Graph Automorphism

Two vertices of a graph are said to be similar if there exists a graph automorphism mapping one of them into the other. Procedures aiming to separate vertices of a graph into equivalence classes accordingly to their similarities are the basis of many isomorphism-testing algorithms. In practice, these procedures efficiently reduce the search space of backtracking-based techniques. We present a new refinement procedure that associates to each pair of vertices the value of an arbitrary predefined invariant. We also propose such a new invariant that we call the least colored shortest path.

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism and isomorphism problems GA and GI, for the directed graph reachability problem GAP, for the determinant function det, and for logspace self-reducible languages we establish the following results:

The problem of lifting graph automorphisms along covering projections is considered in a purely combinatorial setting. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful reexamination of the whole subject, which leads to simpli cation and generalization of several known results. For example, Cayley graphs, including those with involutory or repeated generators, are characterized as regular coverings over one-vertex graphs. The ordinary and the permutation voltage constructions are uni ed into the concept of a voltage space, constituting the crucial tool for combinatorialization of the lifting problem. Particular attention is paid to the structure of lifted groups, with focus on split extensions having a nice geometrical and combinatorial description. Some applications of these results to regular maps on surfaces are given.

A graph is called a semi–regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi–regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation–preserving automorphisms of maps on surfaces (orientable and non–orientable) or just orientable surfaces with a given underlying semi–regular graph Γ are determined. Formulas for the numbers of non–equivalent embeddings of this kind of graphs on surfaces (orientable, non–orientable or both) are established, and especially, the non–equivalent embeddings of circulant graphs of a prime order on orientable, non–orientable and general surfaces are enumerated.

Automatic symmetry detection has received a signican t amount of interest, which has re- sulted in a large number of proposed methods. This paper reports on our experiences while im- plementing the approach of (9). In particular, it proposes a modication of the approach to deal with general expressions, discusses the in- sights gained, and gives the results of a

In this paper we demonstrate a potential extension of formal verification methodology in order to deal with time-domain properties of analog and mixed-signal circuits whose dynamic behavior is described by differential algebraic equations. To model and analyze such circuits under all possible input signals and all values of parameters, we build upon two techniques developed in the context of hybrid (discrete-continuous) control systems. First, we extend our algorithm for approximating sets of reachable sets for dense-time continuous systems to deal with differential algebraic equations (DAEs) and apply it to a biquad low-pass filter. To analyze more complex circuits, we resort to bounded horizon verification. We use optimal control techniques to check whether a Δ-Σ modulator, modeled as a discrete-time hybrid automaton, admits an input sequence of bounded length that drives it to saturation.

The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G = K 2 , K 3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [1] on powers of prime graphs, and results of Klavžar and Zhu . More generally, we also prove that d(G H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2 |H| − |H|. Under additional conditions similar results hold for strong and direct products.

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with nontrivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.

Ontology classification—the computation of subsumption hierarchies for classes and properties—is one of the most important tasks for OWL reasoners. Based on the algorithm by Shearer and Horrocks [9], we present a new classification procedure that addresses several open issues of the original algorithm, and that uses several novel optimisations in order to achieve superior performance. We also consider the classification of (object and data) properties. We show that algorithms commonly used to implement that task are incomplete ...

The problem of lifting graph automorphisms along covering projections is considered in a purely combinatorial setting. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful reexamination of the whole subject, which leads to simpli cation and generalization of several known results. For example, Cayley graphs, including those with involutory or repeated generators, are characterized as regular coverings over one-vertex graphs. The ordinary and the permutation voltage constructions are uni ed into the concept of a voltage space, constituting the crucial tool for combinatorialization of the lifting problem. Particular attention is paid to the structure of lifted groups, with focus on split extensions having a nice geometrical and combinatorial description. Some applications of these results to regular maps on surfaces are given. 1 Supported in part by \Ministrstvo za znanost in tehnologijo Slovenije", proj. no. P1-0208-1010-94. 2 Supported in part by VEGA, grant no. 1/331/96. 3 Supported in part by VEGA, garnt no. 1/331/96.

