Approximate Envy-Freeness in Graphical Cake Cutting
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913391451308032 |
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| author | Yuen, Sheung Man Suksompong, Warut |
| author_facet | Yuen, Sheung Man Suksompong, Warut |
| contents | We study the problem of fairly allocating a divisible resource in the form of a graph, also known as graphical cake cutting. Unlike for the canonical interval cake, a connected envy-free allocation is not guaranteed to exist for a graphical cake. We focus on the existence and computation of connected allocations with low envy. For general graphs, we show that there is always a $1/2$-additive-envy-free allocation and, if the agents' valuations are identical, a $(2+ε)$-multiplicative-envy-free allocation for any $ε> 0$. In the case of star graphs, we obtain a multiplicative factor of $3+ε$ for arbitrary valuations and $2$ for identical valuations. We also derive guarantees when each agent can receive more than one connected piece. All of our results come with efficient algorithms for computing the respective allocations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_11659 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximate Envy-Freeness in Graphical Cake Cutting Yuen, Sheung Man Suksompong, Warut Computer Science and Game Theory Discrete Mathematics We study the problem of fairly allocating a divisible resource in the form of a graph, also known as graphical cake cutting. Unlike for the canonical interval cake, a connected envy-free allocation is not guaranteed to exist for a graphical cake. We focus on the existence and computation of connected allocations with low envy. For general graphs, we show that there is always a $1/2$-additive-envy-free allocation and, if the agents' valuations are identical, a $(2+ε)$-multiplicative-envy-free allocation for any $ε> 0$. In the case of star graphs, we obtain a multiplicative factor of $3+ε$ for arbitrary valuations and $2$ for identical valuations. We also derive guarantees when each agent can receive more than one connected piece. All of our results come with efficient algorithms for computing the respective allocations. |
| title | Approximate Envy-Freeness in Graphical Cake Cutting |
| topic | Computer Science and Game Theory Discrete Mathematics |
| url | https://arxiv.org/abs/2304.11659 |