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On the Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 44 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
Completeness results for Graph Isomorphism
, 2002
"... We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is manyone complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹hard under AC0reductions ..."
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Cited by 27 (9 self)
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We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is manyone complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹hard under AC0reductions. NC¹completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is Lcomplete. Concerning colored graphs we show that the isomorphism problem for 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 show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.
3connected planar graph isomorphism is in logspace
, 2008
"... We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13]. ..."
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Cited by 13 (3 self)
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We show that the isomorphism of 3connected planar graphs can be decided in deterministic logspace. This improves the previously known bound UL ∩ coUL of [13].
Partially commutative inverse monoids
 PROCEEDINGS OF THE 31TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS 2006), BRATISLAVE (SLOVAKIA), NUMBER 4162 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algo ..."
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Cited by 5 (5 self)
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Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NPcompleteness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.
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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Cited by 2 (2 self)
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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
Monomials, Multilinearity and Identity Testing in Simple ReadRestricted Circuits∗
, 2013
"... We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice or readthrice formulas. In the process, these algorithms also test if the input circuit is compu ..."
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Cited by 1 (1 self)
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We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are readtwice or readthrice formulas. In the process, these algorithms also test if the input circuit is computing a multilinear polynomial. We further study three related computational problems on arithmetic circuits. Given an arithmetic circuit C, 1) ZMC: test if a given monomial in C has zero coefficient or not, 2) MonCount: compute the number of monomials in C, and 3) MLIN: test if C computes a multilinear polynomial or not. These problems were introduced by Fournier, Malod and Mengel [STACS 2012], and shown to characterize various levels of the counting hierarchy (CH). We address the above problems on readrestricted arithmetic circuits and branching programs. We prove several complexity characterizations for the above problems on these restricted classes of arithmetic circuits.
Graph Isomorphism for K 3,3free and K 5free graphs is in Logspace
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
"... Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, [8] proved that planar isomorphism is complete for logspace. We extend this result further to th ..."
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Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, [8] proved that planar isomorphism is complete for logspace. We extend this result further to the classes of graphs which exclude K3,3 or K5 as a minor, and give a logspace algorithm. Our algorithm decomposes K3,3 minorfree graphs into biconnected and those further into triconnected components, which are known to be either planar or K5 components [20]. This gives a triconnected component tree similar to that for planar graphs. An extension of the logspace algorithm of [8] can then be used to decide the isomorphism problem. For K5 minorfree graphs, we consider 3connected components. These are either planar or isomorphic to the fourrung mobius ladder on 8 vertices or, with a further decomposition, one obtains planar 4connected components [9]. We give an algorithm to get a unique decomposition of K5 minorfree graphs into bi, tri and 4connected components, and construct trees, accordingly. Since the algorithm of [8] does not deal with fourconnected component trees, it needs to be modified in a quite nontrivial way.