Oded Goldreich

The Graph Clustering Problem is parameterized by a sequence

of positive integers, $m_1,...,m_t$.

The input is a sequence of $\sum_{i=1}^{t}m_i$ graphs,

and the question is whether the equivalence classes

under the graph isomorphism relation have sizes which match

the sequence of parameters.

In this note

we show ...
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Alfredo De Santis, Giovanni Di Crescenzo, Oded Goldreich, Giuseppe Persiano

The input to the {\em Graph Clustering Problem}\/

consists of a sequence of integers $m_1,...,m_t$

and a sequence of $\sum_{i=1}^{t}m_i$ graphs.

The question is whether the equivalence classes,

under the graph isomorphism relation,

of the input graphs have sizes which match the input sequence of integers.

In this note ...
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Vikraman Arvind, Piyush Kurur

We show that Graph Isomorphism is in the complexity class

SPP, and hence it is in $\ParityP$ (in fact, it is in $\ModkP$ for

each $k\geq 2$). We derive this result as a corollary of a more

general result: we show that a {\em generic problem} $\FINDGROUP$ has

an $\FP^{\SPP}$ ...
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Vikraman Arvind, Piyush Kurur, T.C. Vijayaraghavan

In this paper we study the complexity of Bounded Color

Multiplicity Graph Isomorphism (BCGI): the input is a pair of

vertex-colored graphs such that the number of vertices of a given

color in an input graph is bounded by $b$. We show that BCGI is in the

#L hierarchy ...
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Daniele Micciancio, Shien Jin Ong, Amit Sahai, Salil Vadhan

We provide <i>unconditional</i> constructions of <i>concurrent</i>

statistical zero-knowledge proofs for a variety of non-trivial

problems (not known to have probabilistic polynomial-time

algorithms). The problems include Graph Isomorphism, Graph

Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a

restricted version of Statistical Difference, and approximate

versions of the (<b>coNP</b> forms of the) Shortest Vector ...
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Thomas Thierauf, Fabian Wagner

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC^1.

In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to ... more >>>

Jacobo Toran

We show that several reducibility notions coincide when applied to the

Graph Isomorphism (GI) problem. In particular we show that if a set is

many-one logspace reducible to GI, then it is in fact many-one AC^0

reducible to GI. For the case of Turing reducibilities we show that ...
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Johannes Köbler, Sebastian Kuhnert

We show that k-tree isomorphism can be decided in logarithmic

space by giving a logspace canonical labeling algorithm. This improves

over the previous StUL upper bound and matches the lower bound. As a

consequence, the isomorphism, the automorphism, as well as the

canonization problem for k-trees ...
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Vikraman Arvind, Bireswar Das, Johannes Köbler, Seinosuke Toda

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism} which has running time $b!2^{O(b)}N^{O(1)}$, 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 ... more >>>

Johannes Köbler, Sebastian Kuhnert, Bastian Laubner, Oleg Verbitsky

We present a logspace algorithm for computing a canonical interval representation and a canonical labeling of interval graphs. As a consequence, the isomorphism and automorphism problems for interval graphs are solvable in logspace.

more >>>Fabian Wagner

The group isomorphism problem consists in deciding whether two groups $G$ and $H$

given by their multiplication tables are isomorphic.

An algorithm for group isomorphism attributed to Tarjan runs in time $n^{\log n + O(1)}$, c.f. [Mil78].

Miller and Monk showed in [Mil79] that group isomorphism can be many-one ... more >>>

Joshua Grochow

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set $\mathcal{L}$ of matrices such that $A,B \in \mathcal{L}$ implies$AB - BA \in ... more >>>

Vikraman Arvind, Sebastian Kuhnert, Johannes Köbler, Yadu Vasudev

We study optimization versions of Graph Isomorphism. Given two graphs $G_1,G_2$, we are interested in finding a bijection $\pi$ from $V(G_1)$ to $V(G_2)$ that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an $n^{O(\log n)}$ time approximation scheme that for any constant ... more >>>

Giorgio Ausiello, Francesco Cristiano, Luigi Laura

We investigate the complexity of the syntactic isomorphism problem of CNF Boolean Formulas (CSFI): given two CNF Boolean formulas $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ decide whether there exists a permutation of clauses, a permutation of literals and a bijection between their variables such that $\varphi(a_{1},\ldots,a_{n})$ and $\varphi(b_{1},\ldots,b_{n})$ become syntactically identical. We first ... more >>>

Johannes Köbler, Sebastian Kuhnert, Oleg Verbitsky

We present logspace algorithms for the canonical labeling problem and the representation problem of Helly circular-arc (HCA) graphs. The first step is a reduction to canonical labeling and representation of interval intersection matrices. In a second step, the Delta trees employed in McConnell's linear time representation algorithm for interval matrices ... more >>>

Eric Allender, Bireswar Das

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is

efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP^MCSP.

This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these ... more >>>

Vikraman Arvind, Gaurav Rattan

We study the complexity of Geometric Graph Isomorphism, in

$l_2$ and other $l_p$ metrics: given two sets of $n$ points $A,

B\subset \mathbb{Q}^k$ in

$k$-dimensional euclidean space the problem is to

decide if there is a bijection $\pi:A \rightarrow B$ such that for

...
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Vikraman Arvind, Gaurav Rattan

We give a $O^*(k^{O(k)})$ time isomorphism testing algorithm for graphs of eigenvalue multiplicity bounded by $k$ which improves on the previous

best running time bound of $O^*(2^{O(k^2/\log

k)})$.

Vikraman Arvind, Johannes Köbler, Gaurav Rattan, Oleg Verbitsky

Color refinement is a classical technique used to show that two given graphs $G$ and $H$

are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph $G$ amenable to color refinement if the color-refinement procedure succeeds in distinguishing $G$ from any non-isomorphic ...
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Eric Allender, Joshua Grochow, Cris Moore

We show that the Graph Automorphism problem is ZPP-reducible to MKTP, the problem of minimizing time-bounded Kolmogorov complexity. MKTP has previously been studied in connection with the Minimum Circuit Size Problem (MCSP) and is often viewed as essentially a different encoding of MCSP. All prior reductions to MCSP have applied ... more >>>

Eric Allender, Joshua Grochow, Dieter van Melkebeek, Cris Moore, Andrew Morgan

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions ... more >>>

Oded Goldreich

We consider the problem of testing asymmetry in the bounded-degree graph model, where a graph is called asymmetric if the identity permutation is its only automorphism. Seeking to determine the query complexity of this testing problem, we provide partial results. Considering the special case of $n$-vertex graphs with connected components ... more >>>

Eric Allender, Azucena Garvia Bosshard, Amulya Musipatla

This paper focuses on a variant of the circuit minimization problem (MCSP), denoted MKTP, which studies resource-bounded Kolmogorov complexity in place of circuit size. MCSP is not known to be hard for any complexity class under any kind of m-reducibility, but recently MKTP was shown to be hard for DET ... more >>>

Jacobo Toran, Florian Wörz

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution.

Using this connection, we ... more >>>

Omkar Baraskar, Agrim Dewan, Chandan Saha

An $n$-variate polynomial $g$ of degree $d$ is a $(n,d,t)$ design polynomial if the degree of the gcd of every pair of monomials of $g$ is at most $t-1$. The power symmetric polynomial $\mathrm{PSym}_{n,d} := \sum_{i=1}^{n} x^d_i$ and the sum-product polynomial $\mathrm{SP}_{s,d} := \sum_{i=1}^{s}\prod_{j=1}^{d} x_{i,j}$ are instances of design polynomials ... more >>>