Results 1  10
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19
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
The complexity of Boolean constraint isomorphism
 In Proceedings of the 21st Symposium on Theoretical Aspects of Computer Science
, 2004
"... Abstract. In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems (nowadays usually called Boolean constraint satisfaction problems) and showed that all these satisfiability problems are eit ..."
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Cited by 8 (4 self)
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Abstract. In 1978, Schaefer proved his famous dichotomy theorem for generalized satisfiability problems. He defined an infinite number of propositional satisfiability problems (nowadays usually called Boolean constraint satisfaction problems) and showed that all these satisfiability problems are either in P or NPcomplete. In recent years, similar results have been obtained for quite a few other problems for Boolean constraints. Almost all of these problems are variations of the satisfiability problem. In this paper, we address a problem that is not a variation of satisfiability, namely, the isomorphism problem for Boolean constraints. Previous work by Böhler et al. showed that the isomorphism problem is either coNPhard or reducible to the graph isomorphism problem (a problem that is in NP, but not known to be NPhard), thus distinguishing a hard case and an easier case. However, they did not classify which cases are truly easy, i.e., in P. This paper accomplishes exactly that. It shows that Boolean constraint isomorphism is coNPhard (and GIhard), or equivalent to graph isomorphism, or in P, and it gives simple criteria to determine which case holds.
Planar graph isomorphism is in logspace
 In IEEE Conference on Computational Complexity
, 2009
"... Abstract. 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]. 1 ..."
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Cited by 8 (1 self)
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Abstract. 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]. 1
Computational indistinguishability between quantum states and its cryptographic application
 Advances in Cryptology – EUROCRYPT 2005
, 2005
"... We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is “secure ” against any polynomialtime quantum adversary. Our problem QSCDff is to distinguish between two types of random coset s ..."
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Cited by 7 (5 self)
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We introduce a computational problem of distinguishing between two specific quantum states as a new cryptographic problem to design a quantum cryptographic scheme that is “secure ” against any polynomialtime quantum adversary. Our problem QSCDff is to distinguish between two types of random coset states with a hidden permutation over the symmetric group of finite degree. This naturally generalizes the commonlyused distinction problem between two probability distributions in computational cryptography. As our major contribution, we show three cryptographic properties: (i) QSCDff has the trapdoor property; (ii) the averagecase hardness of QSCDff coincides with its worstcase hardness; and (iii) QSCDff is computationally at least as hard in the worst case as the graph automorphism problem. These cryptographic properties enable us to construct a quantum publickey cryptosystem, which is likely to withstand any chosen plaintext attack of a polynomialtime quantum adversary. We further discuss a generalization of QSCDff, called QSCDcyc, and introduce a multibit encryption scheme relying on the cryptographic properties of QSCDcyc.
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 5 (3 self)
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Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ̸ = 2. 1
The size of SPP
 Theoretical Computer Science
"... Derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class SPP. 1. µp(SPP) = 0. 2. PH ⊆ SPP. In other words, SPP is small by being a negligible subset of exponential time or large by containing the entire polynomialtime ..."
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Cited by 4 (1 self)
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Derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class SPP. 1. µp(SPP) = 0. 2. PH ⊆ SPP. In other words, SPP is small by being a negligible subset of exponential time or large by containing the entire polynomialtime hierarchy. This addresses an open problem about the complexity of the graph isomorphism problem: it is not weakly complete for exponential time unless PH is contained in SPP. It is also shown that the polynomialtime hierarchy is contained in SPP NP if NP does not have pmeasure 0. 1
Degree bounds on polynomials and relativization theory
 In Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science
, 2003
"... Abstract We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turinghard sets in some relativized world, (non)uniform gapdefinability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & ..."
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Cited by 3 (2 self)
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Abstract We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turinghard sets in some relativized world, (non)uniform gapdefinability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & Zimand [HRZ95] and Fenner, Fortnow & Kurtz [FFK94], extend results of Hemaspaandra, Jain & Vereshchagin [HJV93] and construct oracles achieving desired results.
