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On Connected Strongly-Proportional Cake-Cutting

Zsuzsanna Jankó, Attila Joó, Erel Segal-Halevi, Sheung Man Yuen

TL;DR

This work characterizes when connected strongly-proportional allocations exist in cake-cutting under varying entitlements and hunger assumptions, and analyzes the precise query complexity to decide existence and compute such allocations. For hungry agents with equal entitlements, existence is determined by the disagreement of $r$-marks for $r\in\mathcal{T}$, enabling a constructive $O(n^2)$-query algorithm with a matching $\Theta(n^2)$ lower bound; with unequal entitlements, the problem requires $\Theta(n\,2^n)$. For the general (not necessarily hungry) setting, a permutation-based Marking condition yields an upper bound of $O(n\,2^{n-1})$ queries, with a corresponding $\Omega(n\,2^n)$ lower bound, giving a tight $\Theta(n\,2^n)$ complexity overall. A stronger fairness variant preserves the same exponential bound, while the pie variant is provably intractable (no finite algorithm). These results delineate the computational landscape of connected fair allocations and highlight the crucial role of entitlements and topology in cake-cutting questions.

Abstract

We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie.

On Connected Strongly-Proportional Cake-Cutting

TL;DR

This work characterizes when connected strongly-proportional allocations exist in cake-cutting under varying entitlements and hunger assumptions, and analyzes the precise query complexity to decide existence and compute such allocations. For hungry agents with equal entitlements, existence is determined by the disagreement of -marks for , enabling a constructive -query algorithm with a matching lower bound; with unequal entitlements, the problem requires . For the general (not necessarily hungry) setting, a permutation-based Marking condition yields an upper bound of queries, with a corresponding lower bound, giving a tight complexity overall. A stronger fairness variant preserves the same exponential bound, while the pie variant is provably intractable (no finite algorithm). These results delineate the computational landscape of connected fair allocations and highlight the crucial role of entitlements and topology in cake-cutting questions.

Abstract

We investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents who may have different entitlements. We characterize the existence of a connected strongly-proportional allocation -- one in which every agent receives a contiguous piece worth strictly more than their proportional share. The characterization is supplemented with an algorithm that determines its existence using O(n * 2^n) queries. We devise a simpler characterization for agents with strictly positive valuations and with equal entitlements, and present an algorithm to determine the existence of such an allocation using O(n^2) queries. We provide matching lower bounds in the number of queries for both algorithms. When a connected strongly-proportional allocation exists, we show that it can also be computed using a similar number of queries. We also consider the problem of deciding the existence of a connected allocation of a cake in which each agent receives a piece worth a small fixed value more than their proportional share, and the problem of deciding the existence of a connected strongly-proportional allocation of a pie.
Paper Structure (15 sections, 20 theorems, 10 equations, 4 figures, 1 table, 4 algorithms)

This paper contains 15 sections, 20 theorems, 10 equations, 4 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Further Related Work
  4. Preliminaries
  5. Hungry Agents
  6. Equal Entitlements
  7. Possibly Unequal Entitlements
  8. General Agents
  9. Upper Bound
  10. Lower Bound
  11. Stronger than Strongly-Proportional
  12. Pies
  13. Conclusion
  14. Left-Marks and Right-Marks
  15. Proportionality

Key Result

Lemma 3.1

Let an instance with $n$ hungry agents be given. Suppose that we are given a connected proportional allocation $(X_1, \ldots, X_n)$ such that $V_i(X_i) > w_i$ for some $i \in [n]$. Then, there exists a connected strongly-proportional allocation, and such an allocation can be computed using $O(n)$ qu

Figures (4)

  • Figure 1: Agent $i$'s and $j$'s new pieces in the proof of \ref{['lem:hungry_one_agent']}.
  • Figure 2: The $r$-marks of all the agents in the proof of \ref{['thm:hungry_equal_condition']}. The point $x$ is at one of the $r$-marks and divides agents into $N_1$ and $N_2$.
  • Figure 3: Construction of the cake used in the proof of \ref{['thm:general_equal_lb']}.
  • Figure 4: Construction of the cake used in the proof of \ref{['thm:stronger_lb']}.

Theorems & Definitions (35)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Theorem 3.6
  • proof
  • ...and 25 more