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27
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 36 (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.
Randomization and Derandomization in SpaceBounded Computation
 In Proceedings of the 11th Annual IEEE Conference on Computational Complexity
, 1996
"... This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators tha ..."
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Cited by 35 (0 self)
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This is a survey of spacebounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic spacebounded computations, and the application of such generators to obtain deterministic simulations.
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 26 (8 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
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 25 (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.
The directed planar reachability problem
 In Proc. 25th annual Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS), number 1373 in Lecture Notes in Computer Science
, 2005
"... Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the ..."
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Cited by 23 (9 self)
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Abstract. We investigate the stconnectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspacereducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previouslystudied subclass of planar graphs known as grid graphs. We show that the directed planar stconnectivity problem reduces to the reachability problem for directed grid graphs. A special case of the gridgraph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1
THE ISOMORPHISM PROBLEM FOR PLANAR 3CONNECTED GRAPHS IS IN UNAMBIGUOUS LOGSPACE
, 2008
"... The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace ..."
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Cited by 12 (5 self)
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The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class AC¹. In this paper we improve the upper bound for planar 3connected graphs to unambiguous logspace, in fact to UL ∩ coUL. As a consequence of our method we get that the isomorphism problem for oriented graphs is in NL. We also show that the problems are hard for L.
A Short History of Computational Complexity
 IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
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Cited by 11 (1 self)
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this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld
Refining Randomness
, 1996
"... deny we succeeded with Gal. I want to conclude with my family. I will never forget the love and support I received from my brother and sisters even when we deeply disagreed. Nothing will divide us! Many warm wishes to you Deanna and Harry. I appreciate your letting us go our way. Many times when I ..."
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Cited by 8 (2 self)
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deny we succeeded with Gal. I want to conclude with my family. I will never forget the love and support I received from my brother and sisters even when we deeply disagreed. Nothing will divide us! Many warm wishes to you Deanna and Harry. I appreciate your letting us go our way. Many times when I look at Gal I appreciate the sacrifice you have made. We love you very much. We wish our small world was smaller. Finally, to the one who brought light into my lonely life. To the one with whom I share my life, happy or sad. Dear Paula and lovely Gal, my soul and blood  I love you. Contents 1 Introduction 1 1.1 Randomness Has Lots of Structure . . . . . . . . . . . . . . . . . . . 1 1.1.1 An Example: Random Walks . . . . . . . . . . . . . . . . . . 3 1.2 Is Randomness Feasible? . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Chaos, Quantum Mechanics and Crude Randomness . . . . . 5 1.2.2 Refining Crude Randomness . . . . . . . . . . . . . . . . . .
On the Complexity of Matrix Rank and Rigidity
"... We revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computi ..."
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Cited by 5 (1 self)
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We revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computing the permanent and determinant of tridiagonal matrices over Z is in GapNC 1 and is hard for NC 1. We also initiate the study of computing the rigidity of a matrix: the number of entries that needs to be changed in order to bring the rank of a matrix below a given value. We show that some restricted versions of the problem characterize small complexity classes. We also look at a variant of rigidity where there is a bound on the amount of change allowed. Using ideas from the linear interval equations literature, we show that this problem is NPhard over Q and that a certain restricted version is NPcomplete. Restricting the problem further, we obtain variations which can be computed in PL and are hard for C=L. 1