Fairly Allocating Contiguous Blocks of Indivisible Items

Suksompong, Warut

doi:10.1007/978-3-319-66700-3_26

“…On the other hand, our algorithms may leave some agents empty-handed, leading to unbounded multiplicative envy. We also note that additive envy is the more commonly considered form of approximation, both for cake cutting (Deng, Qi, and Saberi 2012;Brânzei and Nisan 2017;2019) and for indivisible items (Lipton et al 2004;Caragiannis et al 2016).…”

Section: Further Related Workmentioning

confidence: 97%

“…Marenco and Tetzlaff (2014) proved that if the items lie on a line and every item is positively valued by at most one agent, a contiguous envy-free allocation is guaranteed to exist. When each item can yield positive value to any number of agents, Barrera et al (2015), Bilò et al (2019), andSuksompong (2019) showed that various relaxations of envy-freeness can be fulfilled. In addition, contiguity has been studied in the more general model where the items lie on an arbitrary graph (Bouveret et al 2017;Igarashi and Peters 2019;Bei et al 2019).…”

Section: Further Related Workmentioning

confidence: 99%

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Goldberg

¹

,

Hollender

²

,

Suksompong

³

2020

AAAI

Self Cite

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We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results strengthening those from prior work.

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“…Marenco and Tetzlaff (2014) proved that if the items lie on a line and every item is positively valued by at most one agent, a contiguous envy-free allocation is guaranteed to exist. When each item can yield positive value to any number of agents, Barrera et al (2015), Bilò et al (2019), andSuksompong (2019) showed that various relaxations of envy-freeness can be fulfilled. Similarly to cake cutting, contiguity has been studied in the more general model where the indivisible items lie on an arbitrary graph (Bouveret et al, 2017;.…”

Section: Further Related Workmentioning

confidence: 99%

“…On the other hand, our algorithms may leave some agents empty-handed, leading to unbounded multiplicative envy. We also note that additive envy is the more commonly considered form of approximation, both for cake cutting (Deng et al, 2012;Brânzei & Nisan, 2017, 2019 and for discrete items (Lipton et al, 2004;Caragiannis et al, 2019). In particular, for discrete items, a significant stream of work in the last few years has focused on the notions envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX).…”

Section: Further Related Workmentioning

confidence: 99%

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Goldberg¹,

Hollender²,

Suksompong

³

2020

jair

Self Cite

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We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

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“…Fair allocation under connectivity constraints is receiving growing attention in the literature (Suksompong 2017;Lonc and Truszczynski 2018;Bouveret, Cechlárová, and Lesca 2018;Igarashi and Peters 2018;Bilò et al 2019). Nevertheless, our knowledge is still quite partial when focusing on maximin share allocations: The problem of deciding whether a maximin share allocation exists is known to be NP-hard, and to belong to the complexity class Δ P 2 (Lonc and Truszczynski 2018).…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Computing Maximin Share Allocations on Graphs

Greco

¹

,

Scarcello

²

2020

AAAI

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Maximin share is a compelling notion of fairness proposed by Buddish as a relaxation of more traditional concepts for fair allocations of indivisible goods. In this paper we consider this notion within a setting where bundles of goods must induce connected subsets over an underlying graph. This setting received much attention in earlier literature, and our study answers a number of questions that were left open. First, we show that computing maximin share allocations is FΔ2P-complete, even when focusing on consistent scenarios, that is, where such allocations are a-priori guaranteed to exist. Moreover, the problem remains intractable if all agents have the same type, i.e., have the same utility functions, and if either the values returned by the utility functions are polynomially bounded, or the underlying graphs have a low degree of cyclicity (more precisely, have bounded treewidth). However, if these conditions hold all together, then computing maximin share allocations (or checking that none exists) becomes tractable. The result is established via machineries based on logspace alternating machines that use partial representations of connected bundles, which are interesting in their own.

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Fairly Allocating Contiguous Blocks of Indivisible Items

Cited by 24 publications

References 22 publications

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

The Complexity of Computing Maximin Share Allocations on Graphs

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