Determinant Sums for Undirected Hamiltonicity

Björklund, Andreas

doi:10.1137/110839229

Cited by 118 publications

(179 citation statements)

References 25 publications

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“…Even with smart algorithms such as dynamic programming, the conventional computers would require approximately 1/8?N 2 2 N operations (additions and comparisons). 20,21 For computers with a clock period t, this process will require at least t?1/8?N 2 2 N s to perform, which indicates that our approach is Nt/8Dt times faster than dynamic programming algorithms. With a pulse duration of 1 ps, a graph with 30 nodes could be solved approximately 375 times faster than with a 10-GHz clock rate computer.…”

Section: An Opticalmentioning

confidence: 99%

“…With a pulse duration of 1 ps, a graph with 30 nodes could be solved approximately 375 times faster than with a 10-GHz clock rate computer. We note that the optical oracle loses to probabilistic Monte Carlo algorithms 21 that can solve the Hamiltonian path problem with a certain degree of uncertainty in time O(1.657 N ). However, our approach completely excludes false predictions.…”

Section: An Opticalmentioning

confidence: 99%

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An optical fiber network oracle for NP-complete problems

Abajo

Soci

et al. 2014

Light Sci Appl

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The modern information society is enabled by photonic fiber networks characterized by huge coverage and great complexity and ranging in size from transcontinental submarine telecommunication cables to fiber to the home and local segments. This world-wide network has yet to match the complexity of the human brain, which contains a hundred billion neurons, each with thousands of synaptic connections on average. However, it already exceeds the complexity of brains from primitive organisms, i.e., the honey bee, which has a brain containing approximately one million neurons. In this study, we present a discussion of the computing potential of optical networks as information carriers. Using a simple fiber network, we provide a proof-of-principle demonstration that this network can be treated as an optical oracle for the Hamiltonian path problem, the famous mathematical complexity problem of finding whether a set of towns can be travelled via a path in which each town is visited only once. Pronouncement of a Hamiltonian path is achieved by monitoring the delay of an optical pulse that interrogates the network, and this delay will be equal to the sum of the travel times needed to visit all of the nodes (towns). We argue that the optical oracle could solve this NP-complete problem hundreds of times faster than brute-force computing. Additionally, we discuss secure communication applications for the optical oracle and propose possible implementation in silicon photonics and plasmonic networks.

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Section: An Opticalmentioning

confidence: 99%

Section: An Opticalmentioning

confidence: 99%

An optical fiber network oracle for NP-complete problems

Abajo

Soci

et al. 2014

Light Sci Appl

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“…However, rather than work with group algebras we find it more convenient to pursue an implementation via multivariate polynomials and inclusion-exclusion sieving by polynomial substitution [4,5,6,7].…”

Section: Motivation and Earliermentioning

confidence: 99%

Engineering Motif Search for Large Graphs

Björklund

Kaski

Kowalik

et al. 2014

2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)

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In the graph motif problem, we are given as input a vertexcolored graph H (the host graph) and a multiset of colors M (the motif). Our task is to decide whether H has a connected set of vertices whose multiset of colors agrees with M . The graph motif problem is NP-complete but known to admit parameterized algorithms that run in linear time in the size of H. We demonstrate that algorithms based on constrained multilinear sieving are viable in practice, scaling to graphs with hundreds of millions of edges as long as M remains small. Furthermore, our implementation is topologyinvariant relative to the host graph H, meaning only the most crude graph parameters (number of edges and number of vertices) suffice in practice to determine the algorithm performance.

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“…We develop an FPT algorithm for (k, l)-Tree in general graphs that relies upon a nontrivial generalization of the technique underlying the Hamiltonicity and k-Path algorithms in [2,4]. As a result, we break the natural barrier of O * (2 n ) in the running time bound for the k-IST problem.…”

Section: Introductionmentioning

confidence: 99%

Spotting Trees with Few Leaves

Björklund

Kamat

Kowalik

et al. 2017

SIAM J. Discrete Math.

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We show two results related to the Hamiltonicity and k-Path algorithms in undirected graphs by Björklund [FOCS'10], and Björklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O * (1.657 k 2 l/2 ) time. It can be applied as a subroutine to solve the k-Internal Spanning Tree (k-IST) problem in O * (min(3.455 k , 1.946 n )) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of O * (2 n ). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree ∆ = 4, . . . , 12 or with vector chromatic number at most 8.

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Determinant Sums for Undirected Hamiltonicity

Cited by 118 publications

References 25 publications

An optical fiber network oracle for NP-complete problems

An optical fiber network oracle for NP-complete problems

Engineering Motif Search for Large Graphs

Spotting Trees with Few Leaves

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