Splitting necklaces, with constraints [article]

Duško Jojić, Gaiane Panina, Rade Živaljević
2020 arXiv   pre-print
We prove several versions of N. Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace (including "degenerate pieces" if they exist), provided the number of thieves r=p^ν is a prime
more » ... wer. (2) The "binary splitting theorem" claims that if r=2^d and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [7]) in the case r=2^d. (3) An interesting variation arises when the thieves have their own individual preferences. We prove several "envy-free fair necklace-splitting theorems" of various level of generality. By specialization we obtain numerous corollaries, among them envy-free versions of (a) "almost equicardinal splitting theorem", (b) "necklace-splitting theorem for r-unavoidable preferences", (c) "envy-free binary splitting theorem", etc. As a corollary we also obtain a recent result of Avvakumov and Karasev [1] about envy-free divisions where players may prefer an empty part of the necklace.
arXiv:1907.09740v4 fatcat:5jeoa4g5nrfk5hjzilje7twrqu