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The weighted sitting closer to friends than enemies problem in the line

Author

Listed:
  • Julio Aracena

    (Universidad de Concepción)

  • Christopher Thraves Caro

    (Univiversidad de Concepción)

Abstract

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. The Seriation problem is to find a simultaneous reordering of the rows and columns of a symmetric matrix such that the entries are monotone nondecreasing in rows and columns when moving towards the diagonal. If such a reordering exists, it is called a Robinson ordering. In this work, we establish a connection between the SCFE problem and the Seriation problem. We show that if the extended adjacency matrix of a given weighted graph G has no Robinson ordering then G has no injection in $$\mathbb {R}$$ R that solves the SCFE problem. On the other hand, if the extended adjacency matrix of G has a Robinson ordering, we construct a polyhedron that is not empty if and only if there is an injection of the vertex set of G in $$\mathbb {R}$$ R that solves the SCFE problem. As a consequence of these results, we conclude that deciding the existence of (and constructing) such an injection in $$\mathbb {R}$$ R for a given complete weighted graph can be done in polynomial time. On the other hand, we show that deciding if an incomplete weighted graph has such an injection in $$\mathbb {R}$$ R is NP-Complete.

Suggested Citation

  • Julio Aracena & Christopher Thraves Caro, 2023. "The weighted sitting closer to friends than enemies problem in the line," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-21, January.
  • Handle: RePEc:spr:jcomop:v:45:y:2023:i:1:d:10.1007_s10878-022-00953-z
    DOI: 10.1007/s10878-022-00953-z
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    References listed on IDEAS

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    1. Victor Chepoi & Bernard Fichet & Morgan Seston, 2009. "Seriation in the Presence of Errors: NP-Hardness of l ∞ -Fitting Robinson Structures to Dissimilarity Matrices," Journal of Classification, Springer;The Classification Society, vol. 26(3), pages 279-296, December.
    2. D. Fortin, 2017. "Robinsonian Matrices: Recognition Challenges," Journal of Classification, Springer;The Classification Society, vol. 34(2), pages 191-222, July.
    3. Pascal Préa & Dominique Fortin, 2014. "An Optimal Algorithm To Recognize Robinsonian Dissimilarities," Journal of Classification, Springer;The Classification Society, vol. 31(3), pages 351-385, October.
    4. Victor Chepoi & Bernard Fichet, 1997. "Recognition of Robinsonian dissimilarities," Journal of Classification, Springer;The Classification Society, vol. 14(2), pages 311-325, September.
    5. Laurent, Monique & Seminaroti, M. & Tanigawa, Shin-ichi, 2017. "A structural characterization for certifying robinsonian matrices," Other publications TiSEM 5ecebfb8-804e-4267-8c12-b, Tilburg University, School of Economics and Management.
    6. Eduardo G. Pardo & Mauricio Soto & Christopher Thraves, 2015. "Embedding signed graphs in the line," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 451-471, February.
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