data clustering with overlaps

We introduce overlap cluster graph modification problems where, other than in most previous works, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (that is, maximal cliques) that may overlap up to a certain amount specified by the overlap number s . In the case of s -vertex-overlap, each vertex may be part of at most s maximal cliques; s -edge-overlap is analogously defined in terms of edges. We provide a complexity dichotomy (polynomial-time solvable versus NP-hard) for the underlying edge modification problems, develop forbidden subgraph characterizations of “cluster graphs with overlaps”, and study the parameterized complexity in terms of the number of allowed edge modifications, achieving fixed-parameter tractability (in case of constant s -values) and parameterized hardness (in case of unbounded s -values).

Polynomial kernels for 3-leaf power graph modification problems

A graph G = ( V , E ) is a 3-leaf power iff there exists a tree T the leaf set of which is V and such that u v ∈ E iff u and v are at distance at most 3 in T . The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3- leaf power), completion and edge-deletion are FPT when parameterized by the size of the edge set modification. However, polynomial kernels were known for none of these three problems. For each of them, we provide kernels with O ( k 3 ) vertices that can be computed in linear time. We thereby answer an open problem first mentioned by Dom et al. (2004) [8].

An explicit formula for differences between Laplacian-eigenvector components using coalesced graphs

We obtain an explicit formula for the absolute difference between two eigenvector components for a weighted graph’s Laplacian matrix, in terms of the Laplacian’s eigenvalues as well as the eigenvalues of matrices associated with certain coalesced graphs. We then briefly illustrate two uses of this formula, in analyzing graph modifications.

Global and Local Graph Modifiers

We define two modal logics that allow to reason about modifications of graphs. Both have a universal modal operator. The first one only involves global modifications (of some state label, or of some edge label) everywhere in the graph. The second one also allows for modifications that are local to states. The global version generalizes logics of public announcements and public assignments, as well as a logic of preference modification introduced by van Benthem et Liu. By means of reduction axioms we show that it is just as expressive as the underlying logic without global modifiers. We then show that adding local modifiers dramatically increases the power of the logic: the logic of global and local modifiers is undecidable. We finally study its relation with hybrid logic with binder.

On a graph’s security number

A secure set S ⊆ V of graph G = ( V , E ) is a set whose every nonempty subset can be successfully defended from an attack, under appropriate definitions of “attack” and “defended.” The set S is secure when | N [ X ] ∩ S | ≥ | N [ X ] − S | for every X ⊆ S . The smallest cardinality of a secure set in G is the security number of G . New bounds for the security number are established, the effect of some graph modifications on the security number is investigated, and the exact value of the security number for some families of graphs is given.

A brief survey on anonymization techniques for privacy preserving publishing of social network data

Nowadays, partly driven by many Web 2.0 applications, more and more social network data has been made publicly available and analyzed in one way or another. Privacy preserving publishing of social network data becomes a more and more important concern. In this paper, we present a brief yet systematic review of the existing anonymization techniques for privacy preserving publishing of social network data. We identify the new challenges in privacy preserving publishing of social network data comparing to the extensively studied relational case, and examine the possible problem formulation in three important dimensions: privacy, background knowledge, and data utility. We survey the existing anonymization methods for privacy preservation in two categories: clustering-based approaches and graph modification approaches.

Modifying quantum walks: a scattering theory approach

We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering theory. The probability of entering a graph from one tail and leaving from another can be found from the scattering matrix of the graph. We show how the scattering matrix of a graph that is an automorphic image of the original is related to the scattering matrix of the original graph, and we show how the scattering matrix of the reverse graph is related to that of the original graph. Modifications of graphs and the effects of these modifications are then considered. In particular we show how the scattering matrix of a graph is changed if we remove two tails and replace them with an edge or cut an edge and add two tails. This allows us to combine graphs, that is if we connect two graphs we can construct the scattering matrix of the combined graph from those of its parts. Finally, using these techniques, we show how two graphs can be compared by constructing a larger graph in which the two original graphs are in parallel, and performing a quantum walk on the larger graph. This is a kind of quantum walk interferometry.

On bicyclic reflexive graphs

A simple graph is said to be reflexive if the second largest eigenvalue of a ( 0 , 1 ) -adjacency matrix does not exceed 2. We use graph modifications involving Smith trees to construct four classes of maximal bicyclic reflexive graphs.

Gradient Estimate of the Heat Kernel on Modified Graphs

We obtain a condition on the modification of graphs which guarantees the preservation of the Gaussian upper bound for the gradient of the heat kernel.

Speaking Stata: Graphs for all Seasons

Time series showing seasonality—marked variation with time of year—are of interest to many scientists, including climatologists, other environmental scientists, epidemiologists, and economists. The usual graphs plotting response variables against time, or even time of year, are not always the most effective at showing the fine structure of seasonality. I survey various modifications of the usual graphs and other kinds of graphs with a range of examples. Although I introduce here two new Stata commands, cycleplot and sliceplot, I emphasize exploiting standard functions, data management commands, and graph options to get the graphs desired.

Schema versioning in data warehouses: Enabling cross-version querying via schema augmentation

As several mature implementations of data warehousing systems are fully operational, a crucial role in preserving their up-to-dateness is played by the ability to manage the changes that the data warehouse (DW) schema undergoes over time in response to evolving business requirements. In this paper we propose an approach to schema versioning in DWs, where the designer may decide to undertake some actions on old data aimed at increasing the flexibility in formulating cross-version queries, i.e., queries spanning multiple schema versions. First, we introduce a representation of DW schemata as graphs of simple functional dependencies, and discuss its properties. Then, after defining an algebra of schema graph modification operations aimed at creating new schema versions, we discuss how augmented schemata can be introduced to increase flexibility in cross-version querying. Next, we show how a history of versions for DW schemata is managed and discuss the relationship between the temporal horizon spanned by a query and the schema on which it can consistently be formulated.

Cluster graph modification problems

In a clustering problem one has to partition a set of elements into homogeneous and well-separated subsets. From a graph theoretic point of view, a cluster graph is a vertex-disjoint union of cliques. The clustering problem is the task of making the fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that Cluster Editing is NP-complete, Cluster Deletion is NP-hard to approximate to within some constant factor, and Cluster Completion is polynomial. When the desired solution must contain exactly p clusters, we show that Cluster Editing is NP-complete for every p ⩾ 2 ; Cluster Deletion is polynomial for p = 2 but NP-complete for p > 2 ; and Cluster Completion is polynomial for any p. We also give a constant factor approximation algorithm for a variant of Cluster Editing when p = 2 .