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The Kissing Problem: How to End a Gathering When Everyone Kisses Everyone Else Goodbye

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Abstract

This paper introduces the kissing problem: given a rectangular room with n people in it, what is the most efficient way for each pair of people to kiss each other goodbye? The room is viewed as a set of pixels that form a subset of the integer grid. At most one person can stand on a pixel at once, and people move horizontally or vertically. In order to move into a pixel in time step t, the pixel must be empty in time step t−1.

The paper gives one algorithm for kissing everyone goodbye.

  1. (1)

    This algorithm is a 4+o(1)-approximation algorithm in a crowded room (e.g., only one unoccupied pixel).

  2. (2)

    It is a 45+o(1)-approximation algorithm for kissing in a comfortable room (e.g., at most half the pixels are empty).

  3. (3)

    It is a 25+o(1)-approximation for kissing in a sparse room (more than half the pixels are empty) with two people abutting the far walls of the room.

This paper gives optimal solutions for small cases, which were found using a heuristic state space search (IDA*).

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Notes

  1. April seventeenth. Lip service to Haiku Day. Just an FYI.

  2. Two pixels are said to be adjacent if they share an edge.

  3. Look it up

    t’nod uoy fi

    know what

    -ehportsoub

    onic means.

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Authors and Affiliations

  1. Department of Computer Science, Stony Brook University, Stony Brook, NY, 11794-4400, USA

    Michael A. Bender, Rezaul Chowdhury & Samuel McCauley

  2. Tokutek, Inc., Lexington, MA, 02421, USA

    Michael A. Bender

  3. Department of Computer Science, University of Rochester, Rochester, NY, 14627, USA

    Ritwik Bose

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  1. Michael A. Bender
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  2. Ritwik Bose
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  3. Rezaul Chowdhury
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  4. Samuel McCauley
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Corresponding author

Correspondence to Samuel McCauley.

Additional information

This research was supported in part by NSF Grants CCF 0937822, CCF 1114809, CCF 0634793, IIS 1247726, and DOE Grant DE-FG02-08ER25853.

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Bender, M.A., Bose, R., Chowdhury, R. et al. The Kissing Problem: How to End a Gathering When Everyone Kisses Everyone Else Goodbye. Theory Comput Syst 54, 715–730 (2014). https://doi.org/10.1007/s00224-013-9484-x

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