Fair Splitting Research Articles

Splitting Necklaces, with Constraints

We prove several versions of 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 almost the same number of pieces of the necklace (including “degenerate pieces” if they exist), provided the number of thieves $r=p^\nu$ is a prime power. By “almost the same” we mean that for each pair of thieves one of them can be given at most one piece more (one piece less) than the other. (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” in the case $r=2^d$ from Conjecture 2.11 in [M. Asada et al., SIAM J. Discrete Math., 32 (2018), pp. 591--610]. (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, as illustrated by the envy-free versions of (a) Alon's original necklace-splitting theorem, (b) the almost equicardinal splitting theorem, and (c) the binary splitting theorem, etc.

Fair Splitting of Colored Paths

This paper deals with two problems about splitting fairly a path with colored vertices, where "fairly" means that each part contains almost the same amount of vertices in each color.Our first result states that it is possible to remove one vertex per color from a path with colored vertices so that the remaining vertices can be fairly split into two independent sets of the path. It implies in particular a conjecture of Ron Aharoni and coauthors. The proof uses the octahedral Tucker lemma.Our second result is the proof of a particular case of a conjecture of Dömötör Pálvölgyi about fair splittings of necklaces for which one can decide which thieves are advantaged. The proof is based on a rounding technique introduced by Noga Alon and coauthors to prove the discrete splitting necklace theorem from the continuous one.

Caching games between Content Providers and Internet Service Providers

We consider a scenario where an Internet Service Provider (ISP) serves users that choose digital content among M Content Providers (CP). In the status quo, these users pay both access fees to the ISP and content fees to each chosen CP; however, neither the ISP nor the CPs share their profit. We revisit this model by introducing a different business model where the ISP and the CP may have motivation to collaborate in the framework of caching. The key idea is that the ISP deploys a cache for a CP provided that they share both the deployment cost and the additional profit that arises due to caching. Under the prism of coalitional games, our contributions include the application of the Shapley value for a fair splitting of the profit, the stability analysis of the coalition and the derivation of closed-form formulas for the optimal caching policy.Our model captures not only the case of non-overlapping contents among the CPs, but also the more challenging case of overlapping contents; for the latter case, a non-cooperative game among the CPs is introduced and analyzed to capture the negative externality on the demand of a particular CP when caches for other CPs are deployed.

Obstacles for splitting multidimensional necklaces

The well-known “necklace splitting theorem” of Alon (1987) asserts that every k k -colored necklace can be fairly split into q q parts using at most t t cuts, provided k ( q − 1 ) ≤ t k(q-1)\leq t . In a joint paper with Alon et al. (2009) we studied a kind of opposite question. Namely, for which values of k k and t t is there a measurable k k -coloring of the real line such that no interval has a fair splitting into 2 2 parts with at most t t cuts? We proved that k > t + 2 k>t+2 is a sufficient condition (while k > t k>t is necessary). We generalize this result to Euclidean spaces of arbitrary dimension d d , and to arbitrary number of parts q q . We prove that if k ( q − 1 ) > t + d + q − 1 k(q-1)>t+d+q-1 , then there is a measurable k k -coloring of R d \mathbb {R}^d such that no axis-aligned cube has a fair q q -splitting using at most t t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k ( q − 1 ) > t k(q-1)>t implied by Alon’s 1987 work. Moreover, for d = 1 , q = 2 d=1,q=2 we get exactly the result of the 2009 work. Additionally, we prove that if a stronger inequality k ( q − 1 ) > d t + d + q − 1 k(q-1)>dt+d+q-1 is satisfied, then there is a measurable k k -coloring of R d \mathbb {R}^d with no axis-aligned cube having a fair q q -splitting using at most t t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.

Splitting Multidimensional Necklaces and Measurable Colorings of Euclidean Spaces

A necklace splitting theorem of Goldberg and West asserts that any $k$-colored (continuous) necklace can be fairly split using at most $k$ cuts. Motivated by the problem of Erdos on strongly nonrepetitive sequences, Alon et al. proved that there is a $(t+3)$-coloring of the real line in which no necklace has a fair splitting using at most $t$ cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is $k$-coloring of $\mathbb{R}^{d}$ such that no cube has a fair splitting of size $t$, i.e., for each of the axes using at most $t$ hyperplanes orthogonal to it, provided $k\geq (t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3$. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljevic. The question of how many axes-aligned hyperplanes are needed for a fair splitting of a $d$-dimensional $k$-colored cube remains open.

