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Detecting Disjoint Shortest Paths in Linear Time and More

Authors Shyan Akmal , Virginia Vassilevska Williams, Nicole Wein

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Author Details

Shyan Akmal
  • MIT, EECS and CSAIL, Cambridge, MA, USA
Virginia Vassilevska Williams
  • MIT, EECS and CSAIL, Cambridge, MA, USA
Nicole Wein
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Akmal, Shyan: Supported by Virginia Vassilevska Williams' Simons Investigator Award.
  • Vassilevska Williams, Virginia: Supported by NSF Grant CCF-2330048, BSF Grant 2020356, and a Simons Investigator Award.

Acknowledgements

We thank an anonymous reviewer for pointing out issues with a proof in a previous version of this work, and another anonymous reviewer for simplifying our construction of covering families. We thank Malte Renken and André Nichterlein for answering questions about previous work, as well as Matthias Bentert for telling us about his and his coauthors' work [Matthias Bentert et al., 2024], which independently proves Theorem 6 from this paper. The first author additionally thanks Ryan Williams and Zixuan Xu for conversations about early versions of ideas presented here, and Yinzhan Xu for nice discussions about the arguments in this paper.

Cite AsGet BibTex

Shyan Akmal, Virginia Vassilevska Williams, and Nicole Wein. Detecting Disjoint Shortest Paths in Linear Time and More. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.9

Abstract

In the k-Disjoint Shortest Paths (k-DSP) problem, we are given a graph G (with positive edge weights) on n nodes and m edges with specified source vertices s_1, … , s_k, and target vertices t_1, … , t_k, and are tasked with determining if G contains vertex-disjoint (s_i,t_i)-shortest paths. For any constant k, it is known that k-DSP can be solved in polynomial time over undirected graphs and directed acyclic graphs (DAGs). However, the exact time complexity of k-DSP remains mysterious, with large gaps between the fastest known algorithms and best conditional lower bounds. In this paper, we obtain faster algorithms for important cases of k-DSP, and present better conditional lower bounds for k-DSP and its variants. Previous work solved 2-DSP over weighted undirected graphs in O(n⁷) time, and weighted DAGs in O(mn) time. For the main result of this paper, we present optimal linear time algorithms for solving 2-DSP on weighted undirected graphs and DAGs. Our linear time algorithms are algebraic however, and so only solve the detection rather than search version of 2-DSP (we show how to solve the search version in O(mn) time, which is faster than the previous best runtime in weighted undirected graphs, but only matches the previous best runtime for DAGs). We also obtain a faster algorithm for k-Edge Disjoint Shortest Paths (k-EDSP) in DAGs, the variant of k-DSP where one seeks edge-disjoint instead of vertex-disjoint paths between sources and their corresponding targets. Algorithms for k-EDSP on DAGs from previous work take Ω(m^k) time. We show that k-EDSP can be solved over DAGs in O(mn^{k-1}) time, matching the fastest known runtime for solving k-DSP over DAGs. Previous work established conditional lower bounds for solving k-DSP and its variants via reductions from detecting cliques in graphs. Prior work implied that k-Clique can be reduced to 2k-DSP in DAGs and undirected graphs with O((kn)²) nodes. We improve this reduction, by showing how to reduce from k-Clique to k-DSP in DAGs and undirected graphs with O((kn)²) nodes (halving the number of paths needed in the reduced instance). A variant of k-DSP is the k-Disjoint Paths (k-DP) problem, where the solution paths no longer need to be shortest paths. Previous work reduced from k-Clique to p-DP in DAGs with O(kn) nodes, for p = k + k(k-1)/2. We improve this by showing a reduction from k-Clique to p-DP, for p = k + ⌊k²/4⌋. Under the k-Clique Hypothesis from fine-grained complexity, our results establish better conditional lower bounds for k-DSP for all k ≥ 4, and better conditional lower bounds for p-DP for all p ≤ 4031. Notably, our work gives the first nontrivial conditional lower bounds 4-DP in DAGs and 4-DSP in undirected graphs and DAGs. Before our work, nontrivial conditional lower bounds were only known for k-DP and k-DSP on such graphs when k ≥ 6.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • disjoint shortest paths
  • algebraic graph algorithms
  • disjoint paths
  • fine-grained complexity
  • clique

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