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Graph Similarity and Approximate Isomorphism
[article]

2018
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arXiv
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pre-print

The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=_πA_G^π-A_H_F, where π ranges over all permutations of the vertex set of G, where A_G^π denotes the matrix obtained from

arXiv:1802.08509v1
fatcat:6ysczo3kinfqxg57hticjxtlqu
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... _G by permuting rows and columns according to π, and where M_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by SIM (WSIM), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that SIM (WSIM) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSIM is best done in the framework of linear algebra. We show that WSIM is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where one of the matrices has a bounded clustering number, a parameter arising from spectral graph drawing techniques.##
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On the Complexity of Noncommutative Polynomial Factorization
[article]

2015
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arXiv
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pre-print

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F〈 x_1,x_2,...,x_n〉 of polynomials over the field F and noncommuting variables x_1,x_2,...,x_n. Our main results are the following. Although F〈 x_1,x_2,...,x_n 〉 is not a unique factorization ring, we note that variable-disjoint factorization in F〈 x_1,x_2,...,x_n 〉 has the uniqueness property. Furthermore, we prove that computing the variable-disjoint factorization is polynomial-time

arXiv:1501.00671v1
fatcat:55m3f6bpgje6lg2oskfubzzpdi
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... to Polynomial Identity Testing (both when the input polynomial is given by an arithmetic circuit or an algebraic branching program). We also show that variable-disjoint factorization in the black-box setting can be efficiently computed (where the factors computed will be also given by black-boxes, analogous to the work [KT91] in the commutative setting). As a consequence of the previous result we show that homogeneous noncommutative polynomials and multilinear noncommutative polynomials have unique factorizations in the usual sense, which can be efficiently computed. Finally, we discuss a polynomial decomposition problem in F〈 x_1,x_2,...,x_n〉 which is a natural generalization of homogeneous polynomial factorization and prove some complexity bounds for it.##
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Homomorphism Tensors and Linear Equations
[article]

2022
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arXiv
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pre-print

Lovász (1967) showed that two graphs G and H are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph F, the number of homomorphisms from F to G equals the number of homomorphisms from F to H. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on

arXiv:2111.11313v2
fatcat:7xwmantsajayxivhhel5avf2we
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... s arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).##
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Weisfeiler–Leman and Graph Spectra
[article]

2022
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arXiv
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pre-print

We devise a hierarchy of spectral graph invariants, generalising the adjacency spectra and Laplacian spectra, which are commensurate in power with the hierarchy of combinatorial graph invariants generated by the Weisfeiler–Leman (WL) algorithm. More precisely, we provide a spectral characterisation of k-WL indistinguishability after d iterations, for k,d ∈ℕ. Most of the well-known spectral graph invariants such as adjacency or Laplacian spectra lie in the regime between 1-WL and 2-WL. We show

arXiv:2103.02972v2
fatcat:ipgpn6ttnzhrbn56ql5o5qylcu
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... at individualising one vertex plus running 1-WL is already more powerful than all such spectral invariants in terms of their ability to distinguish non-isomorphic graphs. Building on this result, we resolve an open problem of Fürer (2010) about spectral invariants and strengthen a result due to Godsil (1981) about commute distances.##
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Lovász Meets Weisfeiler and Leman
[article]

2018
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arXiv
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pre-print

In this paper, we relate a beautiful theory by Lov\'asz with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs G and H are indistinguishable by the color refinement algorithm if and only if, for all trees T, the number Hom(T,G) of homomorphisms from T to G equals the corresponding number Hom(T,H) for H. There is a natural system of linear

arXiv:1802.08876v2
fatcat:clduredu25azpfhzroddazj6ha
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... ations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The nonnegative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the nonnegativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all t, the two graphs have the same number of length-t walks. We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of bounded path width and solutions to our system where we drop the nonnegativity constraints. A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs G and H, there is a tree T with Hom(T,G) = Hom(T,H).##
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Ordered Subgraph Aggregation Networks
[article]

2022
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arXiv
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pre-print

Numerous subgraph-enhanced graph neural networks (GNNs) have emerged recently, provably boosting the expressive power of standard (message-passing) GNNs. However, there is a limited understanding of how these approaches relate to each other and to the Weisfeiler–Leman hierarchy. Moreover, current approaches either use all subgraphs of a given size, sample them uniformly at random, or use hand-crafted heuristics instead of learning to select subgraphs in a data-driven manner. Here, we offer a

arXiv:2206.11168v2
fatcat:y7y5glkspzdclopkgw3ikrk4je
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... fied way to study such architectures by introducing a theoretical framework and extending the known expressivity results of subgraph-enhanced GNNs. Concretely, we show that increasing subgraph size always increases the expressive power and develop a better understanding of their limitations by relating them to the established k-𝖶𝖫 hierarchy. In addition, we explore different approaches for learning to sample subgraphs using recent methods for backpropagating through complex discrete probability distributions. Empirically, we study the predictive performance of different subgraph-enhanced GNNs, showing that our data-driven architectures increase prediction accuracy on standard benchmark datasets compared to non-data-driven subgraph-enhanced graph neural networks while reducing computation time.##
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Faster FPT Algorithm for Graph Isomorphism Parameterized by Eigenvalue Multiplicity
[article]

