Submodular Max-Min Allocation under Identical Valuations
Pith reviewed 2026-05-10 14:37 UTC · model grok-4.3
The pith
A greedy algorithm achieves a 0.4-approximation for submodular max-min allocation under identical valuations.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
When all players share an identical monotone submodular valuation, the greedy procedure that repeatedly assigns an item to the player whose current bundle has the lowest value produces an allocation whose minimum satisfaction is at least 0.4 times optimal. Mapping any integral solution of the configuration LP to dual variables shows that the integrality gap is bounded by a fixed constant independent of the instance size.
What carries the argument
The greedy augmentation rule that always gives the next item to the player with the smallest current valuation, together with the explicit dual-variable assignment built from any integral primal solution.
If this is right
- The 0.4 ratio strictly improves the earlier 10/27 guarantee.
- The configuration LP admits a constant integrality gap under identical valuations, opening the door to LP-based methods.
- The same dual-construction technique yields an O(k)-estimation algorithm for the version constrained by k matroids.
- The results apply directly to any monotone submodular function that is identical across players.
Where Pith is reading between the lines
- The dual-from-integral mapping may extend to heterogeneous valuations if the symmetry argument can be replaced by a different bounding technique.
- Tighter analysis of the greedy choices could improve the constant 0.4 toward the best possible ratio.
- Analogous primal-to-dual constructions could bound integrality gaps for other symmetric submodular objectives.
- Empirical tests on large identical-preference instances would show how the worst-case 0.4 ratio translates to average-case behavior.
Load-bearing premise
All players share exactly the same monotone submodular valuation function.
What would settle it
An explicit set of items and identical submodular valuation where the greedy allocation's minimum value falls below 0.4 times the true optimal minimum value.
read the original abstract
In the problem of Submodular Max-Min Allocation, we are given a set of items, a set of players, and monotone submodular valuation functions that represent the satisfaction of a player with a certain subset of items. The goal is to find an allocation of the items to the players that maximizes the lowest satisfaction among all players. We study this problem in the special case where all players have the same valuation function. We devise a greedy algorithm which gives a $0.4$-approximation, improving the previously best factor of $\frac{10}{27} \approx 0.37$ by Uziahu and Feige. Furthermore, we study the integrality gap of the \emph{configuration LP} when players have identical valuations. By constructing a variable assignment to the dual from a primal integral solution, we give the first constant upper bound on the integrality gap for submodular valuations. Generalizing the result to the case where players' allocations must be independent in $k$ given matroids, we derive a $\mathcal{O}(k)$-estimation algorithm for max-min allocation subject to $k$ matroid constraints under identical valuations.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies the submodular max-min fair allocation problem when all players have identical monotone submodular valuation functions. It presents a greedy algorithm achieving a 0.4-approximation (improving on the prior 10/27 factor), establishes the first constant upper bound on the integrality gap of the configuration LP via a dual solution constructed directly from an optimal integral primal allocation, and generalizes the results to k matroid constraints per player to obtain an O(k)-estimation algorithm.
Significance. If the 0.4 guarantee and constant integrality-gap bound hold, the work provides a modest but concrete algorithmic improvement and the first constant bound on the configuration-LP gap for identical submodular valuations. The dual-from-primal construction technique, if verified, is a reusable idea for identical-valuation settings; the matroid generalization extends applicability. These are solid contributions to submodular fair allocation.
major comments (1)
- [Proof of the integrality-gap upper bound (dual construction from primal integral solution)] The constant upper bound on the integrality gap (the central theoretical claim) rests on constructing dual variables from an optimal integral primal solution so that every configuration-LP dual constraint is satisfied. The manuscript invokes monotonicity and identical valuations to bound the dual objective, but does not explicitly verify that the chosen multipliers remain non-negative and satisfy all covering inequalities for arbitrary monotone submodular functions whose marginal returns are non-uniform across bundles. A concrete check (or additional argument) confirming dual feasibility for general v is required; otherwise the gap bound is not established.
minor comments (1)
- [Abstract] The abstract states an 'O(k)-estimation algorithm' for the matroid generalization; clarifying whether this is a true approximation algorithm or merely an estimation procedure that bounds the optimum would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and valuable feedback on our manuscript. The concern regarding explicit verification of dual feasibility is well-taken, and we will strengthen the presentation in the revision.
read point-by-point responses
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Referee: [Proof of the integrality-gap upper bound (dual construction from primal integral solution)] The constant upper bound on the integrality gap (the central theoretical claim) rests on constructing dual variables from an optimal integral primal solution so that every configuration-LP dual constraint is satisfied. The manuscript invokes monotonicity and identical valuations to bound the dual objective, but does not explicitly verify that the chosen multipliers remain non-negative and satisfy all covering inequalities for arbitrary monotone submodular functions whose marginal returns are non-uniform across bundles. A concrete check (or additional argument) confirming dual feasibility for general v is required; otherwise the gap bound is not established.
Authors: We agree that an explicit verification of dual feasibility would improve clarity. In the current manuscript the dual variables are constructed by assigning to each configuration in the optimal integral allocation a multiplier derived from the common valuation function v, scaled by the minimum value in the allocation; monotonicity ensures non-negativity of these multipliers, while identical valuations across players ensure that the same multipliers cover the configuration-LP dual constraints uniformly. Submodularity is used to bound the objective value relative to the primal. Nevertheless, to address the referee’s request for a concrete check that works for arbitrary monotone submodular v (including those with non-uniform marginal returns), we will add a dedicated lemma in the revised version. The lemma will enumerate the dual constraints, show non-negativity directly from the construction, and verify the covering inequalities by considering an arbitrary bundle and applying the submodular inequality to the marginal gains relative to the integral allocation. This addition will make the constant integrality-gap bound fully rigorous without altering the claimed result. revision: yes
Circularity Check
No significant circularity; explicit greedy and dual constructions are self-contained
full rationale
The paper presents an explicit greedy algorithm whose 0.4-approximation is derived from direct analysis of marginal gains under identical monotone submodular valuations, and separately constructs dual variables from any given integral primal allocation to certify an upper bound on the configuration-LP value. Neither step defines a quantity in terms of itself, fits parameters to a target outcome, nor relies on a load-bearing self-citation whose validity is assumed rather than proven. The dual feasibility argument uses only monotonicity, submodularity, and identical valuations to verify the covering constraints; this is a standard primal-to-dual mapping whose correctness is independent of the claimed gap bound. No renaming of known results or smuggling of ansatzes occurs. The derivation chain is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Valuation functions are monotone and submodular
Reference graph
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discussion (0)
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