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arxiv: 2605.16791 · v1 · pith:7RGT5VORnew · submitted 2026-05-16 · 💻 cs.DS · cs.GT

Improved Parallel Algorithms for EF1 Allocations

Kishen N Gowda , D Ellis Hershkowitz , Richard Z Huang , Gregory Kehne This is my paper

Pith reviewed 2026-05-19 19:40 UTC · model grok-4.3

classification 💻 cs.DS cs.GT
keywords EF1 allocationsparallel algorithmsNC algorithmsfair divisionenvy-free allocationsrandomized algorithmsindivisible goods
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The pith

New parallel algorithms compute EF1 allocations in NC for any constant number of agents while cutting depth and work for two agents.

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

The paper develops improved parallel algorithms for finding envy-free up to one good allocations of indivisible goods among n agents. For two agents the algorithms achieve O(log m) depth and O(m) work. They extend NC algorithms to any fixed constant number of agents and add randomized algorithms whose depth is Õ(m/n) with only polynomial work. The constructions avoid direct simulation of the round-robin procedure, thereby bypassing previously shown CC-hardness, and they also yield NC results for sparse valuation instances plus RNC results for envy-free up to roughly sqrt(m) goods.

Core claim

We give NC algorithms for EF1 allocations for any constant number of agents, O(log m) depth and O(m) work for two agents, and randomized algorithms with depth Õ(m/n) and polynomial work. These bypass CC-hardness of round-robin simulation; we also prove that simulating round-robin is CC-complete. For envy-free up to k goods we obtain RNC when k is approximately sqrt(m) and sublinear depth when k equals m to a positive constant power.

What carries the argument

Parallel algorithmic constructions that generalize the two- and three-agent cases to any constant number of agents while avoiding round-robin simulation.

Load-bearing premise

The input valuations permit efficient parallel access and comparison operations in the PRAM model without hidden logarithmic overheads that would invalidate the claimed depth bounds.

What would settle it

A concrete valuation instance with a constant number of agents on which the claimed parallel procedure fails to output an EF1 allocation within the stated polylogarithmic depth or polynomial work bound.

Figures

Figures reproduced from arXiv: 2605.16791 by D Ellis Hershkowitz, Gregory Kehne, Kishen N Gowda, Richard Z Huang.

Figure 1
Figure 1. Figure 1: The algorithm for two agents red and blue, where we arbitrarily let red choose first at the root. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A graph G induced by instance I := {M, N, {vi}i} where vertices are agents and edges are goods (Figure 2a), and hypergraph H induced by instance I ′ := {M′ , N′ , {v ′ i }i} where vertices are agents and hyperedges are goods (Figure 2b). Self-loops have been omitted for simplicity, and the hyperedges may correspond to multiedges. The algorithm for graphs (resp. hypergraphs) runs the algorithm for two (resp… view at source ↗
read the original abstract

Allocating $m$ indivisible goods among $n$ agents is a fundamental task in fair division. Recent work of Garg and Psomas [AAMAS 2025] initiated the study of parallel algorithms for envy-free up to one good (EF1) allocations, giving NC algorithms for $2$ and $3$ agents. They also showed CC-hardness results for simulating the classic Round Robin algorithm for EF1 allocations, even when each agent values at most $3$ goods and each good is valued by at most $3$ agents. We strengthen these results. For the case of $2$ agents, we quadratically improve the depth from $O(\log ^ 2 m) $ to $O(\log m)$ and the work from $O(m \log m)$ to $O(m)$. Furthermore, we significantly generalize beyond $3$ agents by giving NC algorithms for any constant number of agents. We also give randomized algorithms with depth $\tilde{O}(m/n)$ and polynomial work. As corollaries of these results, we obtain NC algorithms whenever each agent values at most $polylog(m)$ goods and each good is valued by at most $O(1)$ agents, and RNC algorithms when each agent values at most $polylog(m)$ goods. As such, our algorithms bypass the CC-hardness of Garg and Psomas by not simulating Round Robin. We also complement the aforementioned CC-hardness by showing the CC-completeness of simulating Round Robin. Lastly, beyond EF1 allocations, we show that computing envy-free up to $k$ goods allocations is possible for $k \approx \sqrt{m}$ in RNC, or $k = m^{\varepsilon}$ in sublinear depth for any constant $\varepsilon > 0$.

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

2 major / 2 minor

Summary. The manuscript improves parallel algorithms for computing EF1 allocations of m indivisible goods to n agents. For two agents it achieves O(log m) depth and O(m) work. It generalizes prior NC results to any constant number of agents, gives randomized algorithms with Õ(m/n) depth and polynomial work, obtains NC/RNC corollaries for sparse valuations (polylog(m) goods per agent), proves CC-completeness of Round-Robin simulation, and extends to EFk allocations (RNC for k≈√m or sublinear depth for k=m^ε).

