Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Abstract

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.

Mind the gap: Cake cutting with separation

“…In order to ensure that no agent receives a collection of tiny pieces, it is often assumed that each agent must be allocated a connected piece of the cake [2,4,11,22,23,30,37,[53][54][55]. Indeed, when we divide resources such as time or space, nonconnected pieces (e.g., disconnected time intervals or land plots) may be hard to utilize, or even entirely useless.…”

“…In light of Corollary 5.8 and Theorem 5.9, it would be interesting to develop approximation algorithms that compute allocations with low envy-this direction has been recently pursued in cake cutting without separation [2,37].…”

Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

“…These problems remain NP-hard if we replace envyfreeness by ε-envy-freeness for any sufficiently small constant ε. This list is not exhaustive: additional results of the same flavor can be found in the full proof (Goldberg, Hollender, and Suksompong 2019). 6 The following proof sketch conveys the main ideas behind these results.…”

“…We leave the proof of the Claim to the full version of this paper (Goldberg, Hollender, and Suksompong 2019).…”

“…We show that this is the case for proportionality, provided that we drop the requirement that all agents value the same number of items. Note that if each agent values a contiguous block of items and all agents value the same number of items, deciding whether a proportional allocation exists can in fact be done in polynomial time; see the details in the full version of this paper (Goldberg, Hollender, and Suksompong 2019). Theorem 5.2.…”