Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Property Testing in Bounded Degree Graphs

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far'' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/? , always accept the graph when it has the tested property, and reject with high probability if the graph is ? -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ? dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

Distribution-Free Connectivity Testing for Sparse Graphs

We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area—graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs:

Algorithmic Meta-theorems for Restrictions of Treewidth

Possibly the most famous algorithmic meta-theorem is Courcelle??s theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time??s dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees. We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.     

Normal eulerian clique-covering and hamiltonicity

A connected graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as an NP-complete problem and no satisfactory characterization for these graphs has been found.Since the seminal work of Dirac in 1952 many sufficient conditions were found. In 1974, Goodman and Hedetniemi gave such a condition based on the existence of a clique-covering of the graph. This condition was recently generalized using the notion of eulerian clique-covering. In addition, an algorithm able to find a normal eulerian clique-covering for a large class of graphs was also introduced. A normal clique-covering has additional properties, making the search for such a covering easier than in the general case.In this article, we prove several properties of normal clique-coverings. In particular we prove that there exists an eulerian clique-covering of a graph if and only if there exists a normal one. Using this result, the search for an eulerian clique-covering as a sufficient condition for hamiltonicity can be reduced to the normal case.     

A random walk model for infection on graphs: spread of epidemics &; rumours with mobile agents

We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end, we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. More precisely, we prove that for homogeneous graphs including regular graphs and the classical Erdős–Rényi model, the coincidence time is inversely proportional to the number of nodes in the graph. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution. We finally undertake a preliminary analysis for the case with k random walkers and provide upper bounds on the convergence time for both the complete graph and regular graphs.     

A characterization of span program size and improved lower bounds for monotone span programs

We give a characterization of span program size by a combinatorial-algebraic measure. The measure we consider is a generalization of a measure on covers which has been used to prove lower bounds on formula size and has also been studied with respect to communication complexity.?In the monotone case our new methods yield lower bounds for the monotone span program complexity of explicit Boolean functions in n variables over arbitrary fields, improving the previous lower bounds on monotone span program size. Our characterization of span program size implies that any matrix with superpolynomial separation between its rank and cover number can be used to obtain superpolynomial lower bounds on monotone span program size. We also identify a property of bipartite graphs that is suficient for constructing Boolean functions with large monotone span program complexity. Received: September 30, 2000.     

A general framework for types in graph rewriting

We investigate a general framework which can be instantiated in order to obtain type systems for graph rewriting, allowing us to statically infer behavioural properties of a graph. We describe conditions such as the subject reduction property and compositionality that should be satisfied by such a framework. We present a methodology for proving these conditions, specifically we prove that it is sufficient to show properties that are local to graph transformation rules. In order to show the applicability of this framework, we describe in several case studies how to integrate existing type systems (for the π-calculus and the λ-calculus) and a system for typing acyclic graphs. This is a completely revised and extended version of a paper of which an earlier version has appeared in FSTTCS '00.     

Predicting metamorphic relations for testing scientific software: a machine learning approach using graph kernels

Comprehensive, automated software testing requires an oracle to check whether the output produced by a test case matches the expected behaviour of the programme. But the challenges in creating suitable oracles limit the ability to perform automated testing in some programmes, and especially in scientific software. Metamorphic testing is a method for automating the testing process for programmes without test oracles. This technique operates by checking whether the programme behaves according to properties called metamorphic relations. A metamorphic relation describes the change in output when the input is changed in a prescribed way. Unfortunately, finding the metamorphic relations satisfied by a programme or function remains a labour‐intensive task, which is generally performed by a domain expert or a programmer. In this work, we propose a machine learning approach for predicting metamorphic relations that uses a graph‐based representation of a programme to represent control flow and data dependency information. In earlier work, we found that simple features derived from such graphs provide good performance. An analysis of the features used in this earlier work led us to explore the effectiveness of several representations of those graphs using the machine learning framework of graph kernels, which provide various ways of measuring similarity between graphs. Our results show that a graph kernel that evaluates the contribution of all paths in the graph has the best accuracy and that control flow information is more useful than data dependency information. The data used in this study are available for download at http://www.cs.colostate.edu/saxs/MRpred/functions.tar.gz to help researchers in further development of metamorphic relation prediction methods. Copyright © 2015 John Wiley & Sons, Ltd.     