Mantle flow beneath La Réunion hotspot track from SKS splitting

If upper mantle anisotropy beneath fast-moving oceanic plates is expected to align the fast azimuths close to the plate motion directions, the upper mantle flow pattern beneath slow-moving oceanic plates will reflect the relative motion between the moving plate and the underlying large-scale convecting mantle. In addition to the non-correlation of the fast azimuths with the plate motion direction, the flow and anisotropy pattern may be locally perturbed by other factors such as the upwelling and the sublithospheric spreading of mantle plumes. Investigating such plume–lithosphere interaction is strongly dependent on the available seismological data, which are generally sparse in oceanic environment. In this study, we take the opportunity of recent temporary deployments of 15 seismic stations and 5 permanent stations on the Piton de la Fournaise volcano, the active locus of La Réunion hotspot and of 6 permanent stations installed along or close to its fossil track of about 3700km in length, to analyze azimuthal anisotropy detected by SKS wave splitting and to decipher the various possible origins of anisotropy beneath the Western Indian Ocean. From about 150 good and fair splitting measurements and more than 1000 null splitting measurements, we attempt to distinguish between the influence of a local plume signature and large-scale mantle flow. The large-scale anisotropy pattern obtained at the SW-Indian Ocean island stations is well explained by plate motion relative to the deep mantle circulation. By contrast, stations on La Réunion Island show a complex signature characterized by numerous “nulls” and by fast split shear wave polarizations trending normal to the plate motion direction and obtained within a small backazimuthal window, that cannot be explained by either a single or two anisotropic layers. Despite the sparse spatial coverage which precludes a unique answer, we show that such pattern may be compatible with a simple model of sublithospheric spreading of La Réunion plume characterized by a conduit located at 100–200km north of La Réunion Island. Anisotropy beneath the new GEOSCOPE station in Rodrigues Island does not appear to be influenced by La Réunion plume-spreading signature but is fully compatible with either a model of large-scale deep mantle convection pattern and/or with a channeled asthenospheric flow beneath the Rodrigues ridge.

Seismic anisotropy in the south western Pacific region from shear wave splitting

We perform shear‐wave splitting measurements to determine seismic anisotropy in the upper mantle under the remote and still not well‐understood South Western Pacific Region, more precisely the New Hebrides subduction zone. We obtained 29 good and 35 fair splitting measurements at 8 stations and 93 good and fair Null measurements from the data provided by various networks. According to the splitting parameters, the seismic stations are grouped into “eastern” and “western” stations. Anisotropy of the subducting plate (“western” stations) appears to be the result of absolute plate motion (APM) of the Australian plate. From fast orientations alone, it would be difficult to judge which process is responsible for the anisotropic structure: either the absolute plate motion or trench‐parallel “toroidal” flow. Their delay times reveal that these anisotropic parameters can be explained without the presence of the subduction zone, and the APM provides a simple explanation for the observed anisotropy. On the other hand, delay times at “eastern” stations are higher than that of “western” stations. These stations are located close to the subduction zone, and the nearly trench‐parallel orientation of fast polarization axes and their higher splitting delay times indicate the presence of subduction zone. Anisotropy at “eastern” stations is apparently due to the additional effect from “toroidal” flow near the subduction zone, which increases the total anisotropy.

Necklace Bisection with One Cut Less than Needed

A well-known theorem of Goldberg and West states that two thieves can always split a necklace containing an even number of beads from each of $k$ types fairly by at most $k$ cuts. We prove that if we can use at most $k-1$ cuts and fair splitting is not possible then the thieves still have the following option. Whatever way they specify two disjoint sets $D_1, D_2$ of the types of beads with $D_1\cup D_2\neq\emptyset$, it will always be possible to cut the necklace (with $k-1$ cuts) so that the first thief gets more of those types of beads that are in $D_1$ and the second gets more of those in $D_2$, while the rest is divided equally. The proof combines the simple proof given by Alon and West to the original statement with a variant of the Borsuk-Ulam theorem due to Tucker and Bacon.

Dual Electric Power Supply with Increasing Wind Power Generation, Requirement for an Advanced Secondary Control Concept

Based on the example of the East German power system - being coupled to the European interconnected grid - there has been investigated for low load situations- up to which amount wind power generation can be fed in and transported through the high voltage transmission network as well as- up to which amount wind power can be overtaken by reducing as far as possibly the power generation of the normal power plant units.Based on this an advanced secondary control concept (tie line bias control) is introduced for a fair splitting up of the whole renewable power generation, which is fed into the interconnected power system, to the different grid partners.

Regression Trees for Survival Data — an Approach to Select Discontinuous Split Points by Rank Statistics

Regression trees allow to search for meaningful explanatory variables that have a non linear impact on the dependent variable. Often they are used when there are many covariates and one does not want to restrict attention to only few of them. To grow a tree at each stage one has to select a cut point for splitting a group into two subgroups. The basis for this are the maxima of the test statistics related to the possible splits due to every covariate. They or the resulting P-values are compared as measure of importance. If covariates have different numbers of missing values, ties, or even different measurement scales the covariates lead to different numbers of tests. Those with a higher number of tests have a greater chance to achieve a smaller P-value if they are not adjusted. This can lead to erroneous splits even if the P-values are looked at informally. There is some theoretical work by Miller and Siegmund (1982) and Lausen and Schumacher (1992) to give an adjustment rule. But the asymptotic is based on a continuum of split points and may not lead to a fair splitting rule when applied to smaller data sets or to covariates with only few different values. Here we develop an approach that allows determination of P-values for any number of splits. The only approximation that is used is the normal approximation of the test statistics. The starting point for this investigation has been a prospective study on the development of AIDS. This is presented here as the main application.