DDSL: Efficient Subgraph Listing on Distributed and Dynamic Graphs

Subgraph listing is a fundamental problem in graph theory and has wide applications in areas like sociology, chemistry, and social networks. Modern graphs can usually be large-scale as well as highly dynamic, which challenges the efficiency of existing subgraph listing algorithms. Recent works have shown the benefits of partitioning and processing big graphs in a distributed system, however, there is only few work targets subgraph listing on dynamic graphs in a distributed environment. In this paper, we propose an efficient approach, called Distributed and Dynamic Subgraph Listing (DDSL), which can incrementally update the results instead of running from scratch. DDSL follows a general distributed join framework. In this framework, we use a Neighbor-Preserved storage for data graphs, which takes bounded extra space and supports dynamic updating. After that, we propose a comprehensive cost model to estimate the I/O cost of listing subgraphs. Then based on this cost model, we develop an algorithm to find the optimal join tree for a given pattern. To handle dynamic graphs, we propose an efficient left-deep join algorithm to incrementally update the join results. Extensive experiments are conducted on real-world datasets. The results show that DDSL outperforms existing methods in dealing with both static dynamic graphs in terms of the responding time

Polynomial-Time Isomorphism Test of Groups that are Tame Extensions

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 preceding problem in the so-called "tame" case, i.e., when a Sylow p-subgroup of $G/\mathbb{Z}_p^d$ is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases correspond exactly to the group algebra $\overline{\mathbb{F}}_p[G/\mathbb{Z}_p^d]$ being of tame type, as in the celebrated tame-wild dichotomy in representation theory. We then solve new cases of GpI in polynomial time. Our result relies crucially on the divide-and-conquer strategy proposed earlier by the authors (CCC 2014), which splits GpI into two problems, one on group actions (representations), and one on group cohomology. Based on this strategy, we combine permutation group and representation algorithms with new mathematical results, including bounds on the number of indecomposable representations of groups in the tame case, and on the size of their cohomology groups. Finally, we note that when a group extension is not tame, the preceding bounds do not hold. This suggests a precise sense in which the tame-wild dichotomy from representation theory may also be a dividing line between the (currently) easy and hard instances of GpI.Comment: 23 page

Boundaries of VP and VNP

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = VP, where VP is the class of families of polynomials that can be computed by arithmetic circuits of polynomial degree and size, and VP is the class of families of polynomials that can be approximated infinitesimally closely by arithmetic circuits of polynomial degree and size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that VP is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in VP), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP∗. We also introduce analogous degenerations of VNP. We show that Stable-VP ⊆ Newton-VP ⊆ VP∗ ⊆ VNP, and Stable-VNP = Newton-VNP = VNP∗ = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating VP from VP. Although we do not yet construct explicit candidates for the polynomial families in VP \VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP \VP based on semi-invariants of quivers would have to be nongeneric by showing that, for many finite quivers (including some wild ones), Newton degeneration of any generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits

On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials IV: Linear-Length Reductions and Their Applications

By giving new reductions, we show the following algorithmic results and relations between various isomorphism problems: If Graph Isomorphism is in P, then testing equivalence of cubic forms in n variables over a finite field Fq, and testing isomorphism of n-dimensional algebras over Fq, can both be solved in time qO(n), improving from the brute-force upper bound qO(n2) for both of these. Polynomial-time search- and counting-to-decision reduction for testing isomorphism of p-groups of class 2 and exponent p in the Cayley table model. This answers questions of Arvind and Torán (Bull. EATCS, 2005) for this group class, thought to be one of the hardest cases of Group Isomorphism. Combined with the |G|O((log|G|)1/2)-time isomorphism test for p-groups of Frattini class 2 (Ivanyos, Mendoza, Qiao, Sun, and Zhang, FOCS '24), our reductions extend this runtime to p-groups of exponent p and class c q can be solved in time qÕ(n3/2), where n is the side length. This improves the previous state of the art bound, which was the brute force qO(n2), for the isomorphism problems for cubic forms, algebras, tensors, and more. The key to our reductions is to give new gadgets that improve the parameters of previous reductions around Tensor Isomorphism (Grochow and Qiao, ITCS '21; SIAM J. Comp., '23). In particular, several of these previous reductions incurred a quadratic increase in the length of the tensors involved. When the tensors represent p-groups, this corresponds to an increase in the order of the group of the form |G|(log|G|), negating any asymptotic gains in the Cayley table model. We remedy this by presenting a new kind of tensor gadget that allows us to replace those quadratic-length reductions with linear-length ones, yielding the above consequences

