Necklace splitting problem

Necklace splitting is a picturesque name given to several related problems in combinatorics and measure theory. Its name and solutions are due to mathematicians Noga Alon[1] and Douglas B. West.[2]

a stylized picture of a necklace, with 8 red and 6 green pearls. The pearls are threaded on to an incomplete elliptical black curve that represents the string. The gap in the curve represents the clasp (open in the diagram) which may be closed when the necklace is placed around the neck. There are two short heavy lines marking breaks in the necklace string. Starting from the left, the necklace is: RRRGRBRRGRRGGBGG, where "R" means "red pearl", "G" means "green pearl", and "B" means "break". The breaks correspond to those in the text
Example of necklace splitting with k = 2 (i.e. two partners), and t = 2 (i.e. two types of beads, here 8 red and 6 green). A 2-split is shown: one partner receives the largest section, and the other receives the remaining two pieces.

The basic setting involves a necklace with beads of different colors. The necklace should be divided between several partners (e.g. thieves), such that each partner receives the same amount of every color. Moreover, the number of cuts should be as small as possible (in order to waste as little as possible of the metal in the links between the beads).

Variants

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The following variants of the problem have been solved in the original paper:

  1. Discrete splitting:[1]: Th 1.1  The necklace has   beads. The beads come in   different colors. There are   beads of each color  , where   is a positive integer. Partition the necklace into   parts (not necessarily contiguous), each of which has exactly   beads of color i. Use at most   cuts. Note that if the beads of each color are contiguous on the necklace, then at least   cuts must be done inside each color, so   is optimal.
  2. Continuous splitting:[1]: Th 2.1  The necklace is the real interval  . Each point of the interval is colored in one of   different colors. For every color  , the set of points colored by   is Lebesgue-measurable and has length  , where   is a non-negative real number. Partition the interval to   parts (not necessarily contiguous), such that in each part, the total length of color   is exactly  . Use at most   cuts.
  3. Measure splitting:[1]: Th 1.2  The necklace is a real interval. There are   different measures on the interval, all absolutely continuous with respect to length. The measure of the entire necklace, according to measure  , is  . Partition the interval to   parts (not necessarily contiguous), such that the measure of each part, according to measure  , is exactly  . Use at most   cuts. This is a generalization of the Hobby–Rice theorem, and it is used to get an exact division of a cake.

Each problem can be solved by the next problem:

  • Discrete splitting can be solved by continuous splitting, since a discrete necklace can be converted to a coloring of the real interval   in which each interval of length 1 is colored by the color of the corresponding bead. In case the continuous splitting tries to cut inside beads, the cuts can be slid gradually such that they are made only between beads.[1]: 249 
  • Continuous splitting can be solved by measure splitting, since a coloring of an interval in   colors can be converted to a set   measures, such that measure   measures the total length of color  . The opposite is also true: measure splitting can be solved by continuous splitting, using a more sophisticated reduction.[1]: 252–253 

Proof

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The case   can be proved by the Borsuk-Ulam theorem.[2]

When   is an odd prime number, the proof involves a generalization of the Borsuk-Ulam theorem.[3]

When   is a composite number, the proof is as follows (demonstrated for the measure-splitting variant). Suppose  . There are   measures, each of which values the entire necklace as  . Using   cuts, divide the necklace to   parts such that measure   of each part is exactly  . Using   cuts, divide each part to   parts such that measure   of each part is exactly  . All in all, there are now   parts such that measure   of each part is exactly  . The total number of cuts is   plus   which is exactly  .

Further results

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Splitting random necklaces

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In some cases, random necklaces can be split equally using fewer cuts. Mathematicians Noga Alon, Dor Elboim, Gábor Tardos and János Pach studied the typical number of cuts required to split a random necklace between two thieves.[4] In the model they considered, a necklace is chosen uniformly at random from the set of necklaces with t colors and m beads of each color. As m tends to infinity, the probability that the necklace can be split using ⌊(t + 1)/2⌋ cuts or less tends to zero while the probability that it's possible to split with ⌊(t + 1)/2⌋ + 1 cuts is bounded away from zero. More precisely, letting X = X(t,m) be the minimal number of cuts required to split the necklace. The following holds as m tends to infinity. For any  

 

For any  

 

Finally, when   is odd and  

 

One can also consider the case in which the number of colors tends to infinity. When m=1 and the t tends to infinity, the number of cuts required is at most 0.4t and at least 0.22t with high probability. It is conjectured that there exists some 0.22 < c < 0.4 such that X(t,1)/t  converges to c in distribution.