2014
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arXiv
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pre-print

We give a O^*(k^O(k)) time isomorphism testing algorithm for graphs of eigenvalue multiplicity bounded by k which improves on the previous best running time bound of O^*(2^O(k^2/ k)).

arXiv:1408.3510v1
fatcat:r3k5f2ckzrhrdidlzule4oaeqq
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The Parameterized Complexity of Geometric Graph Isomorphism

2015
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Algorithmica
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SpeqNets: Sparsity-aware Permutation-equivariant Graph Networks
[article]

2022
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arXiv
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pre-print

*Gaurav*

*Rattan*is supported by the DFG Research Grants Program-RA 3242/1-1-411032549. Siamak Ravanbakhsh's research is in part supported by CIFAR AI Chairs program. ...

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On the Complexity of Noncommutative Polynomial Factorization
[chapter]

2015
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Lecture Notes in Computer Science
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In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F x 1

doi:10.1007/978-3-662-48054-0_4
fatcat:fdp4wsfyrjd6vper43ptmx7ymu
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The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs
[article]

2016
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arXiv
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pre-print

In this paper we study the complexity of the following problems: Given a colored graph X=(V,E,c), compute a minimum cardinality set S of vertices such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G on [n] given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these

arXiv:1606.04383v1
fatcat:6b5tmtvjpvgnlm4iv35elngdvm
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... ms. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT algorithms. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c) compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time (even in logspace), while starting from color class size 4 they become W[P]-hard.##
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Graph Isomorphism, Color Refinement, and Compactness
[article]

2015
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arXiv
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pre-print

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in distinguishing G from any non-isomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined compact graphs: A graph is compact if its fractional automorphisms polytope is integral. Tinhofer noted that isomorphism

arXiv:1502.01255v3
fatcat:pivgpwx3zbd7pila44ocsuaqty
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... ng for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. Our results are summarized below: - We show that amenable graphs are recognizable in time O((n + m)logn), where n and m denote the number of vertices and the number of edges in the input graph. - We show that all amenable graphs are compact. - We study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. The corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.##
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Weisfeiler and Leman go sparse: Towards scalable higher-order graph embeddings

2020
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Neural Information Processing Systems
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Graph kernels based on the 1-dimensional Weisfeiler-Leman algorithm and corresponding neural architectures recently emerged as powerful tools for (supervised) learning with graphs. However, due to the purely local nature of the algorithms, they might miss essential patterns in the given data and can only handle binary relations. The k-dimensional Weisfeiler-Leman algorithm addresses this by considering k-tuples, defined over the set of vertices, and defines a suitable notion of adjacency

dblp:conf/nips/0001RM20
fatcat:wd3tzhpetffmrgbvwhhfhjzdce
## more »

... these vertex tuples. Hence, it accounts for the higher-order interactions between vertices. However, it does not scale and may suffer from overfitting when used in a machine learning setting. Hence, it remains an important open problem to design WL-based graph learning methods that are simultaneously expressive, scalable, and non-overfitting. Here, we propose local variants and corresponding neural architectures, which consider a subset of the original neighborhood, making them more scalable, and less prone to overfitting. The expressive power of (one of) our algorithms is strictly higher than the original algorithm, in terms of ability to distinguish non-isomorphic graphs. Our experimental study confirms that the local algorithms, both kernel and neural architectures, lead to vastly reduced computation times, and prevent overfitting. The kernel version establishes a new state-of-the-art for graph classification on a wide range of benchmark datasets, while the neural version shows promising performance on large-scale molecular regression tasks.##
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On the isomorphism problem for decision trees and decision lists

2015
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Theoretical Computer Science
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We study the complexity of isomorphism testing for Boolean functions that are represented by decision trees or decision lists. Our results include a 2 √ s(lg s) O(1) time algorithm for isomorphism testing of decision trees of size s. Additionally, we show: • Isomorphism testing of rank-1 decision trees is complete for logspace. • For r ≥ 2, isomorphism testing for rank-r decision trees is polynomialtime equivalent to Graph Isomorphism. As a consequence we obtain a 2 √ s(lg s) O(1) time

doi:10.1016/j.tcs.2015.01.025
fatcat:n3eciswqlrf6dfcmqekd7croei
## more »

... for isomorphism testing of decision trees of size s. • The isomorphism problem for decision lists admits a Schaefer-type dichotomy: depending on the class of base functions, the isomorphism problem is either in polynomial time, or equivalent to Graph Isomorphism, or coNP-hard.##
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Design of a Compact and Versatile Bench Scale Tubular Reactor

2009
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Bulletin of Chemical Reaction Engineering & Catalysis
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