On p-group isomorphism: Search-to-decision, counting-to-decision, and nilpotency class reductions via tensors

In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of “small” class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.)

On the complexity of isomorphism problems for tensors, groups, and polynomials I: Tensor isomorphism-completeness

We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC)

Average-case algorithms for testing isomorphism of polynomials, algebras, and multilinear forms

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.</jats:p

Polynomial-Time Isomorphism Test of Groups that are Tame Extensions (Extended Abstract)

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 Zdp, and given the coset of isomorphisms Iso(G/Zdp,H/Zdp), compute Iso(G,H) in time poly(|G|). Babai&Qiao (STACS 2012) solved this problem when a Sylow p-subgroup of G/Zdp is trivial. In this paper, we solve the preceding problem in the so-called “tame” case, i. e., when a Sylow p-subgroup of G/Zdp is cyclic, dihedral, semi-dihedral, or generalized quaternion. These cases correspond exactly to the group algebra Fp[G/Zdp] being of tame type, as in the celebrated tame-wild dichotomy in representation theory. We then solve new cases of GpI in polynomial time. Our result relies crucially on the divide-and-conquer strategy proposed earlier by the authors (CCC 2014), which splits GpI into two problems, one on group actions (representations), and one on group cohomology. Based on this strategy, we combine permutation group and representation algorithms with new mathematical results, including bounds on the number of indecomposable representations of groups in the tame case, and on the size of their cohomology groups. Finally, we note that when a group extension is not tame, the preceding bounds do not hold. This suggests a precise sense in which the tame-wild dichotomy from representation theory may also be a key barrier to cross to put GpI into P

On p-Group Isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors

In this paper we study some classical complexity-theoretic questions regarding G roup I somorphism (G p I). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneck case for G p I, and work in the model of matrix groups over finite fields. Our main results are as follows. • Although search-to-decision and counting-to-decision reductions have been known for over four decades for G raph I somorphism (GI), they had remained open for G p I, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from T ensor I somorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • D espite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with T ensor I somorphism -completeness results (Grochow & Qiao, ibid.). </jats:p

Algorithms for group isomorphism via group extensions and cohomology

The isomorphism problem for groups given by their multiplication tables (GPI) has long been known to be solvable in nO(log n) time, but only recently has there been significant progress towards polynomial time. For example, Babai et al. (ICALP 2012) gave a polynomial-time algorithm for groups with no abelian normal subgroups. Thus, at present it is crucial to understand groups with abelian normal subgroups to develop no(log n)-time algorithms. Towards this goal we advocate a strategy via the extension theory of groups, which describes how a normal subgroup N is related to G/N via G. This strategy 'splits' GPI into two sub problems: one regarding group actions on other groups, and one regarding group co homology. The solution of these problems is essentially necessary and sufficient to solve GPI. Most previous works naturally align with this strategy, and it thus helps explain in a unified way the recent polynomial-time algorithms for other group classes. In particular, most prior results in the multiplication table model focus on the group action aspect, despite the general necessity of co homology, for example for p-groups of class 2-believed to be the hardest case of GPI. To make progress on the group co homology aspect of 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. Recall that Babai et al. (ICALP 2012) consider the class of groups G such that G itself has no abelian normal subgroups. Following the above strategy, we solve GPI in 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(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 that a nontrivial algorithm with worst-case guarantees has tackled both aspects of GPI-actions and cohomology-simultaneously. Prior to this work, only nO(log n)-time algorithms were known, even for groups with a central radical of constant size, such as Z(G) = ℤ2. To develop these algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work. © 2014 IEEE