Research in algorithms for Boolean satisfiability and their implementations [23, 6] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [10] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve because of their large size, but are still solved in minutes [25]. Yet, small and difficult SAT instances must exist because Boolean satisfiability is NP-complete. We propose an improved construction of symmetry-breaking clauses [9] and apply it to achieve significant speed-ups over current state-ofthe-art in Boolean satisfiability. Our techniques are formulated as preprocessing and can be applied to any SAT solver without changing its source code. We also show that considerations of symmetry may lead to more efficient reductions to SAT in the routing domain. Our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from routing problems. Using an efficient implementation to solve the graph automorphism problem [18, 20, 22], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.

Many symmetry breaking techniques assume that the symmetries of a CSP are given as input in addition to the CSP itself. We present a method that can be used to detect all the symmetries of a CSP. This method constructs a graph that has the same symmetries as the CSP. Then, generators for the symmetry group are computed using a graph automorphism algorithm. This method improves and extends previous work in order to cover global constraints, arithmetic expressions and value symmetries. We show that this method is able to find symmetries for examples that were thought to be too convoluted for automated detection. We also show that the overhead of symmetry detection is quite negligible, even on very large instances. We present a comprehensive set of examples where automated symmetry detection is coupled with symmetry breaking techniques.

This paper is motivated by the theory of sequential dynamical systems (SDS), developed as a basis for a mathematical theory of computer simulation. A sequential dynamical system is a collection of symmetric Boolean local update functions, with the update order determined by a permutation of the Boolean variables. In this paper, the notion of SDS is generalized to allow arbitrary functions over a general finite field, with the update schedule given by an arbitrary word on the variables. The paper contains generalizations of some of the known results about SDS with permutation update schedules. In particular, an upper bound on the number of different SDS over words of a given length is proved and open problems are discussed.

Graph transduction is a popular class of semi-supervised learning techniques, which aims to estimate a classification function defined over a graph of labeled and unlabeled data points. The general idea is to propagate the provided label information to unlabeled nodes in a consistent way. In contrast to the traditional view, in which the process of label propagation is defined as a graph Laplacian regularization, here we propose a radically different perspective that is based on game-theoretic notions. Within our framework, the transduction problem is formulated in terms of a non-cooperative multi-player game where any equilibrium of the proposed game corresponds to a consistent labeling of the data. An attractive feature of our formulation is that it is inherently a multi-class approach and imposes no constraint whatsoever on the structure of the pairwise similarity matrix, being able to naturally deal with asymmetric and negative similarities alike. We evaluated our approach on some real-world problems involving symmetric or asymmetric similarities and obtained competitive results against state-of-the-art algorithms.

In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function which can restrict the output of f (x) to one of b(n) possible values. This paper investigates #GA, the function which computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI.

In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function which can restrict the output of f (x) to one of b(n) possible values. This paper investigates #GA, the function which computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI.

Let $G$ be a finite simple graph of order $n$, maximum degree $\Delta$, and minimum degree $\delta$. A compact regularization of $G$ is a $\Delta$-regular graph $H$ of which $G$ is an induced subgraph: $H$ is symmetric if every automorphism of $G$ can be extended to an automorphism of $H$. The index $|H:G|$ of a regularization $H$ of $G$ is the ratio $|V(H)|/|V(G)|$. Let $\mbox{mcr}(G)$ denote the index of a minimum compact regularization of $G$ and let $\mbox{mcsr}(G)$ denote the index of a minimum compact symmetric regularization of $G$.Erdős and Kelly proved that every graph $G$ has a compact regularization and $\mbox{mcr}(G) \leq 2$. Building on a result of König, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and $\mbox{mcsr}(G) \leq 2^{\Delta - \delta}$.  Using a partial Cartesian product construction, we improve this to $\mbox{mcsr}(G) \leq \Delta - \delta + 2$ and give examples to show this bound cannot be reduced below $\Delta - \delta ...

Ontology classification—the computation of subsumption hierarchies for classes and properties—is one of the most important tasks for OWL reasoners. Based on the algorithm by Shearer and Horrocks [9], we present a new classification procedure that addresses several open issues of the original algorithm, and that uses several novel optimisations in order to achieve superior performance. We also consider the classification of (object and data) properties. We show that algorithms commonly used to implement that task are incomplete ...

Ontology classification—the computation of subsumption hierarchies for classes and properties—is one of the most important tasks for OWL reasoners. Based on the algorithm by Shearer and Horrocks [9], we present a new classification procedure that addresses several open issues of the original algorithm, and that uses several novel optimisations in order to achieve superior performance. We also consider the classification of (object and data) properties. We show that algorithms commonly used to implement that task are incomplete ...

