arXiv subjects

Youming Qiao

Publications and source records attributed to Youming Qiao.

At least 37 records · Page 2Linked to original sources

arXiv2022

The isomorphism problem for plain groups is in $Σ_3^{\mathsf{P}}$

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{é}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of \emph{plain groups}, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in $Σ_3^{\mathsf{P}}$. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).

math.GR↗

arXiv2021

Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras

One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lov'asz (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).

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arXiv2020

Enumerating alternating matrix spaces over finite fields with explicit coordinates

We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating labelled graphs. To support this viewpoint, we present q-analogues of Gilbert's formula for enumerating connected graphs (Can. J. Math., 1956), and Read's formula for enumerating c-colored graphs (Can. J. Math., 1960). We also develop an analogue of Riddell's formula relating the exponential generating function of graphs with that of connected graphs (Riddell's PhD thesis, 1951), building on Eulerian generating functions developed by Srinivasan (Discrete Math., 2006).

math.CO↗

arXiv2020

On the Baer-Lovász-Tutte construction of groups from graphs: isomorphism types and homomorphism notions

Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a $p$-group $P_G$ of class $2$ and exponent $p$ that is naturally associated with $G$. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs $G$ and $H$, $G$ and $H$ are isomorphic as graphs if and only if $P_G$ and $P_H$ are isomorphic as groups. Our second contribution is a new homomorphism notion for graphs. Based on this notion, a category of graphs can be defined, and the Baer-Lovász-Tutte construction naturally leads to a functor from this category of graphs to the category of groups.

math.CO↗

arXiv2019

From independent sets and vertex colorings to isotropic spaces and isotropic decompositions

In the 1970's, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration. This paper is dedicated to the memory of Ker-I Ko.

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arXiv2019

Isomorphism problems for tensors, groups, and cubic forms: completeness and reductions

In this paper we consider the problems of testing isomorphism of tensors, $p$-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as orbit problems on multi-way arrays under different group actions. Our first two main results are: 1. All the aforementioned isomorphism problems are equivalent under polynomial-time reductions, in conjunction with the recent results of Futorny-Grochow-Sergeichuk (Lin. Alg. Appl., 2019). 2. Isomorphism of $d$-tensors reduces to isomorphism of 3-tensors, for any $d \geq 3$. Our results suggest that these isomorphism problems form a rich and robust equivalence class, which we call Tensor Isomorphism-complete, or TI-complete. We then leverage the techniques used in the above results to prove two first-of-their-kind results for Group Isomorphism (GpI): 3. We give a reduction from GpI for $p$-groups of exponent $p$ and small class ($c < p$) to GpI for $p$-groups of exponent $p$ and class 2. The latter are widely believed to be the hardest cases of GpI, but as far as we know, this is the first reduction from any more general class of groups to this class. 4. We give a search-to-decision reduction for isomorphism of $p$-groups of exponent $p$ and class 2 in time $|G|^{O(\log \log |G|)}$. While search-to-decision reductions for Graph Isomorphism (GI) have been known for more than 40 years, as far as we know this is the first non-trivial search-to-decision reduction in the context of GpI. Our main technique for (1), (3), and (4) is a linear-algebraic analogue of the classical graph coloring gadget, which was used to obtain the search-to-decision reduction for GI. This gadget construction may be of independent interest and utility. The technique for (2) gives a method for encoding an arbitrary tensor into an algebra.

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arXiv2019

Group-theoretic generalisations of vertex and edge connectivities

Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $κ(P)$, is the smallest integer $s$ for the existence of a subgroup $S$ of $P$ satisfying (1) $S\cap [P,P]=[S,S]$, (2) $|S/[S,S]|=p^{n-s}$, and (3) $S$ admits a non-trivial central decomposition. The second parameter, $λ(P)$, is the smallest integer $s$ for the existence of a central subgroup $N$ of order $p^s$, such that $P/N$ admits a non-trivial central decomposition. While defined in purely group-theoretic terms, these two parameters generalise respectively the vertex and edge connectivities of graphs: For a simple undirected graph $G$, through the classical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond. Math. Soc., 1947) and Lovász (B. Braz. Math. Soc., 1989), there is a $p$-group of class $2$ and exponent $p$ $P_G$ that is naturally associated with $G$. Our main results show that the vertex connectivity $κ(G)$ is equal to $κ(P_G)$, and the edge connectivity $λ(G)$ is equal to $λ(P_G)$. We also discuss the relation between $κ(P)$ and $λ(P)$ for a general $p$-group $P$ of class $2$ and exponent $p$, as well as the computational aspects of these parameters.

math.CO↗

arXiv2019

General Linear Group Action on Tensors: A Candidate for Post-Quantum Cryptography

Starting from the one-way group action framework of Brassard and Yung (Crypto '90), we revisit building cryptography based on group actions. Several previous candidates for one-way group actions no longer stand, due to progress both on classical algorithms (e.g., graph isomorphism) and quantum algorithms (e.g., discrete logarithm). We propose the general linear group action on tensors as a new candidate to build cryptography based on group actions. Recent works (Futorny--Grochow--Sergeichuk, Lin. Alg. Appl., 2019) suggest that the underlying algorithmic problem, the tensor isomorphism problem, is the hardest one among several isomorphism testing problems arising from areas including coding theory, computational group theory, and multivariate cryptography. We present evidence to justify the viability of this proposal from comprehensive study of the state-of-art heuristic algorithms, theoretical algorithms, and hardness results, as well as quantum algorithms. We then introduce a new notion called pseudorandom group actions to further develop group-action based cryptography. Briefly speaking, given a group $G$ acting on a set $S$, we assume that it is hard to distinguish two distributions of $(s, t)$ either uniformly chosen from $S\times S$, or where $s$ is randomly chosen from $S$ and $t$ is the result of applying a random group action of $g\in G$ on $s$. This subsumes the classical decisional Diffie-Hellman assumption when specialized to a particular group action. We carefully analyze various attack strategies that support the general linear group action on tensors as a candidate for this assumption. Finally, we establish the quantum security of several cryptographic primitives based on the one-way group action assumption and the pseudorandom group action assumption.

