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Tree-Residue Vertex-Breaking: a new tool for proving hardness

Authors Erik D. Demaine, Mikhail Rudoy



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Erik D. Demaine
  • MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA.
Mikhail Rudoy
  • MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA. Now at Google Inc.

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Erik D. Demaine and Mikhail Rudoy. Tree-Residue Vertex-Breaking: a new tool for proving hardness. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SWAT.2018.32

Abstract

In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked "breakable," is it possible to convert G into a tree via a sequence of "vertex-breaking" operations (replacing a degree-k breakable vertex by k degree-1 vertices, disconnecting the k incident edges)? We characterize the computational complexity of TRVB with any combination of the following additional constraints: G must be planar, G must be a simple graph, the degree of every breakable vertex must belong to an allowed list B, and the degree of every unbreakable vertex must belong to an allowed list U. The two results which we expect to be most generally applicable are that (1) TRVB is polynomially solvable when breakable vertices are restricted to have degree at most 3; and (2) for any k >= 4, TRVB is NP-complete when the given multigraph is restricted to be planar and to consist entirely of degree-k breakable vertices. To demonstrate the use of TRVB, we give a simple proof of the known result that Hamiltonicity in max-degree-3 square grid graphs is NP-hard. We also demonstrate a connection between TRVB and the Hypergraph Spanning Tree problem. This connection allows us to show that the Hypergraph Spanning Tree problem in k-uniform 2-regular hypergraphs is NP-complete for any k >= 4, even when the incidence graph of the hypergraph is planar.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • NP-hardness
  • graphs
  • Hamiltonicity
  • hypergraph spanning tree

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References

  1. Erik D. Demaine and Mikhail Rudoy. Hamiltonicity is hard in thin or polygonal grid graphs, but easy in thin polygonal grid graphs. arXiv:1706.10046, 2017. URL: https://arxiv.org/abs/1706.10046.
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  3. Christos H. Papadimitriou and Umesh V. Vazirani. On two geometric problems related to the traveling salesman problem. J. Algorithms, 5(2):231-246, 1984. URL: http://dx.doi.org/10.1016/0196-6774(84)90029-4.
  4. Ján Plesník. The np-completeness of the hamiltonian cycle problem in planar digraphs with degree bound two. Inf. Process. Lett., 8(4):199-201, 1979. URL: http://dx.doi.org/10.1016/0020-0190(79)90023-1.
  5. Hans Jürgen Prömel and Angelika Steger. The Steiner tree problem: a tour through graphs, algorithms, and complexity. Springer Science &Business Media, 2002. Google Scholar
  6. Markus W. Schäffter. Drawing graphs on rectangular grids. Discrete Applied Mathematics, 63(1):75-89, 1995. URL: http://dx.doi.org/10.1016/0166-218X(94)00020-E.
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