License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.8
URN: urn:nbn:de:0030-drops-186618
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Akmal, Shyan ; Wein, Nicole

A Local-To-Global Theorem for Congested Shortest Paths

LIPIcs-ESA-2023-8.pdf (0.9 MB)


Amiri and Wargalla proved the following local-to-global theorem about shortest paths in directed acyclic graphs (DAGs): if G is a weighted DAG with the property that for each subset S of 3 nodes there is a shortest path containing every node in S, then there exists a pair (s,t) of nodes such that there is a shortest st-path containing every node in G. We extend this theorem to general graphs. For undirected graphs, we prove that the same theorem holds (up to a difference in the constant 3). For directed graphs, we provide a counterexample to the theorem (for any constant). However, we prove a roundtrip analogue of the theorem which guarantees there exists a pair (s,t) of nodes such that every node in G is contained in the union of a shortest st-path and a shortest ts-path.
The original local-to-global theorem for DAGs has an application to the k-Shortest Paths with Congestion c ((k,c)-SPC) problem. In this problem, we are given a weighted graph G, together with k node pairs (s_1,t_1),… ,(s_k,t_k), and a positive integer c ≤ k, and tasked with finding a collection of paths P_1,… , P_k such that each P_i is a shortest path from s_i to t_i, and every node in the graph is on at most c paths P_i, or reporting that no such collection of paths exists. When c = k, there are no congestion constraints, and the problem can be solved easily by running a shortest path algorithm for each pair (s_i,t_i) independently. At the other extreme, when c = 1, the (k,c)-SPC problem is equivalent to the k-Disjoint Shortest Paths (k-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed k, k-DSP is polynomial-time solvable on DAGs and undirected graphs. Amiri and Wargalla interpolated between these two extreme values of c, to obtain an algorithm for (k,c)-SPC on DAGs that runs in polynomial time when k-c is constant.
In the same way, we prove that (k,c)-SPC can be solved in polynomial time on undirected graphs whenever k-c is constant. For directed graphs, because of our counterexample to the original theorem statement, our roundtrip local-to-global result does not imply such an algorithm (k,c)-SPC. Even without an algorithmic application, our proof for directed graphs may be of broader interest because it characterizes intriguing structural properties of shortest paths in directed graphs.

BibTeX - Entry

  author =	{Akmal, Shyan and Wein, Nicole},
  title =	{{A Local-To-Global Theorem for Congested Shortest Paths}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{8:1--8:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-186618},
  doi =		{10.4230/LIPIcs.ESA.2023.8},
  annote =	{Keywords: disjoint paths, shortest paths, congestion, parameterized complexity}

Keywords: disjoint paths, shortest paths, congestion, parameterized complexity
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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