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Dividing a Graphical Cake

Xiaohui Bei, Warut Suksompong

TL;DR

The paper extends fair division to graphical cakes, where the resource is the edge-set of a graph and each agent’s piece must be connected. It presents constructive, moving-knife–style protocols that achieve strong fairness guarantees, including a universal bound of $\tfrac{1}{2n-1}$ on egalitarian welfare for any graph and a sharp star-based bound $f(n,k)$, with a complete two-agent classification via almost bridgeless graphs. It also shows that allowing multiple connected pieces per agent yields improved guarantees, and it provides a rich parallel analysis for chore division. The results illuminate when proportionality is achievable in networks, offer explicit allocation procedures, and suggest directions for envy-freeness and valuations that mix positive and negative utilities. Overall, the work advances graph-structured fair division with practical implications for networked resources such as roads and cables.

Abstract

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. This allows us to model the division of road or cable 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 tight guarantees 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).

Dividing a Graphical Cake

TL;DR

The paper extends fair division to graphical cakes, where the resource is the edge-set of a graph and each agent’s piece must be connected. It presents constructive, moving-knife–style protocols that achieve strong fairness guarantees, including a universal bound of on egalitarian welfare for any graph and a sharp star-based bound , with a complete two-agent classification via almost bridgeless graphs. It also shows that allowing multiple connected pieces per agent yields improved guarantees, and it provides a rich parallel analysis for chore division. The results illuminate when proportionality is achievable in networks, offer explicit allocation procedures, and suggest directions for envy-freeness and valuations that mix positive and negative utilities. Overall, the work advances graph-structured fair division with practical implications for networked resources such as roads and cables.

Abstract

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. This allows us to model the division of road or cable 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 tight guarantees 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).

Paper Structure

This paper contains 15 sections, 23 theorems, 2 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Further Related Work
  4. Preliminaries
  5. Any Number of Agents
  6. General Guarantees
  7. Stars
  8. Two Agents
  9. Graph-Specific Guarantees
  10. Utility Frontiers
  11. Beyond One Connected Piece
  12. Equitability
  13. Chore Division
  14. Conclusion and Future Work
  15. Chore Division Protocol for Three Agents

Key Result

theorem 1

For any graph $G$, there exists a connected allocation with egalitarian welfare at least $\frac{1}{2n-1}$. On the other hand, there exists a graph $G$ and identical valuations of the agents such that any connected allocation yields egalitarian welfare at most $\frac{1}{2n-1}$.

Figures (6)

  • Figure 1: Examples of graphs that are almost bridgeless but do not admit a bipolar numbering.
  • Figure 2: The red region corresponds to the values of $(\alpha,\beta)$ such that in any instance with two agents, there exists a connected allocation that yields utility $\alpha$ to the first agent and $\beta$ to the second agent.
  • Figure 3: Graph $G$ in the proof of Proposition \ref{['prop:two-agent-alpha-negative']}.
  • Figure 4: The blue region corresponds to the values of $(\alpha,\beta)$ such that in any instance with two agents, there exists a connected allocation that yields utility $\alpha$ to one agent and $\beta$ to the other agent.
  • Figure 5: The graph $G$ in the proof of Theorem \ref{['thm:two-agent-k-connected-upper']} for $k=2$.
  • ...and 1 more figures

Theorems & Definitions (25)

  • theorem 1
  • lemma 1
  • theorem 2
  • definition 1
  • definition 2
  • lemma 2
  • proposition 1
  • theorem 3
  • lemma 3
  • theorem 4
  • ...and 15 more