Approximate Envy-Freeness in Graphical Cake Cutting

TL;DR

The paper tackles graphical cake cutting where the resource is a connected graph and allocations must be connected, showing that exact envy-freeness may fail in this setting. It establishes constructive, polynomial-time algorithms that achieve strong approximate fairness: a $\tfrac{1}{2}$-additive-EF allocation for general graphs, and a $(2+\epsilon)$-EF allocation for identical valuations on general graphs, with even stronger results for star graphs ($3+\epsilon$-EF for arbitrary valuations and $2$-EF for identical valuations). It also develops a layered approach for identical valuations using adaptive thresholds and balance-path techniques to reach $(2+\epsilon)$-EF, and introduces a bag-filling method for stars to obtain exact $2$-EF, all within polynomial time. Beyond single-piece allocations, the work defines the path similarity number to relate connected allocations to interval results, offering a route to multi-piece fair sharing. The study closes with open questions about constant multiplicative envy on general graphs with non-identical valuations and further improvements via path-based reductions.

Abstract

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.

Figures (5)

• Figure 1: (Left) A star graph with three edges of equal length. Two agents with identical valuations distributed uniformly over the three edges cannot each receive a connected piece worth at least $1/2$ of the whole cake at the same time. (Right) A star graph with many edges to be divided between two agents. If sharing of vertices is disallowed, then the agent who does not receive the center vertex will be restricted to at most one edge, and will incur envy equal to almost the value of the entire cake.
• Figure 2: (a) The points $x_k$ are found, where $[x_k, v]$ is worth at most $\epsilon'/m$ to every agent. (b) The unallocated intervals (dotted lines) are the ones to be considered in Phase 2a. (c) The unallocated intervals (dotted lines) are appended leftwards in $v_k$'s direction, except for the one containing $v_k$ which is appended rightwards. (d) The remaining unallocated portion $H$ (bold lines) is a share connected by $v$.
• Figure 3: (Left) Segments of value $1/n$ each (dotted lines) are allocated to the agents. The remaining stubs (solid lines) have value less than $1/n$ each. (Right) Two groups of the lowest value are merged together repeatedly. The figure shows three final groups: one group of three stubs on the right and two groups of one stub each on the left.
• Figure 4: (a) A star graph with six edges and (b) its corresponding path graph. A segment $[x, y]$ along the path graph (double lines) corresponds to at most two connected pieces in the star graph.
• Figure 5: (a) A tree $G$ of height $3$ and (b) its corresponding path graph $\phi(G)$ based on a depth-first search of $G$. A segment $[x, y]$ along the path graph (double lines) corresponds to at most four connected pieces in the tree.

Theorems & Definitions (30)

• Proposition 1
• Lemma 2
• Theorem 3
• proof
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7