8,671 research outputs found

    On the equivalence between graph isomorphism testing and function approximation with GNNs

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    Graph neural networks (GNNs) have achieved lots of success on graph-structured data. In the light of this, there has been increasing interest in studying their representation power. One line of work focuses on the universal approximation of permutation-invariant functions by certain classes of GNNs, and another demonstrates the limitation of GNNs via graph isomorphism tests. Our work connects these two perspectives and proves their equivalence. We further develop a framework of the representation power of GNNs with the language of sigma-algebra, which incorporates both viewpoints. Using this framework, we compare the expressive power of different classes of GNNs as well as other methods on graphs. In particular, we prove that order-2 Graph G-invariant networks fail to distinguish non-isomorphic regular graphs with the same degree. We then extend them to a new architecture, Ring-GNNs, which succeeds on distinguishing these graphs and provides improvements on real-world social network datasets

    Partially Symmetric Functions are Efficiently Isomorphism-Testable

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    Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.Comment: 22 page

    Stability of relative equilibria with singular momentum values in simple mechanical systems

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    A method for testing GμG_\mu-stability of relative equilibria in Hamiltonian systems of the form "kinetic + potential energy" is presented. This method extends the Reduced Energy-Momentum Method of Simo et al. to the case of non-free group actions and singular momentum values. A normal form for the symplectic matrix at a relative equilibrium is also obtained.Comment: Partially rewritten. Some mistakes fixed. Exposition improve

    P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

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    While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside [0,1]n[0,1]^n, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in {0,1}n\{0,1\}^n, we get geometric formulations as the question if plane or sphere intersects with {0,1}n\{0,1\}^n. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of 02πicos(φki)dφ\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi fourier-type integral for some natural kik_i. The last discussed approach is using anti-commuting Grassmann numbers θi\theta_i, making (Adiag(θi))n(A \cdot \textrm{diag}(\theta_i))^n nonzero only if AA has a Hamilton cycle. Hence, the P\neNP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

    Algorithms for group isomorphism via group extensions and cohomology

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    The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in nlogn+O(1)n^{\log n+O(1)} time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N is abelian, this strategy is well-known. Our first contribution is to extend this strategy to handle the case when N is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. (SODA 2011): the class of groups such that G mod its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. (ICALP 2012), namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in nO(loglogn)n^{O(\log \log n)} time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in nO(loglogn)n^{O(\log\log n)} time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on a no(logn)n^{o(\log n)}-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously.Comment: 54 pages + 14-page appendix. Significantly improved presentation, with some new result

    Testing isomorphism of graded algebras

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    We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that often dramatically improve the performance of the algorithm and report on an implementation in Magma

    An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

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    The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the kk-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
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