One cut fewer than needed

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In the case of two thieves [i.e. k = 2] and t colours, a fair split would require at most t cuts. If, however, only t − 1 cuts are available, Hungarian mathematician Gábor Simonyi[5] shows that the two thieves can achieve an almost fair division in the following sense.

If the necklace is arranged so that no t-split is possible, then for any two subsets D1 and D2 of { 1, 2, ...,  t }, not both empty, such that  , a (t − 1)-split exists such that:

  • If colour  , then partition 1 has more beads of colour i than partition 2;
  • If colour  , then partition 2 has more beads of colour i than partition 1;
  • If colour i is in neither partition, both partitions have equally many beads of colour i.

This means that if the thieves have preferences in the form of two "preference" sets D1 and D2, not both empty, there exists a (t − 1)-split so that thief 1 gets more beads of types in his preference set D1 than thief 2; thief 2 gets more beads of types in her preference set D2 than thief 1; and the rest are equal.

Simonyi credits Gábor Tardos with noticing that the result above is a direct generalization of Alon's original necklace theorem in the case k = 2. Either the necklace has a (t − 1)-split, or it does not. If it does, there is nothing to prove. If it does not, we may add beads of a fictitious colour to the necklace, and make D1 consist of the fictitious colour and D2 empty. Then Simonyi's result shows that there is a t-split with equal numbers of each real colour.

Negative result

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For every   there is a measurable  -coloring of the real line such that no interval can be fairly split using at most   cuts.[6]

Splitting multidimensional necklaces

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The result can be generalized to n probability measures defined on a d dimensional cube with any combination of n(k − 1) hyperplanes parallel to the sides for k thieves.[7]

Approximation algorithm

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An approximation algorithm for splitting a necklace can be derived from an algorithm for consensus halving.[8]

See also

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References

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  1. ^ a b c d e f Alon, Noga (1987). "Splitting Necklaces". Advances in Mathematics. 63 (3): 247–253. doi:10.1016/0001-8708(87)90055-7.
  2. ^ a b Alon, Noga; West, Douglas B. (December 1986). "The Borsuk-Ulam theorem and bisection of necklaces". Proceedings of the American Mathematical Society. 98 (4): 623–628. doi:10.1090/s0002-9939-1986-0861764-9.
  3. ^ I.Barany and S.B.Shlosman and A.Szucs (1981). "On a topological generalization of a theorem of Tverberg". Journal of the London Mathematical Society. 2 (23): 158–164. CiteSeerX 10.1.1.640.1540. doi:10.1112/jlms/s2-23.1.158.
  4. ^ Alon, Noga; Elboim, Dor; Tardos, Gábor; Pach, János (2021). "Random Necklaces require fewer cuts". arXiv:2112.14488 [math.CO].
  5. ^ Simonyi, Gábor (2008). "Necklace bisection with one cut less than needed". Electronic Journal of Combinatorics. 15 (N16). doi:10.37236/891.
  6. ^ Alon, Noga (November 25, 2008). "Splitting necklaces and measurable colorings of the real line". Proceedings of the American Mathematical Society. 137 (5): 1593–1599. arXiv:1412.7996. doi:10.1090/s0002-9939-08-09699-8. ISSN 1088-6826.
  7. ^ de Longueville, Mark; Rade T. Živaljević (2008). "Splitting multidimensional necklaces". Advances in Mathematics. 218 (3): 926–939. arXiv:math/0610800. doi:10.1016/j.aim.2008.02.003.
  8. ^ Simmons, Forest W.; Su, Francis Edward (February 2003). "Consensus-halving via theorems of Borsuk-Ulam and Tucker". Mathematical Social Sciences. 45 (1): 15–25. CiteSeerX 10.1.1.203.1189. doi:10.1016/s0165-4896(02)00087-2.
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  • "Sneaky Topology" on YouTube, an introductory video presenting the problem with its topological solution.