# Mind the Gap: Cake Cutting With Separation

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Abstract:

We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for example, constraints arising from social distancing guidelines. While it is sometimes impossible to...More

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Introduction

- The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street.
- Each vendor naturally has a preference over potential locations, possibly depending on the proximity to certain attractions or the estimated number of customers visiting that space.
- This year is different from previous years due to the social distancing guidelines issued by the government—vendors are required to be placed at least two meters apart.
- Better still, such an allocation can be found by a simple and efficient algorithm (Dubins and Spanier

Highlights

- The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street
- We demonstrate that maximin share fairness is an appropriate substitute for proportionality in cake cutting with separation, and analyze it from a computational perspective
- As is commonly done in cake cutting, we assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake
- We have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting
- Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations
- While the canonical maximin share is a reasonable fairness requirement when agents have equal entitlements to the resource, in certain situations the agents may be endowed with different entitlements (Chakraborty et al 2020; Cseh and Fleiner 2020)

Results

- As is commonly done in cake cutting, the authors assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake.
- In Section 3, the authors begin by proving that an allocation that gives every agent at least her maximin share always exists, meaning that maximin share fairness can be guaranteed.
- Such an allocation can be found by a simple algorithm provided that the algorithm knows the maximin share of each agent.
- The authors design an algorithm that approximates the maximin share up to an arbitrarily small error, which allows them to compute an allocation wherein each

Conclusion

- The authors have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting.
- Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations.
- This is the motivation behind the model of pie cutting that the authors have addressed in Section 4.
- A connected cake allocation may not exist even without separation (Segal-Halevi 2019; Crew, Narayanan, and Spirkl 2020)

Summary

## Introduction:

The end of the year is fast approaching, and members of a city council are busy planning the traditional New Year’s fair on their city’s main street.- Each vendor naturally has a preference over potential locations, possibly depending on the proximity to certain attractions or the estimated number of customers visiting that space.
- This year is different from previous years due to the social distancing guidelines issued by the government—vendors are required to be placed at least two meters apart.
- Better still, such an allocation can be found by a simple and efficient algorithm (Dubins and Spanier
## Objectives:

The authors' goal is to compute Il(k), Ir(k) so that the list Ik = (Il(q), Ir(q))q∈[k] is consistent with some s-separated maximin partition of [0, 1]## Results:

As is commonly done in cake cutting, the authors assume that the cake is represented by an interval and each agent is to be allocated a single subinterval of the cake.- In Section 3, the authors begin by proving that an allocation that gives every agent at least her maximin share always exists, meaning that maximin share fairness can be guaranteed.
- Such an allocation can be found by a simple algorithm provided that the algorithm knows the maximin share of each agent.
- The authors design an algorithm that approximates the maximin share up to an arbitrarily small error, which allows them to compute an allocation wherein each
## Conclusion:

The authors have initiated the study of cake cutting under separation requirements, and established several existence and computational results on maximin share fairness in this setting.- Even though the cake in cake cutting is typically represented by an interval, certain applications of divisible resource allocation may require different representations.
- This is the motivation behind the model of pie cutting that the authors have addressed in Section 4.
- A connected cake allocation may not exist even without separation (Segal-Halevi 2019; Crew, Narayanan, and Spirkl 2020)

- Table1: Summary of the tasks that can and cannot be accomplished by finite algorithms in the Robertson-Webb model for cake cutting and pie cutting. All negative results hold even if the valuations of the agents are piecewise constant (but not given explicitly). The result with an asterisk holds when we do not allow the number of queries that the algorithm makes to depend on the separation parameter s

Related work

- Cake cutting has long been studied by mathematicians and economists, and more recently attracted substantial interest from computer scientists, as it suggests a plethora of computational challenges. In particular, a long line of work in the artificial intelligence community in recent years has focused on cake cutting and its variants (Branzei et al 2016; Alijani et al 2017; Bei et al 2017; Menon and Larson

2017; Bei, Huzhang, and Suksompong

2018; Segal-Halevi 2018; Arunachaleswaran, Barman, and Rathi

2019; Goldberg, Hollender, and Suksompong 2020; Hosseini, Igarashi, and Searns 2020; Bei and Suksompong 2021).

In order to prevent agents from receiving a collection of tiny pieces, it is often assumed that each agent must receive a connected piece of the cake (Dubins and Spanier 1961; Stromquist 1980; Su 1999; Bei et al 2012; Cechlarovaand Pillarova

2012; Cechlarova, Dobos, and Pillarova

2013; Aumann and Dombb 2015; Arunachaleswaran et al.2019; Goldberg, Hollender, and Suksompong 2020). Indeed, when we divide resources such as time or space, non-connected pieces (e.g., disconnected time intervals or land plots) may be hard to use, or even totally useless.

Note that we impose the connectivity constraint not only on the allocation but also in the definition of the maximin share benchmark. Similar conventions have been used in the context of indivisible items, where the items are vertices of an undirected graph and every agent must be allocated a connected subgraph (Bouveret et al 2017; Lonc and Truszczynski 2020).1

Funding

- This work was partially supported by the European Research Council (ERC) under grant number 639945 (ACCORD), by the Israel Science Foundation under grant number 712/20, and by an NUS Start-up Grant

Reference

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