The Monadic Quantifier Alternation Hierarchy over Grids and Graphs

The monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first-order closure of existential monadic second-order logic. We also verify that these graph properties even belong to the complexity class NLOG, which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy.     

Graph Complexity and Slice Functions

   Abstract. A graph-theoretic approach to study the complexity of Boolean functions was initiated by Pudlák, R?dl, and Savicky [PRS] by defining models of computation on graphs. These models generalize well-known models of Boolean complexity such as circuits, branching programs, and two-party communication complexity. A Boolean function f is called a 2-slice function if it evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's f may be nontrivially defined. There is a natural correspondence between 2-slice functions and graphs. Using the framework of graph complexity, we show that sufficiently strong superlinear monotone lower bounds for the very special class of {2-slice functions} would imply superpolynomial lower bounds over a complete basis for certain functions derived from them. We prove, for instance, that a lower bound of n ^1+Ω(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 ^Ω(ℓ) lower bound on the formula size over a complete basis of another explicit function g on l variables, where l=Θ( log n) . We also consider lower bound questions for depth-3 bipartite graph complexity. We prove a weak lower bound on this measure using algebraic methods. For instance, our result gives a lower bound of Ω(( log n) ³ / ( log log n) ⁵ ) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. Lower bounds for depth-3 bipartite graph complexity are motivated by two significant applications: (i) a lower bound of n ^Ω(1) on the depth-3 complexity of an explicit n -vertex bipartite graph would yield superlinear size lower bounds on log-depth Boolean circuits for an explicit function, and (ii) a lower bound of

Formal testing of timed graph transformation systems using metric temporal graph logic

Embedded real-time systems generate state sequences where time elapses between state changes. Ensuring that such systems adhere to a provided specification of admissible or desired behavior is essential. Formal model-based testing is often a suitable cost-effective approach. We introduce an extended version of the formalism of symbolic graphs, which encompasses types as well as attributes, for representing states of dynamic systems. Relying on this extension of symbolic graphs, we present a novel formalism of timed graph transformation systems (TGTSs) that supports the model-based development of dynamic real-time systems at an abstract level where possible state changes and delays are specified by graph transformation rules. We then introduce an extended form of the metric temporal graph logic (MTGL) with increased expressiveness to improve the applicability of MTGL for the specification of timed graph sequences generated by a TGTS. Based on the metric temporal operators of MTGL and its built-in graph binding mechanics, we express properties on the structure and attributes of graphs as well as on the occurrence of graphs over time that are related by their inner structure. We provide formal support for checking whether a single generated timed graph sequence adheres to a provided MTGL specification. Relying on this logical foundation, we develop a testing framework for TGTSs that are specified using MTGL. Lastly, we apply this testing framework to a running example by using our prototypical implementation in the tool AutoGraph.

     

Space-Efficient Counting in Graphs on Surfaces

We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2^k, for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. The techniques used are algebraic topological that may be useful in many other problems involving planar or higher genus graphs – such as higher genus graph recognition in Logspace. In the spirit of counting problems modulo 2^k, we also exhibit a highly parallel ?L\oplus {\bf L} algorithm for finding the value of a permanent modulo 2^k. Previously, the best known result in this direction was Valiant’s result that this problem lies in P. We also show that we can count the number of perfect matchings modulo 2^k in an arbitrary graph in P. This extends Valiant’s result for the permanent, since the Permanent may be modeled as counting the number of perfect matchings in bipartite graphs.     

Graph theoretic closure properties of the family of boundary NLC graph languages

Summary Node label controlled (NLC) grammars are graph grammars (operating on node labeled undirected graphs) which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only. They define (possibly infinite) languages of undirected node labeled graphs (or, if we just omit the labels, languages of unlabeled graphs). Boundary NLC (BNLC) grammars are NLC grammars with the property that whenever — in a graph already generated — two nodes may be rewritten, then these nodes are not adjacent. The graph languages generated by this type of grammars are called BNLC languages. The present paper continues the investigations of basic properties of BNLC grammars and languages where the central question is the following: If L is a BNLC language and P is a graph theoretic property, is the set of all graphs from L satisfying P again a BNLC language? We demonstrate that the class of BNLC languages is very stable in the sense that for almost all properties we consider the resulting languages are BNLC. In particular, the above question gets an affirmative answer, if the property P is: being k-colorable, being connected, having a subgraph homeomorphic to a given graph, and being nonplanar.This research was carried out during the second author's stay at the Rijksuniversiteit Leiden, The Netherlands     

