A Protocol for Serializing Unique Strategies

Crasmaru, Marcel; Glaßer, Christian; Regan, Kenneth W.; Sengupta, Samik

doi:10.1007/978-3-540-28629-5_51

Cited by 8 publications

(4 citation statements)

References 11 publications

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“…The problem of discovering important subgraphs in a large network, however, poses significant computational challenges: the subgraph isomorphism problem, determining whether or not a given graph occurs as a subgraph in a larger graph, is NP-complete [22,115]. On the other hand, graph isomorphism, the special case of subgraph isomorphism when the graphs have the same size, is believed to be an easier problem [96,25], especially given the recent quasi-polynomial time algorithm [2]. A natural algorithm to count k-node motifs is therefore to enumerate subgraphs of size k and test whether they are isomorphic to the given motif.…”

Section: Detecting Network Motifsmentioning

confidence: 99%

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and its Applications

Tillquist,

Frongillo,

Lladser

2021

Preprint

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The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. This survey gives a self-contained introduction to metric dimension and an overview of the quintessential results and applications. We discuss methods for approximating the metric dimension of general graphs, and specific bounds and asymptotic behavior for deterministic and random families of graphs. We conclude with related concepts and directions for future work.

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Section: Detecting Network Motifsmentioning

confidence: 99%

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and its Applications

Tillquist,

Frongillo,

Lladser

2021

Preprint

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“…2 Arvind and Kurur [AK02] showed that the graph isomorphism problem (GI) belongs to SPP, a class introduced in [Gup92, OH93,FFK94]. Subsequently, Crasmaru et al [CGRS04] observed that the proof of classifying GI into SPP, as given by Arvind and Kurur [AK02], actually yields a somewhat improved classification for GI: GI belongs to R p s,T (Promise-UP), a subclass of SPP [CGRS04].…”

Section: Definition 210 (Unambiguity Based Hierarchies [Lr94nr98])mentioning

confidence: 99%

“…They introduced the following unambiguity based hierarchies: AUPH, UPH, and U PH. It is known that AUPH ⊆ UPH ⊆ U PH ⊆ UAP [LR94,CGRS04], where UAP (unambiguous alternating polynomial-time) is the analog of UP for alternating polynomialtime Turing machines. These hierarchies received renewed interests in some recent papers (see, for instance, [ACRW04,CGRS04, ST,GT05]).…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Unambiguity

Spakowski¹,

Tripathi²

2009

SIAM J. Comput.

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We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to generalize known constructs involving relativized unambiguity based complexity classes (UP and Promise-UP) to new constructs involving arbitrary higher levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical arrangement of unambiguous nondeterministic polynomial-time Turing machines, and so they differ substantially, in applicability and in nature, from standard methods (such as the switching lemma [Hås87]), which play roles in carrying out similar generalizations.Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč [CHV93] on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP. * A preliminary version of this paper was presented at the MFCS '06 conference. † Supported in part by the DFG under grants RO 1202/9-1 and RO 1202/9-3. hierarchy: They proved that there is a relativized world where Σ p 2 = Π p 2 . However, Baker and Selman [BS79] noted that their proof techniques do not apply at higher levels of the polynomial hierarchy because of certain constraints in their counting argument. Thus, it required the development of entirely different proof techniques for separating all the levels of the relativized polynomial hierarchy. The landmark paper by Furst, Saxe, and Sipser [FSS84] established the connection between the relativization of the polynomial hierarchy and lower bounds for small depth circuits computing certain functions. Techniques for proving such lower bounds were developed in a series of papers [FSS84, Sip83,Yao85,Hås87], which were motivated by questions about the relativized structure of the polynomial hierarchy. Yao [Yao85] finally succeeded in separating the levels of the relativized polynomial hierarchy by applying these new techniques. Håstad [Hås87] gave the most refined presentation of these techniques via the switching lemma. Even to date, Håstad's switching lemma [Hås87] is used as an essential tool to separate relativized hierarchies, composed of classes stacked one on top of another. (See, for instance,[Hås87,Ko89,BU98,ST] where the switching lemma is used as a strong tool for proving the feasibility of oracle constructions.)A major contribution of our paper lies in demonstrating that known oracle constructions involving the initial levels of the unambiguous polynomial hierarchy (UPH) and the promise unambiguous polynomial hierarchy (U PH), i.e. UP and P Promise-UP s , respectively, can be extended to oracle constructions involving arbitrary higher levels of UPH by application only of pure counting arguments. In fact, it seems implausible to achieve these extensions by well-known techniques from circuit complexity (e.g., the switching lemma [Hås87] and the polynomial method surveyed in [Bei93,Reg97]).

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“…We prove that the computational complexity of QSCD ff is lower-bounded by that of GA. Well-known upper bounds of GA include NP ∩ co-AM [21,54], SPP [5], and UAP [12]; however, GA is not known to sit in NP ∩ co-NP. Notice that, since most cryptographic problems fall in NP ∩ co-NP, very few cryptographic systems are lower-bounded by the worst-case hardness of problems outside of NP ∩ co-NP.…”

Section: Average-case Hardness Versus Worst-case Hardnessmentioning

confidence: 99%

Computational Indistinguishability Between Quantum States and Its Cryptographic Application

Kawachi

Koshiba

Nishimura

et al. 2005

Lecture Notes in Computer Science

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Abstract. We introduce a problem of distinguishing between two quantum states as a new underlying problem to build a computational cryptographic scheme that is "secure" against quantum adversary. Our problem is a natural generalization of the distinguishability problem between two probability distributions, which are commonly used in computational cryptography. More precisely, our problem QSCD ff is the computational distinguishability problem between two types of random coset states with a hidden permutation over the symmetric group. We show that (i) QSCD ff has the trapdoor property; (ii) the average-case hardness of QSCD ff coincides with its worst-case hardness; and (iii) QSCD ff is at least as hard in the worst case as the graph automorphism problem. Moreover, we show that QSCD ff cannot be efficiently solved by any quantum algorithm that naturally extends Shor's factorization algorithm. These cryptographic properties of QSCD ff enable us to construct a public-key cryptosystem, which is likely to withstand any attack of a polynomial-time quantum adversary.

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A Protocol for Serializing Unique Strategies

Cited by 8 publications

References 11 publications

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and its Applications

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and its Applications

Hierarchical Unambiguity

Computational Indistinguishability Between Quantum States and Its Cryptographic Application

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