Isomorphism Testing Research Articles

Nearly Tight Bounds for Testing Function Isomorphism

We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove the following: (1) For most functions $f:\{0,1\}^n \to \{0,1\}$, the query complexity of testing isomorphism to $f$ is $\Omega(n)$. Moreover, the query complexity of testing isomorphism to most $k$-juntas $f:\{0,1\}^n \to \{0,1\}$ is $\Omega(k)$. (2) Isomorphism to any $k$-junta $f:\{0,1\}^n \to \{0,1\}$ can be tested with $O(k \log k)$ queries. (3) For some $k$-juntas $f:\{0,1\}^n \to \{0,1\}$, testing isomorphism to $f$ with one-sided error requires $\Omega(k\log(n/k))$ queries. In particular, testing whether $f:\{0,1\}^n \to \{0,1\}$ is a $k$-parity with one-sided error requires $\Omega(k\log(n/k))$ queries. (4) The query complexity of testing isomorphism between two unknown functions $f,g:\{0,1\}^n \to \{0,1\}$ is $\widetilde{\Theta}(2^{n/2})$. These bounds are tight up to logarithmic factors, and they significantly strengthen the bounds proved by Fischer, Kindler, Ron, Safra, and Samorodnitsky [J. Comput. System Sci., 68 (2004), pp. 753--787] and Blais and O'Donnell [Proceedings of the IEEE Conference on Computational Complexity, 2010, pp. 235--246]. We also obtain results closely related to isomorphism testing, answering a question posed by Diakonikolas, Lee, Matulef, Onak, Rubinfeld, Servedio, and Wan [Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007, pp. 549--558]: testing whether a function $f:\{0,1\}^n \to \{0,1\}$ can be computed by a circuit of size $\le s$ requires $s^{\Omega(1)}$ queries. All of our lower bounds apply to general (adaptive) testers.

Mining functional subgraphs from cancer protein-protein interaction networks

BackgroundProtein-protein interaction (PPI) networks carry vital information about proteins' functions. Analysis of PPI networks associated with specific disease systems including cancer helps us in the understanding of the complex biology of diseases. Specifically, identification of similar and frequently occurring patterns (network motifs) across PPI networks will provide useful clues to better understand the biology of the diseases.ResultsIn this study, we developed a novel pattern-mining algorithm that detects cancer associated functional subgraphs occurring in multiple cancer PPI networks. We constructed nine cancer PPI networks using differentially expressed genes from the Oncomine dataset. From these networks we discovered frequent patterns that occur in all networks and at different size levels. Patterns are abstracted subgraphs with their nodes replaced by node cluster IDs. By using effective canonical labeling and adopting weighted adjacency matrices, we are able to perform graph isomorphism test in polynomial running time. We use a bottom-up pattern growth approach to search for patterns, which allows us to effectively reduce the search space as pattern sizes grow. Validation of the frequent common patterns using GO semantic similarity showed that the discovered subgraphs scored consistently higher than the randomly generated subgraphs at each size level. We further investigated the cancer relevance of a select set of subgraphs using literature-based evidences.ConclusionFrequent common patterns exist in cancer PPI networks, which can be found through effective pattern mining algorithms. We believe that this work would allow us to identify functionally relevant and coherent subgraphs in cancer networks, which can be advanced to experimental validation to further our understanding of the complex biology of cancer.

Algorithms for classifying regular polytopes with a fixed automorphism group

In this paper, various algorithms used in the classifications of regular polytopes for given groups are compared. First computational times and memory usages are analyzed for the original algorithm used in one of these classifications. Second, a possible algorithm for isomorphism testing among polytopes is suggested. Then, two improved algorithms are compared, and finally, results are given for a new classification of all regular polytopes for certain alternating groups and for the sporadic group $Co_3$.

An efficient heuristic approach to detecting graph isomorphism based on combinations of highly discriminating invariants

The search for an easily computable, finite, complete set of graph invariants remains a challenging research topic. All measures characterizing the topology of a graph that have been developed thus far exhibit some degree of degeneracy, i.e., an inability to distinguish between non-isomorphic graphs. In this paper, we show that certain graph invariants can be useful in substantially reducing the computational complexity of isomorphism testing. Our findings are underpinned by numerical results based on a large scale statistical analysis.