Support measures for graph data*

The concept of support is central to data mining. While the definition of support in transaction databases is intuitive and simple, that is not the case in graph datasets and databases. Most mining algorithms require the support of a pattern to be no greater than that of its subpatterns, a property called anti-monotonicity, or admissibility. This paper examines the requirements for admissibility of a support measure. Support measures for mining graphs are usually based on the notion of an instance graph---a graph representing all the instances of the pattern in a database and their intersection properties. Necessary and sufficient conditions for support measure admissibility, based on operations on instance graphs, are developed and proved. The sufficient conditions are used to prove admissibility of one support measure—the size of the independent set in the instance graph. Conversely, the necessary conditions are used to quickly show that some other support measures, such as weighted count of instances, are not admissible. ^*Partially supported by the KITE consortium under contract to the Israeli Ministry of Trade and Industry, and by the Paul Ivanier Center for Robotics and Production Management.     

The lexicographically first maximal subgraph problems:P-completeness andNC algorithms

The lexicographically first maximal (lfm) subgraph problem for a property is to compute the lfm vertex set whose induced subgraph satisfies . The main contribution of this paper is theP-completeness of the lfm subgraph problem for any nontrivial hereditary property. We also observe that most of the lfm subgraph problems are stillP-complete even if the instances are restricted to graphs with degree 3. However, some exceptions are found. For example, it is shown that the lfm 4-cycle free subgraph problem is inNC ² for graphs with degree 3 but turns out to beP-complete for graphs with degree 4. Further, we analyze the complexity of the lfmedge-induced subgraph problem for some graph properties and show that it has a different complexity feature.This work was done while the author visited Universität-GH-Paderborn and a part of this paper was presented at the 14th ICALP, Karlsruhe, 1987.     

Gibbs and Markov properties of graphs

Directed acyclic graphs (DAG's) and, more generally, chain graphs have in recent years been widely used for statistical modelling. Their Gibbs and Markov properties are now well understood and are exploited, e.g., in reducing the complexity encountered in estimating the joint distribution of many random variables. The scope of the models has been restricted to acyclic or recursive processes and this restriction was long considered imperative, due to the supposed fundamentally different nature of processes involving reciprocal interactions between variables. Recently however it was shown independently by Spirtes (Spirtes, 1995) and Koster (Koster, 1996) that graphs containing directed cycles may be given a proper Markov interpretation. This paper further generalizes the scope of graphical models. It studies a class of conditional independence (CI) probability models determined by a general graph which may have directed and undirected edges, and may contain directed cycles. This class of graphical models strictly includes the well-known class of graphical chain models studied by Frydenberg et al., and the class of probability models determined by a directed cyclic graph or a reciprocal graph, studied recently by Spirtes and Koster. It is shown that the Markov property determined by a graph is equivalent to the existence of a Gibbs-factorization of the density (assumed positive). To better understand the structural aspects of the Gibbs and Markov properties embodied by graphs the notion of lattice conditional independence (LCI), introduced by Andersson and Perlman (Andersson and Perlman, 1993), is needed. The Gibbs-factorization has an outer ‘skeleton’ which is determined by the ring of all anterior sets of the graph. This revised version was published online in June 2006 with corrections to the Cover Date.     

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

We explore three important avenues of research in algorithmic graph-minor theory, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties. First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K _3,k -minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixed-parameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. Third, we disprove a variation of Wagner’s Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.     

The parameterized complexity of some minimum label problems

We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π, and we are asked to find a subset of edges satisfying property Π with respect to G that uses the minimum number of labels. These problems have a lot of applications in networking. We show that all the problems under consideration are W[2]-hard when parameterized by the number of used labels, and that they remain W[2]-hard even on graphs whose pathwidth is bounded above by a small constant. On the positive side, we prove that most of these problems are FPT when parameterized by the solution size, that is, the size of the sought edge set. For example, we show that computing a maximum matching or an edge dominating set that uses the minimum number of labels, is FPT when parameterized by the solution size. Proving that some of these problems are FPT requires interesting algorithmic methods that we develop in this paper.     

Uniquely Restricted Matchings

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs. Received April 12, 1998; revised June 21, 1999.