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Level C · Reviewed Algorithms P-shortest-path-length-modulo-m

Shortest paths and cycles with prescribed length modulo m

Is there a polynomial-time algorithm for the shortest s–t path (or cycle) whose length is ℓ mod m? Known for ℓ = 0; open in general.

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Cite 
@misc{cairn-shortest-path-length-modulo-m,
  title        = {Shortest paths and cycles with prescribed length modulo m},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/shortest-path-length-modulo-m}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
}

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The problem

The question

Fix integers ℓ ≥ 0 and m ≥ 2.

  1. Is there a polynomial-time algorithm that, given a graph and vertices s, t, finds a shortest s–t path whose length is congruent to ℓ modulo m, or reports that none exists?
  2. The same for a shortest cycle of length ℓ modulo m.

What is known

The case ℓ = 0 was settled (independently by two groups) as a corollary of work on disjoint paths with group-expressible constraints. The general case is open even for small m (e.g. m = 3, ℓ = 1).

What counts as progress

  • Algorithms for specific (ℓ, m) or graph classes, tested against brute force on small graphs.
  • Hardness results (NP-hardness or conditional lower bounds).

Source. Posed by Chun-Hung Liu (with Youngho Yoo) in an extended abstract of the Oberwolfach workshop Graph Theory (2025), recorded in Oberwolfach Reports 1/2025, p. 35 (EMS Press, DOI 10.4171/OWR/2025/1), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.