Lindex: a lattice-based index for graph databases

Subgraph querying has wide applications in various fields such as cheminformatics and bioinformatics. Given a query graph, q, a subgraph-querying algorithm retrieves all graphs, D(q), which have q as a subgraph, from a graph database, D. Subgraph querying is costly because it uses subgraph isomorphism tests, which are NP-complete. Graph indices are commonly used to improve the performance of subgraph querying in graph databases. Subgraph-querying algorithms first construct a candidate answer set by filtering out a set of false answers and then verify each candidate graph using subgraph isomorphism tests. To build graph indices, various kinds of substructure (subgraph, subtree, or path) features have been proposed with the goal of maximizing the filtering rate. Each of them works with a specifically designed index structure, for example, discriminative and frequent subgraph features work with gIndex, ?-TCFG features work with FG-index, etc. We propose Lindex, a graph index, which indexes subgraphs contained in database graphs. Nodes in Lindex represent key-value pairs where the key is a subgraph in a database and the value is a list of database graphs containing the key. We propose two heuristics that are used in the construction of Lindex that allows us to determine answers to subgraph queries conducting less subgraph isomorphism tests. Consequently, Lindex improves subgraph-querying efficiency. In addition, Lindex is compatible with any choice of features. Empirically, we demonstrate that Lindex used in conjunction with subgraph indexing features proposed in previous works outperforms other specifically designed index structures. As a novel index structure, Lindex (1) is effective in filtering false graphs (2) provides fast index lookups, (3) is fast with respect to index construction and maintenance, and (4) can be constructed using any set of substructure index features. These four properties result in a fast and scalable subgraph-querying infrastructure. We substantiate the benefits of Lindex and its disk-resident variation Lindex+ theoretically and empirically.

NODAR: mining globally distributed substructures from a single labeled graph

Data mining in structured and semi-structured data focuses on frequent data values. However, in graph data mining, the focus is on common specific topologies. Graph mining, although its ubiquity, is a difficult task since it requires subgraph isomorphism which is known to be NP-complete. In order to effectively prune the search space and thereby save computational time, a graph mining algorithm requires that the support measure of a pattern to be no greater than that of its subpatterns. This property of the support measure is referred to in the literature as the down-closure, anti-monotonicity or admissibility. Unfortunately, when mining a single labeled graph, simply counting the occurrences of a graph pattern may not have the down-closure property. For this, most existing approaches mine frequent substructures in a set of labeled graphs (called also the transactional setting) and few efforts have been devoted to mining frequent globally distributed substructures in a single labeled graph. In this paper, we propose a graph mining algorithm, called NODAR(Non-Overlapping embeDding based grAph mineR), for computing common and globally distributed substructures in a single labeled graph. NODAR adopts the Depth-First Search (DFS) strategy and is based on the SMNOES (Size of Maximum Non Overlapping Embedding Set) as support measure. The core idea of NODAR is to automatically extract frequent subpatterns; and thus without frequency computation thanks to the down-closure property of SMNOES. By adopting this strategy in the computation of frequent substructures, NODAR reduces the number of subgraph isomorphism tests needed to compute pattern frequencies. Experimental results on monograph and transactional graph databases; and comparison with well-known probabilistic and exact algorithms; prove the efficacy of NODAR.

On Isomorphism Testing of Groups with Normal Hall Subgroups

A normal Hall subgroup N of a group G is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts. We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall (STACS 2009), we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem, which asks whether two linear subspaces are the same up to permutation of coordinates. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai et al. (SODA 2011). Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations ρ and τ of a group H over \( \mathbb{Z}_p^d \), p a prime, determine if there exists an automorphism : H → H, such that the induced representation ρ𝜙 = ρ ◦ 𝜙 and τ are equivalent, in time poly(|H|, p d).

An Algorithm for Testing Isomorphism of Planer Graph Based on Distant Matrix

The testing for graph isomorphism is one of the many problems in the subject of graph theory. This thesis proposes an algorithm for testing isomorphism of planer graph of polynomial time via structuring characteristics of planer graph based on distance matrix. The algorithm, with a time complexity of O (n^4) and a space complexity of O (n^2), has a great application value.

