Network-Dependent Allocation
Ranking Under Non-Separable Valuation
Publication
Archived on Zenodo
10.5281/zenodo.20680894
June 13, 2026
v1
Yes
Research Publication Notice
This is a research publication and has not been peer-reviewed. It is presented for research discussion and may be cited as a research publication. Findings are subject to revision.
Status: Research Publication — Version 1.0 — June 2026
Epistemic Status
Propositions 1-3 are formally stated and argued with proof sketches, but full formal proof validation is pending peer review.
NP-hardness claims reference established computational complexity classes (Maximum Coverage, Budgeted Maximum Coverage).
Governance implications require empirical validation.
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How to Cite This Paper
Patrone, M. (2026). Network-Dependent Allocation: Ranking Under Non-Separable Valuation. HomeSelf Research. https://doi.org/10.5281/zenodo.20680894
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Abstract
This research publication presents a theoretical framework for analyzing allocation problems where valuations are non-separable. The core problem addresses systems where the value of selecting an artifact depends on which other artifacts are selected simultaneously, violating the independence assumption underlying traditional ranking-based selection.
The framework introduces the Network-Dependent Allocation (NDA) problem: selecting a subset R of artifacts with cardinality constraint K that maximizes a non-separable valuation function V(R).
We establish three core propositions: (1) No ranking function can produce optimal allocations for all capacity constraints under non-separable valuations; (2) NDA generalizes several NP-hard allocation problems; (3) For monotone submodular valuations, the greedy algorithm achieves (1 - 1/e) approximation.
The paper concludes by distinguishing retrieval from allocation as formally distinct problem classes, with implications for the design of selection systems that must reason about network-dependent valuations.
Keywords
JEL Classification
Journal of Economic Literature (JEL) classification codes for this research publication:
Core Allocation Problem
Where R is a subset of artifacts, K is capacity constraint, and V may be non-separable
Introduction
Selection Under Non-Separable Valuation
Traditional selection systems rely on ranking: artifacts are ordered by independent scores, and users consume the top results. This model assumes that the value of including an artifact in a selection set depends only on that artifact's intrinsic properties.
When selection systems reason over structured representations to construct selections, the value of including an artifact may depend on:
- Complementarities: What other artifacts provide complementary information
- Substitutabilities: What artifacts serve as alternatives
- Network effects: How the artifact interacts with the overall selection structure
- Coherence: How well the selection forms a coherent set
Theoretical Framework
Problem Formulation
The Network-Dependent Allocation (NDA) problem is defined as selecting a subset R of artifacts with cardinality constraint K that maximizes a potentially non-separable valuation function V(R).
Non-Separable Valuations
A valuation function V is separable if it can be decomposed into independent artifact values. When this condition fails, the valuation is non-separable, and ranking-based selection cannot preserve optimal allocations.
Core Propositions
Proposition 1: Ranking Incompleteness
For any non-separable valuation V, no ranking function exists that produces optimal allocations for all capacity constraints K.
Proof sketch: Assume such a ranking exists. Consider two capacity constraints K₁ ≠ K₂. Since V is non-separable, optimal selections cannot both correspond to the same top-K rankings. Contradiction.
Proposition 2: NP-Hardness
NDA generalizes several known NP-hard problems: Maximum Coverage, Maximum Weight Independent Set, and Budgeted Maximum Coverage are all special cases of NDA.
Reduction proofs show these problems are special cases of NDA, establishing NP-hardness.
Proposition 3: Approximation Bounds
For monotone submodular valuations, the greedy algorithm achieves a (1 - 1/e)-approximation to the optimal NDA solution.
Reference: Nemhauser et al. (1978) - foundational work on submodular optimization.
Computational Implications
Exact Computation
For small problem sizes (|U| ≤ 20), exhaustive search or branch-and-bound can find optimal allocations. For larger instances, exact computation becomes intractable.
Approximation Algorithms
Greedy Algorithm: Iteratively add the artifact with highest marginal value. Achieves (1 - 1/e) for submodular V.
Local Search: Start with an initial selection and iteratively improve by swaps. Performance depends on neighborhood structure.
Domain-Specific Structure
Real-world applications often have structure that can be exploited: hierarchical artifact categories, geographic or temporal constraints, user-specific preference patterns, and representation quality signals.
Research Stack Positioning
Network-Dependent Allocation is part of the HomeSelf Research ecosystem:
References
- Nemhauser, G. L., Wolsey, L. A., & Fisher, M. L. (1978). An analysis of approximations for maximizing submodular set functions. Mathematical Programming, 14(1), 265-294.
- Feige, U. (1998). A threshold of ln n for approximating set cover. Journal of the ACM, 45(4), 634-652.
- Cornuejols, G., Fisher, M. L., & Nemhauser, G. L. (1977). Location of bank accounts to optimize float: An analytical study of exact and approximate problem formulations. Management Science, 23(8), 789-810.
- Kleinberg, J., & Oren, J. (2019). Mechanisms for skill-based ranking. Proceedings of the 52nd Annual ACM Symposium on Theory of Computing, 915-932.
Positioning Statement
This research publication is:
This paper does NOT claim universal economics theory, AGI philosophy, SEO theory, or legal/financial advice.