Cake Cutting on Graphs: A Discrete and Bounded Proportional Protocol

Abstract

The classical cake cutting problem studies how to find fair allocations of a heterogeneous and divisible resource among multiple agents. Two of the most commonly studied fairness concepts in cake cutting are proportionality and envy-freeness. It is well known that a proportional allocation among n agents can be found efficiently via simple protocols [16]. For envy-freeness, in a recent breakthrough, Aziz and Mackenzie [5] proposed a discrete and bounded envy-free protocol for any number of players. However, the protocol suffers from high multiple-exponential query complexity and it remains open to find simpler and more efficient envy-free protocols.In this paper we consider a variation of the cake cutting problem by assuming an underlying graph over the agents whose edges describe their acquaintance relationships, and agents evaluate their shares relatively to those of their neighbors. An allocation is called locally proportional if each agent thinks she receives at least the average value over her neighbors. Local proportionality generalizes proportionality and is in an interesting middle ground between proportionality and envy-freeness: its existence is guaranteed by that of an envy-free allocation, but no simple protocol is known to produce such a locally proportional allocation for general graphs. Previous works showed locally proportional protocols for special classes of graphs, and it is listed in both [1] and [8] as an open question to design simple locally proportional protocols for more general classes of graphs. In this paper we completely resolved this open question by presenting a discrete and bounded locally proportional protocol for any given graph. Our protocol has a query complexity of only single exponential, which is significantly smaller than the six towers of n query complexity of the envy-free protocol given in [5].

Parameterized complexity of envy-free resource allocation in social networks

“…For brevity and ease of presentation, the term "proportionality" will always refer to the non-local notion of proportionality defined above. Note that this notion is different from the notion of "local proportionality" studied in some works [4]. We can now formalize our problems of interest:…”

Relaxations of Envy-Freeness Over Graphs

“…A few other works have modeled fairness concepts using graphs. One line of work specifies a graph structure over divisible or indivisible goods and insists that agents' bundles correspond to connected subgraphs [Bouveret et al, 2017, Bilò et al, 2022, Igarashi and Peters, 2019, Bei et al, 2020, Tucker-Foltz, 2021. Other works have considered graph structures over agents.…”

Thou Shalt Covet The Average Of Thy Neighbors' Cakes

“…The query complexity of local envy-freeness is clearly the same as for envy-freeness, since the worst case is realized when G is a complete graph. In a recent work, Bei, Sun, Wu, Zhang, Zhang, and Zi [9] give a new algorithm for local proportionality establishing an O(14 n ) upper bound. However, prior to this paper, there were no known lower bounds for local proportionality aside from the trivial bound of Ω(n log n) obtained from the case where G is a complete graph.…”