cs.CR↗

arXiv2019

Incorporating Weisfeiler-Leman into algorithms for group isomorphism

In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically been applied to Graph Isomorphism. In particular, we show how to combine several state-of-the-art GpI algorithms for specific group classes into an algorithm for general GpI, namely: composition series isomorphism (Rosenbaum-Wagner, Theoret. Comp. Sci., 2015; Luks, 2015), recursively-refineable filters (Wilson, J. Group Theory, 2013), and low-genus GpI (Brooksbank-Maglione-Wilson, J. Algebra, 2017). Recursively-refineable filters -- a generalization of subgroup series -- form the skeleton of this framework, and we refine our filter by building a hypergraph encoding low-genus quotients, to which we then apply a hypergraph variant of the k-dimensional Weisfeiler-Leman technique. Our technique is flexible enough to readily incorporate additional hypergraph invariants or additional characteristic subgroups.

cs.CC↗

arXiv2019

Algorithms based on *-algebras, and their applications to isomorphism of polynomials with one secret, group isomorphism, and polynomial identity testing

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 $D$, such that for every $i\in\{1, \dots, m\}$, $AB_iD$ is (skew-)symmetric. We show that this problem can be solved in deterministic polynomial time over fields of characteristic not $2$. For both problems we exploit the structure of the underlying $*$-algebras, and utilize results and methods from the module isomorphism problem. Applications of our results range from multivariate cryptography, group isomorphism, to polynomial identity testing. Specifically, these results imply efficient algorithms for the following problems. (1) Test isomorphism of quadratic forms with one secret over a finite field of odd size. This problem belongs to a family of problems that serves as the security basis of certain authentication schemes proposed by Patarin (Eurocrypto 1996). (2) Test isomorphism of $p$-groups of class 2 and exponent $p$ ($p$ odd) with order $p^k$ in time polynomial in the group order, when the commutator subgroup is of order $p^{O(\sqrt{k})}$. (3) Deterministically reveal two families of singularity witnesses caused by the skew-symmetric structure, which represents a natural next step for the polynomial identity testing problem following the direction set up by the recent resolution of the non-commutative rank problem (Garg et al., FOCS 2016; Ivanyos et al., ITCS 2017).

cs.DS↗

arXiv2018

An improved diameter bound for finite simple groups of Lie type

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n {(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm{diam}(G) < q^{O(n {(\log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $\mathrm{diam}(H) < q^{O( n {(\log_{2}n)}^{2})}$ holds for any subgroup $H$ of $\mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.

math.GR↗

arXiv2018

Characterization of multipartite entanglement in terms of local transformations

The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to construct all homogenous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$-copies of them are LU equivalent for some $r$.

quant-ph↗

arXiv2018

Constructive noncommutative rank computation is in deterministic polynomial time

We extend our techniques developed in our earlier paper appeared in Computational Complexity, 2017 (preprint: arXiv:1508.00690) to obtain a deterministic polynomial time algorithm for computing the non-commutative rank together with certificates of linear spaces of matrices over sufficiently large base fields. The key new idea is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen and Makam. Both methods rely crucially on the regularity lemma in our aforementioned paper, and in this manuscript we also improve that lemma by removing a coprime condition there.

cs.CC↗

arXiv2017

Tripartite-to-bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces

We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate, which turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states, by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time.

quant-ph↗

arXiv2017

On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.

cs.CC↗

arXiv2017

Linear algebraic analogues of the graph isomorphism problem and the Erdős-Rényi model

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 group isomorphism. In 1970's, Babai, Erdős, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, we devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters, in a random model of alternating matrix spaces in vein of the Erdős-Rényi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. As a consequence of the main algorithm, we establish a weaker linear algebraic analogue of Erdős and Rényi's classical result that most graphs have the trivial automorphism group. We finally show that Luks' dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.

cs.DS↗

arXiv2017

On the Complexity of Trial and Error for Constraint Satisfaction Problems

In 2013 Bei, Chen and Zhang introduced a trial and error model of computing, and applied to some constraint satisfaction problems. In this model the input is hidden by an oracle which, for a candidate assignment, reveals some information about a violated constraint if the assignment is not satisfying. In this paper we initiate a {\em systematic} study of constraint satisfaction problems in the trial and error model. To achieve this, we first adopt a formal framework for CSPs, and based on this framework we define several types of revealing oracles. Our main contribution is to develop a \emph{transfer theorem} for each type of the revealing oracle, under a broad class of parameters. To any hidden CSP with a specific type of revealing oracle, the transfer theorem associates another, potentially harder CSP in the normal setting, such that their complexities are polynomial time equivalent. This in principle transfers the study of a large class of hidden CSPs, possibly with a promise on the instances, to the study of CSPs in the normal setting. We then apply the transfer theorems to get polynomial-time algorithms or hardness results for hidden CSPs, including satisfaction problems, monotone graph properties, isomorphism problems, and the exact version of the Unique Games problem. Most of the proofs of these results are short and straightforward, which exhibits the power of the transfer theorems.

cs.CC↗

arXiv2017

Networked Fairness in Cake Cutting

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality to this graphical setting. Given a simple undirected graph G, an allocation is envy-free on G if no agent envies any of her neighbor's share, and is proportional on G if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations open new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm recently designed by Aziz and Mackenzie for complete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on descendant graphs, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs.

cs.DS↗