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30
Search space contraction in canonical labeling of graphs
 CORR
, 2008
"... The individualizationrefinement paradigm for computing a canonical labeling and/or the automorphism group of a graph is investigated. New techniques are introduced with the aim of reducing the size of the associated search space. In particular, a new partition refinement algorithm is proposed, toge ..."
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The individualizationrefinement paradigm for computing a canonical labeling and/or the automorphism group of a graph is investigated. New techniques are introduced with the aim of reducing the size of the associated search space. In particular, a new partition refinement algorithm is proposed, together with graph invariants having a global nature. Experimental results and comparisons with existing tools, such as nauty, reveal that the presented approach produces a huge contraction of the search space. Such reduction will be shown to be exponential for special classes of graphs which are intractable by nauty.
Colored Hypergraph Isomorphism is Fixed Paramter Tractable
 Electronic Colloquium on Computation Complexity
, 2009
"... We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation g ..."
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We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism which has running time 2 O(b) N O(1) , where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm. Fixed parameter tractability, fpt algorithms, graph isomorphism, com
Bounded Color Multiplicity Graph Isomorphism is in the #L Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 121 (2004)
, 2004
"... In this paper we study the complexity of Bounded Color Multiplicity Graph Isomorphism BCGIb: the input is a pair of vertexcolored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGIb is in the #L hierarchy (more precisely, the ModkL hierarch ..."
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In this paper we study the complexity of Bounded Color Multiplicity Graph Isomorphism BCGIb: the input is a pair of vertexcolored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGIb is in the #L hierarchy (more precisely, the ModkL hierarchy for some constant k depending on b). Combined with the fact that Bounded Color Multiplicity Graph Isomorphism is logspace manyone hard for every set in the ModkL hierarchy for any constant k, we get a tight classification of the problem using logspacebounded counting classes.
Two Observations on Probabilistic Primality Testing
, 1987
"... In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's test so good. It turns out that a single iteration fails with a nonnegligible probability on a composite numb ..."
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In this note, we make two loosely related observations on Rabin's probabilistic primality test. The first remark gives a rather strange and provocative reason as to why is Rabin's test so good. It turns out that a single iteration fails with a nonnegligible probability on a composite number of the form 4j +3 only if this number happens to be easy to split. The second observation is much more fundamental because is it not restricted to primality testing: it has profound consequences for the entire field of probabilistic algorithms. There we ask the question: how good is Rabin's algorithm? Whenever one wishes to produce a uniformly distributed random probabilistic prime with a given bound on the error probability, it turns out that the size of the desired prime must be taken into account.
The Complexity of Graph Isomorphism for Colored Graphs with Color Classes of Size 2 and 3
"... We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under manyone reductions. This result improves the existing upper bounds for the problem. We also ..."
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We prove that the graph isomorphism problem restricted to colored graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under manyone reductions. This result improves the existing upper bounds for the problem. We also
A polynomialtime theory of matrix groups and black box groups
 in these Proceedings
"... We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic proble ..."
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We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a byproduct, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order ≥ N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.
A Nonadaptive NC Checker for Permutation Group Intersection
 Theoretical Computer Science
, 1997
"... We design a nonadaptive NC checker for permutation group intersection, sharpening a result of Blum and Kannan [4]. Additionally, we also get nonadaptive NC checkers for some related grouptheoretic problems. Keywords: program checking, interactive proofs, polynomial time. 1. Introduction We des ..."
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We design a nonadaptive NC checker for permutation group intersection, sharpening a result of Blum and Kannan [4]. Additionally, we also get nonadaptive NC checkers for some related grouptheoretic problems. Keywords: program checking, interactive proofs, polynomial time. 1. Introduction We design nonadaptive parallel program checkers for certain permutation grouptheoretic problems. As defined by Blum and Kannan in [4], a program checker for a problem takes a purported program P for that problem and an instance x as input and decides whether the output of P on x is correct. The checker has access to random bits and can query P on instances other than x. We focus on two efficiency parameters of a program checker: its running time and the number of adaptive queries made by the checker to the program. Clearly, the number of query rounds can be a serious bottleneck for the checker's running time. Thus, we consider nonadaptive NC checkers to be an ideal model for efficient parallel ch...
Symmetry and equivalence relations in classical and geometric complexity theory
, 2012
"... This thesis studies some of the ways in which symmetries and equivalence relations arise in classical and geometric complexity theory. The Geometric Complexity Theory program is aimed at resolving central questions in complexity such as P versus NP using techniques from algebraic geometry and repres ..."
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This thesis studies some of the ways in which symmetries and equivalence relations arise in classical and geometric complexity theory. The Geometric Complexity Theory program is aimed at resolving central questions in complexity such as P versus NP using techniques from algebraic geometry and representation theory. The equivalence relations we study are mostly algebraic in nature and we heavily use algebraic techniques to reason about the computational properties of these problems. We first provide a tutorial and survey on Geometric Complexity Theory to provide perspective and motivate the other problems we study. One equivalence relation we study is matrix isomorphism of matrix Lie algebras, which is a problem that arises naturally in Geometric Complexity Theory. For certain cases of matrix isomorphism of Lie algebras we provide polynomialtime algorithms, and for other cases we show that the problem is as hard as graph isomorphism. To our knowledge, this is the first time graph isomorphism has appeared in connection with any lower bounds program. Finally, we study algorithms for equivalence relations more generally (joint work with
LINBOX LIBRARY
, 2002
"... (Under the direction of Erich Kaltofen.) Black box algorithms for exact linear algebra view a matrix as a linear operator on a vector space, gathering information about the matrix only though matrixvector products and not by directly accessing the matrix elements. Wiedemann’s approach to black box ..."
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(Under the direction of Erich Kaltofen.) Black box algorithms for exact linear algebra view a matrix as a linear operator on a vector space, gathering information about the matrix only though matrixvector products and not by directly accessing the matrix elements. Wiedemann’s approach to black box linear algebra uses the fact that the minimal polynomial of a matrix generates the Krylov sequences of the matrix and their projections. By preconditioning the matrix, this approach can be used to solve a linear system, find the determinant of the matrix, or to find the matrix’s rank. This dissertation discusses preconditioners based on Beneˇs networks to localize the linear independence of a black box matrix and introduces a technique to use determinantal divisors to find preconditioners that ensure the cyclicity of nonzero eigenvalues. This technique, in turn, introduces a new determinantpreserving preconditioner for a dense integer matrix determinant algorithm based on the Wiedemann approach to black box linear algebra and relaxes a condition on the preconditioner for the KaltofenSaunders black box rank algorithm. The dissertation also investigates theminimal generating matrix polynomial of Coppersmith’s block Wiedemann algorithm, how to compute it using Beckermann and Labahn’s Fast Power HermitePadé Solver, and a block algorithm for computing the rank of a black box matrix. Finally, it discusses the design of the LinBox library for symbolic linear algebra. BLACK BOX LINEAR ALGEBRA WITH THE
The Complexity of Equivalence Relations
, 2008
"... To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equal ..."
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To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equality of permutation groups given by generators. To determine if two given graphs are cospectral, however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (“Equivalence relations, invariants, and normal forms, I and II”, 1984) ask whether these conditions on equivalence relations – having an FP canonical form, having an FP complete invariant, and simply being in P – are in fact different. They showed that this question requires nonrelativizing techniques to resolve. Here we extend their results using generic oracles, and give new connections to probabilistic and quantum computation.