Characterisations and examples of graph classes with bounded expansion

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Nešetřil and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper, we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters and the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with the Erdös–Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d > 0 , there exists a class with bounded expansion, such that a random graph of order n and edge probability d / n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with ‘linear’ crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.

Mining top-K large structural patterns in a massive network

With ever-growing popularity of social networks, web and bio-networks, mining large frequent patterns from a single huge network has become increasingly important. Yet the existing pattern mining methods cannot offer the efficiency desirable for large pattern discovery. We propose Spider-Mine, a novel algorithm to efficiently mine top- K largest frequent patterns from a single massive network with any user-specified probability of 1 - ∈. Deviating from the existing edge-by-edge ( i.e. , incremental) pattern-growth framework, SpiderMine achieves its efficiency by unleashing the power of small patterns of a bounded diameter, which we call "spiders". With the spider structure, our approach adopts a probabilistic mining framework to find the top- k largest patterns by (i) identifying an affordable set of promising growth paths toward large patterns, (ii) generating large patterns with much lower combinatorial complexity, and finally (iii) greatly reducing the cost of graph isomorphism tests with a new graph pattern representation by a multi-set of spiders. Extensive experimental studies on both synthetic and real data sets show that our algorithm outperforms existing methods.

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward–forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward–backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.

Two-particle quantum walks: Entanglement and graph isomorphism testing

We study discrete-time quantum walks on the line and on general undirected graphs with two interacting or noninteracting particles. We introduce two simple interaction schemes and show that they both lead to a diverse range of probability distributions that depend on the correlations and relative phases between the initial coin states of the two particles. We investigate the characteristics of these quantum walks and the time evolution of the entanglement between the two particles from both separable and entangled initial states. We also test the capability of two-particle discrete-time quantum walks to distinguish nonisomorphic graphs. For strongly regular graphs, we show that noninteracting discrete-time quantum walks can distinguish some but not all nonisomorphic graphs with the same family parameters. By incorporating an interaction between the two particles, all nonisomorphic strongly regular graphs tested are successfully distinguished.

Weisfeiler-Lehman Graph Kernels

In this article, we propose a family of efficient kernels for large graphs with discrete node labels. Key to our method is a rapid feature extraction scheme based on the Weisfeiler-Lehman test of isomorphism on graphs. It maps the original graph to a sequence of graphs, whose node attributes capture topological and label information. A family of kernels can be defined based on this Weisfeiler-Lehman sequence of graphs, including a highly efficient kernel comparing subtree-like patterns. Its runtime scales only linearly in the number of edges of the graphs and the length of the Weisfeiler-Lehman graph sequence. In our experimental evaluation, our kernels outperform state-of-the-art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime. Our kernels open the door to large-scale applications of graph kernels in various disciplines such as computational biology and social network analysis.

Fast graph query processing with a low-cost index

This paper studies the problem of processing supergraph queries, that is, given a database containing a set of graphs, find all the graphs in the database of which the query graph is a supergraph. Existing works usually construct an index and performs a filtering-and-verification process, which still requires many subgraph isomorphism testings. There are also significant overheads in both index construction and maintenance. In this paper, we design a graph querying system that achieves both fast indexing and efficient query processing. The index is constructed by a simple but fast method of extracting the commonality among the graphs, which does not involve any costly operation such as graph mining. Our query processing has two key techniques, direct inclusion and filtering. Direct inclusion allows partial query answers to be included directly without candidate verification. Our filtering technique further reduces the candidate set by operating on a much smaller projected database. Experimental results show that our method is significantly more efficient than the existing works in both indexing and query processing, and our index has a low maintenance cost.

