A Polynomial Time Algorithm for the <i>k</i>-Disjoint Shortest Paths Problem

Lochet, William

doi:10.1137/1.9781611976465.12

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…Therefore, finding many paths for each of the pairs k is "at least of class NP," as this problem can be reduced to k-DPP by adding an appropriate number of paths between distinguished pairs of terminals. However, if k is fixed, polynomial solutions exist for k-DPP and even for shortest k-DPP [20,21], so we can hope to search for a solution while dividing the network into clusters.…”

Section: Multiple Communicating Partiesmentioning

confidence: 99%

Analysis of Multiple Overlapping Paths algorithms for Secure Key Exchange in Large-Scale Quantum Networks

Stępniak¹,

Mielczarek²

2022

Preprint

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Quantum networks open the way to an unprecedented level of communication security. However, due to physical limitations on the distances of quantum links, current implementations of quantum networks are unavoidably equipped with trusted nodes. As a consequence, the quantum key distribution can be performed only on the links. Due to this, some new authentication and key exchange schemes must be considered to fully benefit from the unconditional security of links. One such approach uses Multiple Non-Overlapping Paths (MNOPs) for key exchange to mitigate the risk of an attack on a trusted node. The scope of the article is to perform a security analysis of this scheme for the case of both uncorrelated attacks and correlated attacks with finite resources. Furthermore, our analysis is extended to the case of Multiple Overlapping Paths (MOPs). We prove that introducing overlapping paths allows one to increase the security of the protocol, compared to the non-overlapping case with the same number of additional links added. This result may find application in optimising architectures of large-scale (hybrid) quantum networks.

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Section: Multiple Communicating Partiesmentioning

confidence: 99%

Analysis of Multiple Overlapping Paths algorithms for Secure Key Exchange in Large-Scale Quantum Networks

Stępniak¹,

Mielczarek²

2022

Preprint

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“…In other work of Bérczi et al [3] they showed that the undirected k-DSPP (disjoint shortest paths problem) and the vertex-disjoint version of the directed k-DSPP can be solved in polynomial time if the input graph is planar and k is a fixed constant. Lochet [16] shows that for any fixed k, the disjoint shortest paths problem admits a slicewise polynomial time algorithm.…”

Section: Related Workmentioning

confidence: 99%

Partially Disjoint k Shortest Paths

Dinitz¹,

Kumar²,

Schieber³

2022

Preprint

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A solution of the k shortest paths problem may output paths that are identical up to a single edge. On the other hand, a solution of the k independent shortest paths problem consists of paths that share neither an edge nor an intermediate node. We investigate the case in which the number of edges that are not shared among any two paths in the output k-set is a parameter. We study two main directions: exploring near-shortest paths and exploring exactly shortest paths. We assume that the weighted graph G = (V, E, w) has no parallel edges and that the edge lengths (weights) are positive. Our results are also generalized to the cases of k shortest paths where there are several weights per edge, and the results should take into account the multi-criteria prioritized weight.

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“…At the point of writing, the complexity of the problem was still open for r ≥ 3. In the meantime, it was shown that finding r disjoint shortest paths in undirected, unweighted graphs is indeed polynomial-time solvable for each fixed r [8,42].…”

Section: (|V (G)| + |V (T )| + )mentioning

confidence: 99%

“…5, that finding r disjoint shortest paths is polynomialtime solvable for r = 2 even if the paths are required to be compatible with a transition system. In the case of undirected graphs without forbidden transitions, this problem is polynomial-time solvable for each fixed r [8,42]. However, the problem is open for every r ≥ 3 in the case of directed graphs without forbidden transitions.…”

Section: Lemma 55 In a Directed Acyclic Graph G = (V E) With Transit...mentioning

confidence: 99%

The Complexity of Routing Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

Bellitto

Okrasa

et al. 2022

Algorithmica

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The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. A widely-studied special case are edge-colored graphs, where a compatible walk is forbidden to take two edges of the same color in a row. We initiate the study of fundamental problems on finding paths, cycles and walks in forbidden-transition graphs from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth for finding a compatible Hamiltonian cycle in the edge-colored graph setting.

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A Polynomial Time Algorithm for the k-Disjoint Shortest Paths Problem

Cited by 7 publications

References 9 publications

Analysis of Multiple Overlapping Paths algorithms for Secure Key Exchange in Large-Scale Quantum Networks

Analysis of Multiple Overlapping Paths algorithms for Secure Key Exchange in Large-Scale Quantum Networks

Partially Disjoint k Shortest Paths

The Complexity of Routing Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

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