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arxiv: 1907.05083 · v1 · pith:EY7FSDYHnew · submitted 2019-07-11 · 💻 cs.DS · cs.GT

Cake Cutting on Graphs: A Discrete and Bounded Proportional Protocol

Xiaohui Bei , Xiaoming Sun , Hao Wu , Jialin Zhang , Zhijie Zhang , Wei Zi This is my paper

Pith reviewed 2026-05-24 23:19 UTC · model grok-4.3

classification 💻 cs.DS cs.GT
keywords cake cuttinglocal proportionalitygraph allocationsdiscrete protocolsquery complexityfair divisionresource allocation
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The pith

A discrete bounded protocol produces locally proportional cake allocations on any graph using only single-exponential queries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that local proportionality on an arbitrary acquaintance graph can be achieved by a discrete protocol whose query count grows as a single exponential in the number of agents. Local proportionality requires each agent to value her piece at least as much as the average value received by her graph neighbors. This notion lies strictly between ordinary proportionality and full envy-freeness. The protocol resolves the open question of whether such a method exists for general graphs rather than only for restricted families. It also improves sharply on the tower-of-exponentials query bound known for envy-free cake cutting.

Core claim

The authors give an explicit discrete protocol that, for any undirected graph on n agents, returns an allocation of the interval [0,1] such that every agent i receives a share she values at least as highly as the average value among the pieces of i's neighbors; the protocol uses a number of queries that is single-exponential in n and terminates with a locally proportional allocation.

What carries the argument

A recursive graph-aware division procedure that maintains local average guarantees while trimming and assigning contiguous pieces according to the acquaintance edges.

If this is right

  • Local proportionality is computable by a bounded discrete protocol on every graph.
  • The single-exponential query bound separates local proportionality from the multi-tower complexity of envy-free cake cutting.
  • When the graph is complete the protocol recovers a classical proportional allocation.
  • The same technique yields bounded protocols for any graph that admits an envy-free allocation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Local fairness on networks may be tractable even when global envy-freeness remains hard.
  • The protocol structure suggests similar bounded methods could exist for other graph-based fairness notions such as local envy-freeness.
  • Implementation on real networked resource problems would require only modest query budgets compared with full envy-free methods.

Load-bearing premise

Each agent's value for any subinterval is a positive additive absolutely continuous measure.

What would settle it

A concrete graph together with additive continuous valuations on which every discrete protocol either violates local proportionality or requires more than single-exponential queries.

read the original 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 graphs. 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].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims to resolve an open question by constructing a discrete and bounded protocol that produces a locally proportional allocation (each agent values her piece at least as much as the average of her neighbors) for cake cutting on an arbitrary undirected graph in the Robertson-Webb query model. The protocol works for any number of agents under the standard assumptions of positive, additive, absolutely continuous valuations and achieves single-exponential query complexity, in contrast to the six-tower query complexity of the known envy-free protocol.

Significance. If correct, the result is significant: it supplies the first general-graph protocol for a fairness notion that properly generalizes proportionality while remaining weaker than envy-freeness, and it does so with query complexity that is exponentially better than the best known envy-free protocol. The construction therefore closes a gap left by earlier protocols that were restricted to special graph families.

minor comments (2)
  1. The introduction would benefit from an explicit statement of the precise query model (Robertson-Webb) and the exact definition of local proportionality used in the protocol, even though both are standard.
  2. A short comparison table or paragraph contrasting the new protocol's query bound with the special-graph protocols cited in [1] and [8] would help readers assess the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending acceptance. The referee's summary accurately captures the contribution: a discrete, bounded, single-exponential-query protocol for locally proportional cake cutting on arbitrary graphs.

Circularity Check

0 steps flagged

No circularity in algorithmic construction

full rationale

The paper's central claim is the existence of a new discrete bounded protocol for local proportionality on arbitrary graphs, presented as a constructive algorithmic result in the Robertson-Webb query model. The provided text contains no equations, parameter fittings, predictions, or self-citations that reduce the protocol or its query complexity to the inputs by definition. Citations to [5], [16], [1], and [8] are external references to prior work and the open question; none are load-bearing self-citations from the same authors. The valuation assumptions are the standard ones under which the problem was originally posed. This is a self-contained constructive proof with no reduction to fitted quantities or renamed known results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters or invented entities. Standard cake-cutting domain assumptions (additive continuous valuations, interval resource) are presupposed but not enumerated.

axioms (2)
  • domain assumption Each agent's valuation is a positive additive absolutely continuous measure on [0,1]
    Required for the classical query model and local-proportionality definition referenced via [16] and [5]
  • domain assumption The acquaintance graph is undirected and given as input
    Stated in the problem setup

pith-pipeline@v0.9.0 · 5848 in / 1316 out tokens · 21747 ms · 2026-05-24T23:19:01.412761+00:00 · methodology

discussion (0)

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Reference graph

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