Fair Allocation under Conflict Constraints via Strong Colorability

Abstract: In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results. For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $Δ$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3Δ-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)Δ$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.

• The paper’s main contribution is the full characterization of SD-EF1 allocation thresholds via the strong chromatic number of type 1.
• It refines prior bounds by detailing sufficient conditions for EF1 and EF[1,1] allocations using a hierarchical approach to strong colorability.
• Efficient polynomial-time algorithms are provided under specific graph degree constraints, linking fair allocation with extremal combinatorics.

Fair Allocation under Conflict Constraints via Strong Colorability

Introduction and Problem Statement

The paper "Fair Allocation under Conflict Constraints via Strong Colorability" (2607.01059) investigates fair allocation in the context of indivisible items exhibiting conflict constraints, operationalized through a conflict graph $G=(V,E)$. Vertices represent items, and conflict constraints are encoded as adjacency: no agent may be allocated two adjacent items. The central concern is: For a given graph $G$ and fairness criterion, what is the minimal number of agents $\ell$ such that, for every common agent preference, there exists a feasible (i.e., independent set) partitioning achieving the target level of fairness?

Three fairness concepts are studied under the assumption of common preference (weak order or valuation) among the agents:

• Stochastic-dominance envy-freeness up to one item (SD-EF1): Based on stochastic dominance over weak orders.
• Envy-freeness up to one item (EF1): With monotone additive valuations.
• Envy-freeness up to one item from each side (EF[1,1]): With general additive valuations, allowing both goods and chores.

The authors leverage and extend the theory of strong chromatic numbers, introducing a hierarchy of strong colorability notions, and establish qualitative and quantitative connections between these combinatorial invariants and the existence of fair, feasible allocations.

Hierarchy of Strong Chromatic Numbers

The classical strong chromatic number $\chi_s(G)$, as defined by Alon and Fellows, generalizes traditional coloring by asking: for all equipartitions of the vertices into parts of size $\ell$, does there exist a proper $\ell$-coloring where each part receives all colors? The paper builds upon and refines this notion:

• Strong chromatic number of type $r$ ($\chi_s^r(G)$): Quantifies the minimal integer $\ell$ such that for every partition of $V(G)$ into parts of size at most $G$0, with at most $G$1 parts smaller than $G$2, a proper coloring exists assigning each color at most once per part.

This hierarchy leads to the following chain: $G$3 where the last is Gutner and Tarsi's variant with unbounded deviations in part size.

Main Results

SD-EF1: Characterization via $G$4

The central result for SD-EF1 is a full characterization:

• The number of agents required to guarantee a feasible SD-EF1 allocation (for any common agent weak order) is exactly $G$5.
• Necessity and sufficiency are established for this threshold: for all $G$6 and all common weak orders, such allocations exist; for any $G$7, there exists a weak order for which no feasible SD-EF1 allocation exists.

The reduction takes the following form: arrange vertices by preference, group into blocks of size $G$8, and seek a coloring matching the block structure; such an allocation is fair iff such a coloring exists.

EF1 and EF[1,1]: Sufficient Conditions, Informed by $G$9 and $\ell$0

• For EF1 with monotone additive valuations, $\ell$1 is a sufficient condition for feasible allocations—significantly refining prior bounds in the literature. The threshold is not necessary for general graphs.
• For EF[1,1] with arbitrary additive valuations, a less restrictive notion, the sufficient condition is $\ell$2. This cannot be reduced to $\ell$3, as demonstrated by analysis of $\ell$4, where $\ell$5 but $\ell$6.

Both existential (nonconstructive) and algorithmic consequences are drawn in terms of the graph maximum degree $\ell$7: by Haxell's bound, $\ell$8, and thus $\ell$9 suffices in all cases considered.

Algorithmic Consequences

The structural reductions translate fair allocation into a strong coloring problem parameterized by $\chi_s(G)$0, $\chi_s(G)$1 (type in hierarchy), and the given partition of vertex set. For any fixed $\chi_s(G)$2, deterministic polynomial-time algorithms are constructed (utilizing Harris's algorithmic advances) to find feasible SD-EF1, EF1, and EF[1,1] allocations whenever $\chi_s(G)$3.

For special graph families (e.g., paths, cycles), algorithmic thresholds can be smaller than general bounds and exact match to existential thresholds for small $\chi_s(G)$4 are achieved leveraging deep results from extremal combinatorics.

Structural Analysis and Thresholds

The necessity of thresholds is precisely established for $\chi_s(G)$5: for any such $\chi_s(G)$6, and for EF1 and EF[1,1], fewer than three agents can preclude feasible allocations under some monotone additive valuation. However, for higher chromatic parameters (e.g., $\chi_s(G)$7), the sufficient condition is not necessary; there exist graphs with $\chi_s(G)$8 and feasible EF1 allocations for three agents under any monotone valuation.

Implications and Theoretical Contributions

The paper makes several technical and conceptual contributions:

• Unification: By recasting fair allocation as a strong coloring problem, the authors unify several allocation problems (for various fairness notions) within the combinatorial framework of strong colorability.
• Sharp bounds: The tightness of the SD-EF1 characterization sets a benchmark for future studies of other (possibly more complex or heterogeneous) fairness criteria.
• Algorithmic metareductions: The paper demonstrates that advances or conjectures in extremal coloring (e.g., the strong $\chi_s(G)$9-colorability conjecture) have direct consequences for fair allocation under conflict.
• Generality and limits: Sufficient conditions for EF1 and EF[1,1] can be strictly weaker than necessary, especially for graphs with higher chromatic number, highlighting structural gaps between envy-freeness (valuative) and stochastic-dominance unfairness (ordinal).

Future Directions

Three prominent challenges and research avenues are identified:

• Heterogeneous preferences: The precise combinatorial thresholds for EF1 and EF[1,1] under agent heterogeneity remain open and, as shown, do not reduce to the common-preference case.
• Strong choosability: Since fair allocation reduces to coloring, list colorability (choosability) and its strong version may provide further robustness or improved bounds for allocations under more adversarial or dynamic scenarios.
• Bridging gaps for EF1/EF[1,1]: Narrowing the gap between known sufficient and necessary agent thresholds, especially for the EF[1,1] criterion and its variants, is a direction that would yield more refined combinatorial invariants.

Conclusion

This work establishes deep ties between fair allocation under conflict constraints and a newly formalized hierarchy of strong chromatic numbers. By providing both existential and efficient algorithmic results, it pushes forward the theoretical boundary for fair allocation in discrete, combinatorially constrained environments, while also outlining avenues where advances in structural graph theory will have direct algorithmic import.