On the power of unambiguity in alternating machines
 In Proceedings of the 15th International Symposium on Fundamentals of Computation Theory
, 2004
"... Abstract. Recently, the property of unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globallyunique games by Aida et al. [1] and in the design of efficient protocols involving globallyunique games by Crâsmaru et al. [7]. This paper investig ..."
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Cited by 3 (1 self)
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Abstract. Recently, the property of unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globallyunique games by Aida et al. [1] and in the design of efficient protocols involving globallyunique games by Crâsmaru et al. [7]. This paper investigates the power of unambiguity in alternating Turing machines in the following settings: 1. We construct a relativized world where unambiguity based hierarchies—AUPH, UPH, and UPH—are infinite. We construct another relativized world where UAP (unambiguous alternating polynomialtime) is not contained in the polynomial hierarchy. 2. We define the boundedlevel unambiguous alternating solution class UAS(k), for every k ≥ 1, as the class of sets for which strings in the set are accepted unambiguously by some polynomialtime alternating Turing machine N with at most k alternations, while strings not in the set either are rejected or are accepted with ambiguity by N. We construct a relativized world where, for all k ≥ 1, UP≤k ⊂ UP≤k+1 and UAS(k) ⊂ UAS(k + 1). 3. Finally, we show that robustly klevel unambiguous polynomialtime alternating Turing machines accept languages that are computable in P Σp k ⊕A, for every oracle A. This generalizes a result of Hartmanis
A protocol for serializing unique strategies
 In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science. SpringerVerlag Lecture Notes in Computer Science #3153
, 2004
"... Abstract. We devise an efficient protocol by which a series of twoperson games Gi with unique winning strategies can be combined into a single game G with unique winning strategy, even when the result of G is a nonmonotone function of the results of the Gi that is unknown to the players. In computa ..."
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Cited by 3 (0 self)
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Abstract. We devise an efficient protocol by which a series of twoperson games Gi with unique winning strategies can be combined into a single game G with unique winning strategy, even when the result of G is a nonmonotone function of the results of the Gi that is unknown to the players. In computational complexity terms, we show that the class UAP of Niedermeier and Rossmanith [NR98] of languages accepted by unambiguous polynomialtime alternating TMs is selflow, i.e., UAP UAP = UAP. It follows that UAP contains the Graph Isomorphism problem, nominally improving the problem’s classification into SPP by Arvind and Kurur [AK02] since UAP is a subclass of SPP [NR98]. We give some other applications, oracle separations, and results on problems related to uniquealternation formulas. 1
Complexity results in graph reconstruction
 In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science
"... We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertexdeleted or edgedeleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related ..."
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Cited by 1 (0 self)
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We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertexdeleted or edgedeleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c ≥ 1 of deletion: 1. GI ≡l iso VDCc, GI≡l iso EDCc, GI≤l m LVDc, andGI ≡ p iso LEDc. 2. For all k ≥ 2, GI ≡ p iso kVDCc and GI ≡ p iso kEDCc. 3. For all k ≥ 2, GI ≤l m kLVDc. 4. GI ≡ p iso 2LVDc. 5. For all k ≥ 2, GI ≡ p iso kLEDc. For many of these results, even the c = 1 case was not previously known. Similar to the definition of reconstruction numbers vrn∃(G) [HP85]andern∃(G) (see p. 120 of [LS03]), we introduce two new graph parameters, vrn∀(G) andern∀(G), and give an example of a family {Gn}n≥4 of graphs on n vertices for which vrn∃(Gn) < vrn∀(Gn). For every k ≥ 2andn ≥ 1, we show that there exists a collection of k graphs on (2k−1 +1)n + k vertices with 2n 1vertexpreimages, i.e., one has families of graph collections whose number of 1vertexpreimages is huge relative to the size of the graphs involved.