232,069 research outputs found
Isomorphism
Isomorphism: Initially referring to the structuralist, in particular glossematic, hypothesis that the expression and meaning of linguistic signs show structural parallelisms, the term isomorphism is used in current linguistic theory to designate the one-to-one correspondence between expression and meaning
On the Complexity of Polytope Isomorphism Problems
We show that the problem to decide whether two (convex) polytopes, given by
their vertex-facet incidences, are combinatorially isomorphic is graph
isomorphism complete, even for simple or simplicial polytopes. On the other
hand, we give a polynomial time algorithm for the combinatorial polytope
isomorphism problem in bounded dimensions. Furthermore, we derive that the
problems to decide whether two polytopes, given either by vertex or by facet
descriptions, are projectively or affinely isomorphic are graph isomorphism
hard.
The original version of the paper (June 2001, 11 pages) had the title ``On
the Complexity of Isomorphism Problems Related to Polytopes''. The main
difference between the current and the former version is a new polynomial time
algorithm for polytope isomorphism in bounded dimension that does not rely on
Luks polynomial time algorithm for checking two graphs of bounded valence for
isomorphism. Furthermore, the treatment of geometric isomorphism problems was
extended.Comment: 16 pages; to appear in: Graphs and Comb.; replaces our paper ``On the
Complexity of Isomorphism Problems Related to Polytopes'' (June 2001
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
Good potentials for almost isomorphism of countable state Markov shifts
Almost isomorphism is an equivalence relation on countable state Markov
shifts which provides a strong version of Borel conjugacy; still, for mixing
SPR shifts, entropy is a complete invariant of almost isomorphism. In this
paper, we establish a class of potentials on countable state Markov shifts
whose thermodynamic formalism is respected by almost isomorphism
Reduction Techniques for Graph Isomorphism in the Context of Width Parameters
We study the parameterized complexity of the graph isomorphism problem when
parameterized by width parameters related to tree decompositions. We apply the
following technique to obtain fixed-parameter tractability for such parameters.
We first compute an isomorphism invariant set of potential bags for a
decomposition and then apply a restricted version of the Weisfeiler-Lehman
algorithm to solve isomorphism. With this we show fixed-parameter tractability
for several parameters and provide a unified explanation for various
isomorphism results concerned with parameters related to tree decompositions.
As a possibly first step towards intractability results for parameterized graph
isomorphism we develop an fpt Turing-reduction from strong tree width to the a
priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure
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