13,985 research outputs found
Quantum Algorithms for Tree Isomorphism and State Symmetrization
The graph isomorphism problem is theoretically interesting and also has many
practical applications. The best known classical algorithms for graph
isomorphism all run in time super-polynomial in the size of the graph in the
worst case. An interesting open problem is whether quantum computers can solve
the graph isomorphism problem in polynomial time. In this paper, an algorithm
is shown which can decide if two rooted trees are isomorphic in polynomial
time. Although this problem is easy to solve efficiently on a classical
computer, the techniques developed may be useful as a basis for quantum
algorithms for deciding isomorphism of more interesting types of graphs. The
related problem of quantum state symmetrization is also studied. A polynomial
time algorithm for the problem of symmetrizing a set of orthonormal states over
an arbitrary permutation group is shown
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
A Polynomial Time Algorithm for Graph Isomorphism
Algorithms testing two graphs for isomorphism known as yet in computer
science have exponential worst case complexity. In this paper we propose an
algorithm that has polynomial complexity and constructively supplies the
evidence that the graph isomorphism lies in P
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
The Graph Isomorphism Problem and approximate categories
It is unknown whether two graphs can be tested for isomorphism in polynomial
time. A classical approach to the Graph Isomorphism Problem is the
d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can
distinguish many pairs of graphs, but the pairs of non-isomorphic graphs
constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed,
then the WL-algorithm runs in polynomial time. We will formulate the Graph
Isomorphism Problem as an Orbit Problem: Given a representation V of an
algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie
in the same G-orbit. Then we attack the Orbit Problem by constructing certain
approximate categories C_d(V), d=1,2,3,... whose objects include the elements
of V. We show that v_1 and v_2 are not in the same orbit by showing that they
are not isomorphic in the category C_d(V) for some d. For every d this gives us
an algorithm for isomorphism testing. We will show that the WL-algorithms
reduce to our algorithms, but that our algorithms cannot be reduced to the
WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can
distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page
Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth
We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and
develop a quasipolynomial-time algorithm for the multiple-coset isomorphism
problem. The algorithm for the multiple-coset isomorphism problem allows to
exploit graph decompositions of the given input graphs within Babai's
group-theoretic framework.
We use it to develop a graph isomorphism test that runs in time
where is the number of vertices and is
the minimum treewidth of the given graphs and is
some polynomial in . Our result generalizes Babai's
quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur
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