We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.
Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on the hypercube with a sparse Fourier expansion must either be Boolean or far from Boolean. In particular, we show that a multilinear polynomial with at most k terms must either be Boolean, or output values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its domain. It follows that given oracle access to f, together with the guarantee that its …
In this work, we introduce bidirectional collision detection --- a new algorithmic tool that applies to the collision problems that arise in many isomorphism problems. For the group isomorphism problem, we show that bidirectional collision detection yields a deterministic n^((1 / 2) log n + O(1)) time algorithm whereas previously the n^(log n + O(1)) generator-enumeration algorithm was the best result for several decades. For the hard special case of solvable groups, we combine bidirectional collision detection with methods from the author's previous work to obtain a deterministic square-root …
A method for improving the efficiency of graph isomorphism testing is presented. The method uses the structure of the graph colored by vertex hash codes as a means of partitioning vertices into equivalence classes, which in turn reduces the combinatorial burden of isomorphism testing. Unrolling the graph into a tree at each vertex allows structurally different regular graphs to be discriminated, a capability that the color refinement algorithm cannot do.
We give an algorithm that, for every fixed k, decides isomorphism of graphs of rank width at most k in polynomial time. As the clique width of a graph is bounded in terms of its rank width, we also obtain a polynomial time isomorphism test for graph classes of bounded clique width.
We consider the problem of testing if two input forests are isomorphic or are far from being so. An algorithm is called an $\varepsilon$-tester for forest-isomorphism if given an oracle access to two forests $G$ and $H$ in the adjacency list model, with high probability, accepts if $G$ and $H$ are isomorphic and rejects if we must modify at least $\varepsilon n$ edges to make $G$ isomorphic to $H$. We show an $\varepsilon$-tester for forest-isomorphism with a query complexity $\mathrm{polylog}(n)$ and a lower bound of $Ω(\sqrt{\log{n}})$. Further, with the aid of the tester, we show that every…
We engineer a new probabilistic Monte-Carlo algorithm for isomorphism testing. Most notably, as opposed to all other solvers, it implicitly exploits the presence of symmetries without explicitly computing them. We provide extensive benchmarks, showing that the algorithm outperforms all state-of-the-art solutions for isomorphism testing on most inputs from the de facto standard benchmark library for isomorphism testing. On many input types, our data not only show improved running times by an order of magnitude, but also reflect a better asymptotic behavior. Our results demonstrate that, with cu…
The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we intro…
The group isomorphism problem asks whether two given groups are isomorphic or not. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. In this paper we study this problem for a class of groups corresponding to one of the simplest ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group A by a cyclic group of order m. We present an efficient algorithm solving the group isomorphism problem for all the groups of …
The Graph Isomorphism problem has both theoretical and practical interest. In this paper we present an algorithm, called conauto-1.2, that efficiently tests whether two graphs are isomorphic, and finds an isomorphism if they are. This algorithm is an improved version of the algorithm conauto, which has been shown to be very fast for random graphs and several families of hard graphs. In this paper we establish a new theorem that allows, at very low cost, the easy discovery of many automorphisms. This result is especially suited for graphs with regularly connected components, and can be applied …
We consider two basic algorithmic problems concerning tuples of (skew-)symmetric matrices. The first problem asks to decide, given two tuples of (skew-)symmetric matrices $(B_1, \dots, B_m)$ and $(C_1, \dots, C_m)$, whether there exists an invertible matrix $A$ such that for every $i\in\{1, \dots, m\}$, $A^tB_iA=C_i$. We show that this problem can be solved in randomized polynomial time over finite fields of odd size, the real field, and the complex field. The second problem asks to decide, given a tuple of square matrices $(B_1, \dots, B_m)$, whether there exist invertible matrices $A$ and $D…
A graph is said to be circular-arc if the vertices can be associated with arcs of a circle so that two vertices are adjacent if and only if the corresponding arcs overlap. It is proved that the isomorphism of circular-arc graphs can be tested by the Weisfeiler-Leman algorithm after individualization of two vertices.
In this paper, we show that Graph Isomorphism (GI) is not $\textsf{AC}^{0}$-reducible to several problems, including the Latin Square Isotopy problem, isomorphism testing of several families of Steiner designs, and isomorphism testing of conference graphs. As a corollary, we obtain that GI is not $\textsf{AC}^{0}$-reducible to isomorphism testing of Latin square graphs and strongly regular graphs arising from special cases of Steiner $2$-designs. We accomplish this by showing that the generator-enumeration technique for each of these problems can be implemented in $β_{2}\textsf{FOLL}$, which c…
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…
Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in circular-arc graphs and can be seen as their canonical representations; in particular, every intersection model can be easily transformed into a normalized one. Our work adapts and appropriately extends th…
In this paper we present an algorithm, called conauto-2.0, that can efficiently compute a set of generators of the automorphism group of a graph, and test whether two graphs are isomorphic, finding an isomorphism if they are. This algorithm uses the basic individualization/refinement technique, and is an improved version of the algorithm conauto, which has been shown to be very fast for random graphs and several families of hard graphs. In this paper, it is proved that, under some circumstances, it is not only possible to prune the search space (using already found generators of the automorphi…
The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order $n$, an algorithm with $n^{(\log n + O(1))}$ running time, attributed to Tarjan, was proposed in the 1970s [Mil78]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an $n^{(1 / 4 + o(1))\log n}$ running time [Ros13]. The isomorphism testing for $p$-groups of (nilpotent) class 2 and exponent $p$ has been identified as a major barrier to obtaining an $n^{o(\log n)}$ time algorithm for the group isomorphism problem. Alt…
A function $f\colon \{-1,1\}^n \to \{-1,1\}$ is a $k$-junta if it depends on at most $k$ of its variables. We consider the problem of tolerant testing of $k$-juntas, where the testing algorithm must accept any function that is $ε$-close to some $k$-junta and reject any function that is $ε'$-far from every $k'$-junta for some $ε'= O(ε)$ and $k' = O(k)$. Our first result is an algorithm that solves this problem with query complexity polynomial in $k$ and $1/ε$. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardina…
An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that the algorithm actually performs the test is presented.
