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The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs

Authors Archit Chauhan, Samir Datta , Chetan Gupta , Vimal Raj Sharma

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Author Details

Archit Chauhan
  • Chennai Mathematical Institute, India
Samir Datta
  • Chennai Mathematical Institute & UMI ReLaX, India
Chetan Gupta
  • Indian Institute of Technology, Roorkee, India
Vimal Raj Sharma
  • Indian Institute of Technology, Jodhpur, India

Funding

  • Chauhan, Archit: Partially supported by a grant from Infosys foundation and TCS PhD fellowship.
  • Datta, Samir: Partially supported by a grant from Infosys foundation.

Acknowledgements

We would like to thank anonymous reviewers for their useful comments and corrections in past submissions. We would also like to thank Martin Grohe for pointing out some references. We are also grateful to Geevarghese Philip and Vishwa Prakash H .V. for their help in reading some sections and improving the presentation of this paper.

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Archit Chauhan, Samir Datta, Chetan Gupta, and Vimal Raj Sharma. The Even-Path Problem in Directed Single-Crossing-Minor-Free Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.43

Abstract

Finding a simple path of even length between two designated vertices in a directed graph is a fundamental NP-complete problem [Andrea S. LaPaugh and Christos H. Papadimitriou, 1984] known as the EP problem. Nedev [Zhivko Prodanov Nedev, 1999] proved in 1999, that for directed planar graphs, the problem can be solved in polynomial time. More than two decades since then, we make the first progress in extending the tractable classes of graphs for this problem. We give a polynomial time algorithm to solve the EP problem for classes of H-minor-free directed graphs, where H is a single-crossing graph. 
We make two new technical contributions along the way, that might be of independent interest. The first, and perhaps our main, contribution is the construction of small, planar, parity-mimicking networks. These are graphs that mimic parities of all possible paths between a designated set of terminals of the original graph.
Finding vertex disjoint paths between given source-destination pairs of vertices is another fundamental problem, known to be NP-complete in directed graphs [Steven Fortune et al., 1980], though known to be tractable in planar directed graphs [Alexander Schrijver, 1994]. We encounter a natural variant of this problem, that of finding disjoint paths between given pairs of vertices, but with constraints on parity of the total length of paths. The other significant contribution of our paper is to give a polynomial time algorithm for the 3-disjoint paths with total parity problem, in directed planar graphs with some restrictions (and also in directed graphs of bounded treewidth).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Algorithms
  • EvenPath
  • Polynomial-time Algorithms
  • Reachability

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