2 Search Results for "Akhmedov, Maxim"

Document

DOI: 10.4230/LIPIcs.ITCS.2026.39

Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions

Authors: Keerti Choudhary, Amit Kumar, and Lakshay Saggi

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We study the 2-Disjoint Shortest Paths (2-DSP) problem: given a directed weighted graph and two terminal pairs (s₁,t₁) and (s₂,t₂), decide whether there exist vertex-disjoint shortest paths between each pair. Building on recent advances in disjoint shortest paths for DAGs and undirected graphs (Akmal et al. 2024), we present an O(mn log n)-time algorithm for this problem in weighted directed graphs that do not contain negative or zero weight cycles. This algorithm presents a significant improvement over the previously known O(m⁵n)-time bound (Berczi et al. 2017). Our approach exploits the algebraic structure of polynomials that enumerate shortest paths between terminal pairs. A key insight is that these polynomials admit a recursive decomposition, enabling efficient evaluation via dynamic programming over fields of characteristic two. Furthermore, we demonstrate how to report the corresponding paths in O(mn² log n)-time. In addition, we extend our techniques to a more general setting: given two terminal pairs (s₁, t₁) and (s₂, t₂) in a directed graph, find the minimum possible number of vertex intersections between any shortest path from s₁ to t₁ and s₂ to t₂. We call this the Minimum 2-Disjoint Shortest Paths (Min-2-DSP) problem. We provide in this paper the first efficient algorithm for this problem, including an O(m² n³)-time algorithm for directed graphs with positive edge weights, and an O(m+n)-time algorithm for DAGs and undirected graphs. Moreover, if the number of intersecting vertices is at least one, we show that it is possible to report the paths in the same O(m+n)-time. This is somewhat surprising, as there is no known o(mn) time algorithm for explicitly reporting the paths if they are vertex-disjoint, and is left as an open problem in (Akmal et al. 2024).

Cite as

Keerti Choudhary, Amit Kumar, and Lakshay Saggi. Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{choudhary_et_al:LIPIcs.ITCS.2026.39,
  author =	{Choudhary, Keerti and Kumar, Amit and Saggi, Lakshay},
  title =	{{Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{39:1--39:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.39},
  URN =		{urn:nbn:de:0030-drops-253267},
  doi =		{10.4230/LIPIcs.ITCS.2026.39},
  annote =	{Keywords: Disjoint paths, Disjoint shortest paths, Algebraic graph algorithms}
}

Document

DOI: 10.4230/LIPIcs.STACS.2023.5

Packing Odd Walks and Trails in Multiterminal Networks

Authors: Maxim Akhmedov and Maxim Babenko

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Let G be an undirected network with a distinguished set of terminals T ⊆ V(G) and edge capacities cap: E(G) → ℝ_+. By an odd T-walk we mean a walk in G (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd T-walks subject to capacities cap. First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd T-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if cap(δ(v)) is divisible by 4 for any v ∈ V(G)-T, our algorithm constructs an integer packing. Second, we establish and prove the corresponding min-max relation. Third, if G is inner Eulerian (i.e. degrees of all nodes in V(G)-T are even) and cap(e) = 2 for all e ∈ E, we show that there exists an integer packing of odd T-trails (i.e. odd T-walks with no repeated edges) of the same value as in case of odd T-walks, and this packing can be found in polynomial time. To achieve the above goals, we establish a connection between packings of odd T-walks and T-trails and certain multiflow problems in undirected and bidirected graphs.

Cite as

Maxim Akhmedov and Maxim Babenko. Packing Odd Walks and Trails in Multiterminal Networks. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{akhmedov_et_al:LIPIcs.STACS.2023.5,
  author =	{Akhmedov, Maxim and Babenko, Maxim},
  title =	{{Packing Odd Walks and Trails in Multiterminal Networks}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.5},
  URN =		{urn:nbn:de:0030-drops-176570},
  doi =		{10.4230/LIPIcs.STACS.2023.5},
  annote =	{Keywords: Odd path, signed and bidirected graph, multiflow, polynomial algorithm}
}

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2 Search Results for "Akhmedov, Maxim"

Efficient Algorithms for the Disjoint Shortest Paths Problem and Its Extensions

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Packing Odd Walks and Trails in Multiterminal Networks

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