It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for this problem are either not efficient or not exact. In this paper, we present a new algorithm which solves this equivalent formulation via semidefinite programming. Specifically, we show that the problem of determining whether the auxiliary graph contains a clique of specific order can be formulated as a semidefinite programming problem, and can thus be (almo…
In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the nodes in the network, or alternatively, only to a single node. We first consider the decision variant of the problem in which the algorithm distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for the setting in which $G_K$ is given only to a single node. We prove that for…
An $H$-graph is one representable as the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$, introduced by Biró, Hujter and Tuza (1992). An $H$-graph is proper if the representing subgraphs of $H$ can be chosen incomparable by the inclusion. In this paper, we focus on the isomorphism problem for $S_d$-graphs and $T$-graphs, where $S_d$ is the star with $d$ rays and $T$ is an arbitrary fixed tree. Answering an open problem of Chaplick, Töpfer, Voborn\'ık and Zeman (2016), we provide an FPT-time algorithm for testing isomorphism and computing the automorphis…
We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G cong H. For several decades, the n^(log_p n + O(1)) generator-enumeration bound (where p is the smallest prime dividing the order of the group) has been the best worst-case result for general groups. In this work, we show the first improvement over the generator-enumeration bound for p-groups, which are believed to be the hard case of the group isomorphism problem. We start by giving a Turing reduction from group isomorphism to n^((1 / 2) log_p n + O(1)) instances of…
We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n^(log_p n + O(1)) generator-enumeration algorithm. In previous work with Fabian Wagner, we showed an n^((1 / 2) log_p n + O(log n / log log n)) time algorithm for testing isomorphism of p-groups by building graphs with degree bounded by p + O(1) that represent composition series for the groups and applying Luks' algorithm for testing isomorphism of bounded degree graphs. In this work, we extend this improvement to the more gen…
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 nega…
An $H$-graph is an intersection graph of connected subgraphs of a suitable subdivision of a fixed graph $H$. Many important classes of graphs, including interval graphs, circular-arc graphs, and chordal graphs, can be expressed as $H$-graphs, and, in particular, every graph is an $H$-graph for a suitable graph $H$. An $H$-graph is called proper if it has a representation where no subgraph properly contains another. We consider the recognition and isomorphism problems for proper $U$-graphs where $U$ is a unicylic graph. We prove that testing whether a graph is a (proper) $U$-graph, for some $U$…
A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpoint leads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of …
Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2
University of Colorado Boulder（科罗拉多大学博尔德分校）；HUN-REN Institute for computer Science and Control（匈牙利科学院计算机与控制研究所）；University of Technology Sydney（悉尼科技大学）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）
In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Z_m$ with the order of $A$ coprime with $m$. More precisely, the runni…
While efficient algorithms are known for solving many important problems related to groups, no efficient algorithm is known for determining whether two arbitrary groups are isomorphic. The particular case of 2-nilpotent groups, a special type of central extension, is widely believed to contain the essential hard cases. However, looking specifically at central extensions, the natural formulation of being "the same" is not isomorphism but rather "equivalence," which requires an isomorphism to preserves the structure of the extension. In this paper, we show that equivalence of central extensions …
We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width $k$. Our algorithm then makes heavy use of the group theoretic techniq…
We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices $V$ and a permutation group $Γ$ over domain $V$, and asking whether there is a permutation $γ\in Γ$ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on $d$ points, this problem can be solved in time $(n+m)^{O((\log d)^{c})}$ for some absolute constant $c$ where $n$ denotes the number of vertices and $m$ the number of hyperedges. In particular, this gives the currentl…
We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation groups, tree decompositions, and so on. Our approach allows for a systematic transfer of the techniques that have been developed for isomorphism testing to canonization. We use it to design a canonization algorithm for combinatorial objects in general. This result gives new fastest canonization algorithms with an asymptotic running time matching the best known iso…
We give new polynomial-time algorithms for testing isomorphism of a class of groups given by multiplication tables (GpI). Two results (Cannon & Holt, J. Symb. Comput. 2003; Babai, Codenotti & Qiao, ICALP 2012) imply that GpI reduces to the following: given groups G, H with characteristic subgroups of the same type and isomorphic to $\mathbb{Z}_p^d$, and given the coset of isomorphisms $Iso(G/\mathbb{Z}_p^d, H/\mathbb{Z}_p^d)$, compute Iso(G, H) in time poly(|G|). Babai & Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of $G/\mathbb{Z}_p^d$ is trivial. In this paper, we solve the …
We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this…
Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing
Indian Statistical Institute（印度统计研究所）；Université Paris Cité, CNRS, IRIF（巴黎西岱大学、法国国家科学研究中心、信息研究与信息系统研究所）；Indian Institute of Technology, Hyderabad（印度理工学院海德拉巴分校）