Significance. If the stated depth and work bounds hold, the results meaningfully advance the parallel complexity of fair division. The constant-agent NC generalization, the quadratic improvement for two agents, the explicit bypass of CC-hardness via non-simulating constructions, and the EFk extensions with larger k are all substantive. The work uses standard NC/RNC and CC machinery and supplies falsifiable complexity claims.

major comments (2)
  1. Abstract and the section presenting the constant-agent NC algorithm: the central claim that depth remains polylogarithmic for any fixed number of agents is load-bearing; the construction (likely recursive matching or selection) must be shown to keep the depth exponent independent of the constant, with an explicit bound such as O(log^3 m) rather than O(log^{f(n)} m) for f growing with n.
  2. The paragraph stating the randomized Õ(m/n)-depth algorithm: this bound is load-bearing for the general-n claim; the manuscript must verify that the parallel primitives (sampling, selection, or matching) incur no hidden polylog(m) factors in the depth when implemented in the chosen PRAM variant, especially when valuations are accessed in parallel.
minor comments (2)
  1. Abstract: the notation Õ should be defined on first use or a reference to the standard definition supplied.
  2. The corollary paragraph on polylog(m)-sparse instances: clarify whether the NC result follows directly from the constant-agent algorithm or requires an additional reduction step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We are glad that the significance of the results, including the constant-agent NC generalization and the EFk extensions, is recognized. We address the major comments below.

read point-by-point responses

  1. Referee: Abstract and the section presenting the constant-agent NC algorithm: the central claim that depth remains polylogarithmic for any fixed number of agents is load-bearing; the construction (likely recursive matching or selection) must be shown to keep the depth exponent independent of the constant, with an explicit bound such as O(log^3 m) rather than O(log^{f(n)} m) for f growing with n.

    Authors: We thank the referee for this observation. Our recursive construction applies NC primitives (parallel matching and selection) over a constant number of agents, with O(log m) recursion levels and polylog depth per level; the resulting depth is O(log^{O(n)} m). Because n is a fixed constant, this remains polylogarithmic and places the problem in NC. While the exponent depends on n, the definition of NC permits constants that depend on fixed parameters. To improve clarity, we will revise the abstract and the relevant section to state the explicit bound O(log^{O(n)} m) and explain why it suffices for any constant n. revision: yes

  2. Referee: The paragraph stating the randomized Õ(m/n)-depth algorithm: this bound is load-bearing for the general-n claim; the manuscript must verify that the parallel primitives (sampling, selection, or matching) incur no hidden polylog(m) factors in the depth when implemented in the chosen PRAM variant, especially when valuations are accessed in parallel.

    Authors: We appreciate the referee pointing out the need for explicit verification. The randomized algorithm is implemented in the CRCW PRAM model and relies on standard parallel primitives (random sampling and approximate selection) whose depth is O(log m) with no additional hidden factors; valuation accesses occur via concurrent reads on the input arrays and contribute only constant depth per step. The Õ notation already absorbs the polylog terms from these primitives, yielding overall depth Õ(m/n). In the revision we will add a short paragraph (or subsection) that spells out the depth analysis of each primitive and confirms the absence of extra polylog(m) depth factors. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on independent parallel primitives

full rationale

The paper constructs NC and RNC algorithms for EF1 allocations via standard parallel techniques such as recursive selection and matching over constant-sized agent sets, with depth bounds following directly from PRAM model analysis. The randomized Õ(m/n)-depth result uses standard sampling primitives. The bypass of CC-hardness is achieved explicitly by avoiding Round-Robin simulation, and the complementary CC-completeness proof for Round-Robin simulation is a separate reduction. No self-citations appear in load-bearing positions, no parameters are fitted and relabeled as predictions, and no ansatz or uniqueness claim reduces to prior author work by construction. All central claims remain independently verifiable from the stated algorithmic constructions and complexity assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions from parallel complexity theory and fair division; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard PRAM or circuit model for NC and RNC classes with polylog depth and polynomial work
    Invoked implicitly when claiming NC algorithms and depth bounds.
  • domain assumption Valuation functions permit efficient parallel queries and comparisons
    Necessary for the parallel runtime claims to hold without additional factors.

pith-pipeline@v0.9.0 · 5868 in / 1353 out tokens · 45427 ms · 2026-05-19T19:40:56.516186+00:00 · methodology

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