We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, P-selective sets and membership comparable sets. For the graph automorphism and isomorphism problems GA and GI, for the directed graph reachability problem GAP, for the determinant function det, and for logspace self-reducible languages we establish the following results: 1. If GA is ≤ p tt-reducible to a P-selective set, then GA ∈ P. 2. If GI is O(log n)-membership comparable, then GI ∈ RP. 3. If GAP is logspace O(1)-membership comparable, then GAP ∈ L. 4. If det is ≤ log T-reducible to an L-selective set, then det ∈ FL. 5. If A is logspace self-reducible and ≤ log T-reducible to an L-selective set, then A ∈ L. The last result is a strong logspace version of the characterisation of P as the class of self-reducible P-selective languages. As P and NL have logspace self-reducible complete sets, it also establishes a logspace analogue of the conjecture that if SAT is ≤ p T-reducible to a P-selective set, then SAT ∈ P.

We show that the graph isomorphism problem is low for PP and for C = P, i.e., it does not provide a PP or C = P computation with any additional power when used as an oracle. Furthermore, we show that graph isomorphism belongs to the class LWPP (see Fenner, Fortnow, Kurtz [12]). A similar result holds for the (apparently more difficult) problem Group Factorization. The problem of determining whether a given graph has a nontrivial automorphism, Graph Automorphism, is shown to be in SPP, and is therefore low for PP, C = P, and Mod k P, k ≥ 2.

This paper is motivated by the theory of sequential dynamical systems (SDS), developed as a basis for a mathematical theory of computer simulation. A sequential dynamical system is a collection of symmetric Boolean local update functions, with the update order determined by a ...

Boolean satisfiability (SAT) solvers have experienced dramatic improvements in their performance and scalability over the last several years [5, 7] and are now routinely used in diverse EDA applications. Nevertheless, a number of practical SAT instances remain difficult to solve [9] and continue to defy even the best available SAT solvers [5, 7]. Recent work pointed out that symmetries in the Boolean search space are often to blame. A theoretical framework for detecting and breaking such symmetries was introduced in [2]. This framework was subsequently extended, refined, and empirically shown to yield significant speed-ups for a large number of benchmark classes in [1]. Symmetries in the search space are broken by adding appropriate symmetry-breaking predicates (SBPs) to a SAT instance in conjunctive normal form (CNF). The SBPs prune the search space by acting as a filter that confines the search to non-symmetric regions of the space without affecting the satisfiability of the CNF formula. For symmetry breaking to be effective in practice, the computational overhead of generating and manipulating the SBPs must be significantly less than the run time savings they yield due to search space pruning. In this paper we present several new constructions of SBPs that improve on previous work. Specifically, we give a linearsized CNF formula that selects lex-leaders (among others) for single permutations. We also show how that formula can be simplified by taking advantage of the sparsity of permutations. We test these improvements against earlier constructions and show that they yield smaller SBPs and lead to run time reductions on many benchmarks.

A systematic approach is developed for enumerating congruence classes of 2-cell imbeddings of connected graphs on closed orientable 2-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration.

A systematic approach is developed for enumerating congruence classes of 2-cell imbeddings of connected graphs on closed orientable 2-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration.

This paper reports on an on-going project to investigate techniques to diagnose complex dynamical systems that are modeled as hybrid systems. In particular, we examine continuous systems with embedded supervisory controllers that experience abrupt, partial or full failure of component devices. We cast the diagnosis problem as a model selection problem. To reduce the space of potential models under consideration, we exploit techniques from qualitative reasoning to conjecture an initial set of qualitative candidate diagnoses, which induce a smaller set of models. We refine these diagnoses using parameter estimation and model fitting techniques. As a motivating case study, we have examined the problem of diagnosing NASA’s Sprint AERCam, a small spherical robotic camera unit with 12 thrusters that enable both linear and rotational motion.

Contextual ontologies are ontologies that characterize a concept by a set of properties that vary according to context. Contextual ontologies are now crucial for users who intend to exchange information in a domain. Existing ontology languages are not capable of defining such type of ontologies. The objective of this paper is to formally define a contextual ontology language to support the development of contextual ontologies. In this paper, we use description logics as an ontology language and then we extend it by introducing a new contextual constructor.

