Graph isomorphism is low for PP

Köbler, Johannes; Schöning, Uwe; Torán, Jacobo

doi:10.1007/3-540-55210-3_200

Cited by 10 publications

(7 citation statements)

References 33 publications

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“…It is known that all graphs, except for an exponentially small family, have no symmetries [5, p. 1461]. No general worst-case polynomial-time algorithms are known for this problem, but it is commonly believed not to be NP-complete (unless, of course, P=NP) [30]. Polynomial-time algorithms are available in many special cases [5, p. 1511], in particular for graphs of bounded degrees [33,3].…”

Section: A Subgroup Is a Subset Of A Group That Is Closed Under The Gmentioning

confidence: 99%

Solving difficult instances of boolean satisfiability in the presence of symmetry

Aloul

Ramani

Markov

et al. 2003

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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Research in algorithms for Boolean satisfiability (SAT) and their implementations [45,41,10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39,50,51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.We point out that a previously published symmetry-detection mechanism [18] based on a reduction to the graph automorphism problem often produces many spurious symmetries. Our work contributes two new reductions to graph automorphism, which detect all correct symmetries detected previously [18] as well as phase-shift symmetries not detected earlier. The correctness of our reductions is rigorously proven, and they are evaluated empirically.We also formulate an improved construction of symmetry-breaking clauses in terms of permutation cycles and propose to use only generators of symmetries in this process. These ideas are implemented in a fully automated flow that first detects symmetries in a given SAT instance, pre-processes it by adding symmetry-breaking clauses and then calls a state-of-the-art backtrack SAT solver. Significant speed-ups are shown on many benchmarks versus direct application of the solver.In an attempt to further improve the practicality of our approach, we propose a scheme for fast "opportunistic" symmetry detection and also show that considerations of symmetry may lead to more efficient reductions to SAT in the VLSI routing domain.

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Section: A Subgroup Is a Subset Of A Group That Is Closed Under The Gmentioning

confidence: 99%

Solving difficult instances of boolean satisfiability in the presence of symmetry

Aloul

Ramani

Markov

et al. 2003

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.

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“…Of course, it is highly improbable that the brain can identify arbitrary graph symmetries without additional assumptions: There is no algorithm known which solves this problem in complete generality in polynomial time, let alone in a biologically plausible way [29]. Yet it is possible that some approximation scheme has evolved which is effective in uncovering those invariances that are encoded in natural stimuli.…”

Section: The Read-out Mechanismmentioning

confidence: 99%

Does the Brain Infer Invariance Transformations from Graph Symmetries?

Linde¹

2021

Preprint

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The invariance of natural objects under perceptual changes is possibly encoded in the brain by symmetries in the graph of synaptic connections. The graph can be established via unsupervised learning in a biologically plausible process across different perceptual modalities. This hypothetical encoding scheme is supported by the correlation structure of naturalistic audio and image data and it predicts a neural connectivity architecture which is consistent with many empirical observations about primary sensory cortex.

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“…This problem is polynomially equivalent to the well-known graph isomorphism problem [33,40,19,32]. The graph isomorphism problem is clearly in the class of NP, and it is not known whether it is in P. It is also unknown whether the problem is NP-complete.…”

Section: Graph Isomorphism Problemmentioning

confidence: 99%

Contextualization as an independent abstraction mechanism for conceptual modeling

Analyti

Theodorakis²,

Spyratos

et al. 2007

Information Systems

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The notion of context appears in computer science, as well as in several other disciplines, in various forms. In this paper, we present a general framework for representing the notion of context in information modeling. First, we define a context as a set of objects, within which each object has a set of names and possibly a reference: the reference of the object is another context which "hides" detailed information about the object. Then, we introduce the possibility of structuring the contents of a context through the traditional abstraction mechanisms, i.e. classification, generalization, and attribution. We show that, depending on the application, our notion of context can be used as an independent abstraction mechanism, either in an alternative or a complementary capacity with respect to the traditional abstraction mechanisms. We also study the interactions between contextualization and the traditional abstraction mechanisms, as well as the constraints that govern such interactions. Finally, we present a theory for contextualized information bases. The theory includes a set of validity constraints, a model theory, as well as a set of sound and complete inference rules. We show that our core theory can be easily extended to support embedding of particular information models in our contextualization framework.

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Graph isomorphism is low for PP

Cited by 10 publications

References 33 publications

Solving difficult instances of boolean satisfiability in the presence of symmetry

Solving difficult instances of boolean satisfiability in the presence of symmetry

Does the Brain Infer Invariance Transformations from Graph Symmetries?

Contextualization as an independent abstraction mechanism for conceptual modeling

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