Connectedness in fair division of circle and star cakes between two agents with unequal entitlements

Austin's moving knife procedure is used to divide a cake equitably between two agents: each agent believes that they received exactly half of the cake. This short note deals with the case when the two agents have unequal (rational) entitlements to the cake and seek a weighted equitable division -- one where each agent believes that they received exactly the share that they are entitled to -- and also minimizes the number of connected components that each agent receives. First, we adapt Austin's moving knife procedure to produce a weighted equitable division of a circular cake that gives exactly one connected piece to each agent (recovering a result of Shishido and Zeng's). Next, we use it to produce a weighted equitable division of a star graph cake that gives at most two connected pieces to each agent -- and show that this bound on the number of connected pieces is tight

The Complexity of Envy-Free Graph Cutting

We consider the problem of fairly dividing a set of heterogeneous divisible resources among agents with different preferences. We focus on the setting where the resources correspond to the edges of a connected graph, every agent must be assigned a connected piece of this graph, and the fairness notion considered is the classical envy freeness. The problem is NP-complete, and we analyze its complexity with respect to two natural complexity measures: the number of agents and the number of edges in the graph. While the problem remains NP-hard even for instances with 2 agents, we provide a dichotomy characterizing the complexity of the problem when the number of agents is constant based on structural properties of the graph. For the latter case, we design a polynomial-time algorithm when the graph has a constant number of edges.Comment: Short version appeared at IJCAI 202

Mind the Gap: Cake Cutting With Separation

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 allocate a proportional share to every agent under the separation requirement, we show that the well-known criterion of maximin share fairness can always be attained. We then establish several computational properties of maximin share fairness -- for instance, the maximin share of an agent cannot be computed exactly by any finite algorithm, but can be approximated with an arbitrarily small error. In addition, we consider the division of a pie (i.e., a circular cake) and show that an ordinal relaxation of maximin share fairness can be achieved. We also prove that an envy-free or equitable allocation that allocates the maximum amount of resource exists under separation.Comment: Appears in the 35th AAAI Conference on Artificial Intelligence (AAAI), 202

Dividing a Graphical Cake

We consider the classical cake-cutting problem where we wish to fairly divide a heterogeneous resource, often modeled as a cake, among interested agents. Work on the subject typically assumes that the cake is represented by an interval. In this paper, we introduce a generalized setting where the cake can be in the form of the set of edges of an undirected graph, allowing us to model the division of road networks. Unlike in the canonical setting, common fairness criteria such as proportionality cannot always be satisfied in our setting if each agent must receive a connected subgraph. We determine the optimal approximation of proportionality that can be obtained for any number of agents with arbitrary valuations, and exhibit a tight guarantee for each graph in the case of two agents. In addition, when more than one connected piece per agent is allowed, we establish the best egalitarian welfare guarantee for each total number of connected pieces. We also study a number of variants and extensions, including when approximate equitability is considered, or when the item to be divided is undesirable (also known as chore division)