We study Facility Location games, where a number of facilities are placed in a metric space based on locations reported by strategic agents. A mechanism maps the agents’ locations to a set of facilities. The agents seek to minimize their connection cost, namely the distance of their true location to the nearest facility, and may misreport their location. We are interested in mechanisms that are strategyproof, i.e., ensure that no agent can benefit from misreporting her location, do not resort to monetary transfers, and approximate the optimal social cost. We focus on the closely related problems of k-Facility Location and Facility Location with a uniform facility opening cost, and mostly study winner-imposing mechanisms, which allocate facilities to the agents and require that each agent allocated a facility should connect to it. We show that the winner-imposing version of the Proportional Mechanism (Lu et al., EC ’10) is stategyproof and 4k-approximate for the k-Facility Location game. For the Facility Location game, we show that the winner-imposing version of the randomized algorithm of (Meyerson, FOCS ’01), which has an approximation ratio of 8, is strategyproof. Furthermore, we present a deterministic non-imposing group strategyproof O(logn)-approximate mechanism for the Facility Location game on the line.

In this paper we demonstrate a potential extension of formal verification methodology in order to deal with time-domain properties of analog and mixed-signal circuits whose dynamic behavior is described by differential algebraic equations. To model and analyze such circuits under all possible input signals and all values of parameters, we build upon two techniques developed in the context of hybrid (discrete-continuous) control systems. First, we extend our algorithm for approximating sets of reachable sets for dense-time continuous systems to deal with differential algebraic equations (DAEs) and apply it to a biquad low-pass filter. To analyze more complex circuits, we resort to bounded horizon verification. We use optimal control techniques to check whether a Δ-Σ modulator, modeled as a discrete-time hybrid automaton, admits an input sequence of bounded length that drives it to saturation.

Differential Evolution (DE) is generally considered as a reliable, accurate, robust and fast optimization technique. DE has been successfully applied to solve a wide range of numerical optimization problems. However, the user is required to set the values of the control parameters of DE for each problem. Such parameter tuning is a time consuming task. In this paper, a self-adaptive DE (SDE) is proposed where parameter tuning is not required. The performance of SDE is investigated and compared with other versions of DE. The experiments conducted show that SDE outperformed the other DE versions in all the benchmark functions.

We show that the graph isomorphism problem is low for PP and for C = P, i.e., it does not provide a PP or C = P computation with any additional power when used as an oracle. Furthermore, we show that graph isomorphism belongs to the class LWPP (see Fenner, Fortnow, Kurtz [12]). A similar result holds for the (apparently more difficult) problem Group Factorization. The problem of determining whether a given graph has a nontrivial automorphism, Graph Automorphism, is shown to be in SPP, and is therefore low for PP, C = P, and Mod k P, k ≥ 2.

A novel solution to the problem of virtual craniofacial reconstruction using computer vision, graph theory and geometric constraints is proposed. Virtual craniofacial reconstruction is modeled along the lines of the well-known problem of rigid surface registration. The Iterative Closest Point (ICP) algorithm is first employed with the closest set computation performed using the Maximum Cardinality Minimum Weight (MCMW) bipartite graph matching algorithm. Next, the bounding boxes of the fracture surfaces, treated as cycle graphs, are employed to generate multiple candidate solutions based on the concept of graph automorphism. The best candidate solution is selected by exploiting local and global geometric constraints. Finally, the initialization of the ICP algorithm with the best candidate solution is shown to improve surface reconstruction accuracy and speed of convergence. Experimental results on Computed Tomography (CT) scans of real patients are presented.

Cryptography is one of the prime techniques of secured symbolic data transmission over any communication channel. Security is the most challenging and essential aspects in today's internet and network applications. Thus, design of a secure encryption algorithm is very necessary which can protect the unauthorized attacks. An encryption algorithm is computationally secure if it cannot be intruded with the standard resources. The algorithm proposed here is graph based. Its efficiency surpasses the standard DES algorithm in general. Graphs can be used for designing block ciphers, stream ciphers or public-key ciphers. The algorithm is graph automorphism based partial symmetric key algorithm and it is not fully depended on secret key and produces different cipher text by applying same key on the same plain text.