22,698 research outputs found
Beating the Generator-Enumeration Bound for -Group Isomorphism
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 p-group
composition-series isomorphism. By showing a Karp reduction from p-group
composition-series isomorphism to testing isomorphism of graphs of degree at
most p + O(1) and applying algorithms for testing isomorphism of graphs of
bounded degree, we obtain an n^(O(p)) time algorithm for p-group
composition-series isomorphism. Combining these two results yields an algorithm
for p-group isomorphism that takes at most n^((1 / 2) log_p n + O(p)) time.
This algorithm is faster than generator-enumeration when p is small and slower
when p is large. Choosing the faster algorithm based on p and n yields an upper
bound of n^((1 / 2 + o(1)) log n) for p-group isomorphism.Comment: 15 pages. This is an updated and improved version of the results for
p-groups in arXiv:1205.0642 and TR11-052 in ECC
Faster Isomorphism for -Groups of Class 2 and Exponent
The group isomorphism problem determines whether two groups, given by their
Cayley tables, are isomorphic. For groups with order , an algorithm with
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 running time
[Ros13].
The isomorphism testing for -groups of (nilpotent) class 2 and exponent
has been identified as a major barrier to obtaining an time
algorithm for the group isomorphism problem. Although the -groups of class 2
and exponent have much simpler algebraic structures than general groups,
the best-known isomorphism testing algorithm for this group class also has an
running time.
In this paper, we present an isomorphism testing algorithm for -groups of
class 2 and exponent with running time for any
prime . Our result is based on a novel reduction to the skew-symmetric
matrix tuple isometry problem [IQ19]. To obtain the reduction, we develop
several tools for matrix space analysis, including a matrix space
individualization-refinement method and a characterization of the low rank
matrix spaces.Comment: Accepted to STOC 202
On the complexity of isomorphism problems for tensors, groups, and polynomials IV: linear-length reductions and their applications
Many isomorphism problems for tensors, groups, algebras, and polynomials were
recently shown to be equivalent to one another under polynomial-time
reductions, prompting the introduction of the complexity class TI (Grochow &
Qiao, ITCS '21; SIAM J. Comp., '23). Using the tensorial viewpoint, Grochow &
Qiao (CCC '21) then gave moderately exponential-time search- and
counting-to-decision reductions for a class of -groups. A significant issue
was that the reductions usually incurred a quadratic increase in the length of
the tensors involved. When the tensors represent -groups, this corresponds
to an increase in the order of the group of the form ,
negating any asymptotic gains in the Cayley table model.
In this paper, we present a new kind of tensor gadget that allows us to
replace those quadratic-length reductions with linear-length ones, yielding the
following consequences:
1. Combined with the recent breakthrough -time
isomorphism-test for -groups of class 2 and exponent (Sun, STOC '23),
our reductions extend this runtime to -groups of class and exponent
where .
2. Our reductions show that Sun's algorithm solves several TI-complete
problems over , such as isomorphism problems for cubic forms, algebras,
and tensors, in time .
3. Polynomial-time search- and counting-to-decision reduction for testing
isomorphism of -groups of class and exponent in the Cayley table
model. This answers questions of Arvind and T\'oran (Bull. EATCS, 2005) for
this group class, thought to be one of the hardest cases of Group Isomorphism.
4. If Graph Isomorphism is in P, then testing equivalence of cubic forms and
testing isomorphism of algebra over a finite field can both be solved in
time , improving from the brute-force upper bound
Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers
We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn\u27s theorem on semisimple algebras, and M.E. Harris\u27s 1980
work on p\u27-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest
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.)
Improved algorithms for alternating matrix space isometry: From theory to practice
Motivated by testing isomorphism of p-groups, we study the alternating matrix space isometry problem (AltMatSpIso), which asks to decide whether two m-dimensional subspaces of n × n alternating (skew-symmetric if the field is not of characteristic 2) matrices are the same up to a change of basis. Over a finite field F with some prime p 6= 2, solving AltMatSpIso in time p is equivalent to testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. The latter problem has long been considered a bottleneck case for the group isomorphism problem. Recently, Li and Qiao presented an average-case algorithm for AltMatSpIso in time p when n and m are linearly related (FOCS’17). In this paper, we present an average-case algorithm for AltMatSpIso in time p . Besides removing the restriction on the relation between n and m, our algorithm is considerably simpler, and the average-case analysis is stronger. We then implement our algorithm, with suitable modifications, in Magma. Our experiments indicate that it improves significantly over default (brute-force) algorithms for this problem. p O (n+m) O (n) O (n+m
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