IGraph

Graphs are of growing importance in modeling complex structures such as chemical compounds, proteins, images, and program dependence. Given a query graph Q , the subgraph isomorphism problem is to find a set of graphs containing Q from a graph database, which is NP-complete. Recently, there have been a lot of research efforts to solve the subgraph isomorphism problem for a large graph database by utilizing graph indexes. By using a graph index as a filter, we prune graphs that are not real answers at an inexpensive cost. Then, we need to use expensive subgraph isomorphism tests to verify filtered candidates only. This way, the number of disk I/Os and subgraph isomorphism tests can be significantly minimized. The current practice for experiments in graph indexing techniques is that the author of a newly proposed technique does not implement existing indexes on his own code base, but instead uses the original authors' binary executables and reports only the wall clock time. However, we observe this practice may result in several problems. In order to address these problems, we have made significant efforts in implementing all representative indexing methods on a common framework called iGraph. Unlike existing implementations which either use (full or partial) in-memory representations or rely on OS file system cache without guaranteeing real disk I/Os, we have implemented these indexes on top of a storage engine that guarantees real disk I/Os. Through extensive experiments using many synthetic and real datasets, we also provide new empirical findings in the performance of the full disk-based implementations of these methods.

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of t- ( v , k , λ ) designs. For this class of highly regular graphs, we obtain a worst-case running time of O ( v log v + O ( 1 ) ) for bounded parameters t, k, λ. In a first step, our approach makes use of the Babai–Luks algorithm to compute canonical forms of t-designs. In a second step, we show that t-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.

Così Fan Tutte? Adoption and Rejection of Performance-Related Pay in Italian Municipalities: A Cross-Sector Test of Isomorphism

Using data from Italian municipalities, nonprofits, and business firms, this article explores two sets of relationships. First, sectoral differences in the exposure to institutional forces triggering the adoption of performance-related pay are investigated. Second, this study estimates sectoral differences in the sensitivity to those isomorphic pressures. Coercive pressure is the strongest on municipalities and the weakest on for-profits. Exposition to mimetic pressure tends to be stronger for business establishments compared with both municipalities and nonprofits. For-profits generally face weaker normative forces relative to municipalities and nonprofits. Coercive pressure increases the probability that an organization adopts performance-related pay schemes more in business firms than in nonprofits and more in nonprofits than in municipalities. The effect of normative pressure on the probability of adoption is higher in both municipalities and nonprofits compared with for-profit establishments. Coercive pressure increases the percentage of performance-related pay more in business firms than in nonprofits and more in nonprofits than in municipalities.

On the Complexity of Matroid Isomorphism Problem

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_{2}^{p}$ . In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be $\Sigma_{2}^{p}$ -complete and is co NP-hard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism, and matroid isomorphism are both polynomial time many-one equivalent to graph isomorphism. We give a polynomial time Turing reduction from graphic matroid isomorphism problem to the graph isomorphism problem. Using this, we are able to show that graphic matroid isomorphism testing for planar graphs can be done in deterministic polynomial time. We then give a polynomial time many-one reduction from bounded rank matroid isomorphism problem to graphic matroid isomorphism, thus showing that all the above problems are polynomial time equivalent. Further, for linear and graphic matroids, we prove that the automorphism problems are polynomial time equivalent to the corresponding isomorphism problems. In addition, we give a polynomial time membership test algorithm for the automorphism group of a graphic matroid.

Approach for Efficient Subgraph Isomorphism Testing for Multiple Graphs

This paper proposes an approach for subgraph isomorphism testing from a set of priori given small graphs to a large graph issued on-line.In order to reduce the computational cost,based on DFS Code,an elaborate organization of a set of graphs is presented,and a look-ahead-pruning based algorithm of subgraph isomorphism testing from multiple small graphs to a large graph is proposed.Moreover,an index technique based on data mining is introduced.Analytical and experimental results show the on-line computational cost of the proposed method is much less than the state-of-the-art method and it is about one order of magnitude faster than the existing method with more than one order of magnitude less off-line construction time.

Algorithm for determining whether various two-level fractional factorial split-plot row–column designs are non-isomorphic

The confounding and aliasing scheme for fractional factorial split-plot designs with the units within each wholeplot arranged in rows and columns is described and illustrated. Isomorphism for this design type is described, together with a procedure which considers extensions of the concepts of wordlength patterns and letter patterns that can be used to test isomorphism between designs. Using in part this isomorphism testing procedure, a construction algorithm that may be used to obtain a complete set of such non-isomorphic two-level designs is described. Software based on this construction algorithm was used to obtain a complete set of non-isomorphic designs for up to five wholeplot factors, five subplot factors and up to 64 runs, which is presented as a table of designs. To aid the experimenter in distinguishing between competing designs, the estimation capacity sequence for each design is presented.