Fair Allocation Under Conflict Constraints

• The paper introduces a graph-theoretic model for fair allocation, framing conflict-free independent-set allocations within fairness and efficiency objectives.
• It details various fairness criteria—such as EF1, max–min welfare, and SD-EF1—and presents algorithmic strategies for two-agent cases along with complexity results for multi-agent scenarios.
• It explores parameterized and structural tractability, offering dynamic programming approaches, pseudo-polynomial algorithms, and FPTAS techniques on structured graph classes.

Searching arXiv for recent and related work on fair allocation under conflict constraints. Fair allocation under conflict constraints concerns the division of indivisible items when incompatibilities between pairs of items are encoded by a graph and each agent must receive a conflict-free bundle, typically an independent set. In this model, fairness is no longer studied on arbitrary set partitions but on graph-feasible allocations, so the subject lies at the intersection of fair division, graph coloring, independent-set optimization, and scheduling with incompatibilities. The literature has developed several distinct objective families: max–min welfare, which maximizes the minimum attained utility; envy-based notions such as envy-freeness up to one item (EF1) and its variants; efficiency refinements such as maximality; and relaxations in which conflicts may be violated at controlled cost. Recent work has substantially clarified the landscape: two-agent EF1 plus maximality is now constructive under monotone valuations on arbitrary graphs, whereas for three or more agents hard conflicts already produce nonexistence and NP-hardness; in parallel, the max–min formulation exhibits broad hardness together with pseudo-polynomial dynamic programs and FPTASs on structured graph classes (Igarashi et al., 17 Jun 2025, Chiarelli et al., 2020, Equbal et al., 11 May 2026).

1. Formal model and graph-theoretic formulation

The standard hard-conflict model represents items by the vertex set of an undirected graph $G=(M,E)$. An edge $\{g,g'\}\in E$ means that $g$ and $g'$ cannot be assigned to the same agent. For agents $N=\{1,\dots,n\}$, an allocation is an ordered tuple $A=(A_1,\dots,A_n)$ such that the bundles are pairwise disjoint and each $A_i$ is an independent set in $G$. In many formulations one allows $\bigcup_i A_i\subseteq M$, so some items may remain unassigned; this is naturally viewed as a partial coloring of the conflict graph. In complete-allocation variants, every item must be assigned. Valuations are most often additive, $v_i(S)=\sum_{g\in S}v_i(\{g\})$, but monotone set valuations and mixed-sign additive valuations also appear (Igarashi et al., 17 Jun 2025, Chiarelli et al., 2020).

This graph formulation has several exact equivalences that shape the complexity theory. With one agent, feasibility reduces to choosing a maximum-value independent set. With an edgeless graph, the conflict-free condition disappears and one recovers classical fair division or Santa-Claus-style max–min allocation. For max–min welfare, feasible allocations are partial $\{g,g'\}\in E$0-colorings whose color classes carry agent-specific weights, and the objective is

$\{g,g'\}\in E$1

This simultaneously generalizes Partition and Weighted Independent Set (Chiarelli et al., 2020).

Several important variants extend the base model. Budget-feasible egalitarian allocation associates to each agent-item pair both a satisfaction $\{g,g'\}\in E$2 and a cost $\{g,g'\}\in E$3, constrains $\{g,g'\}\in E$4, and still maximizes the minimum total satisfaction under conflict-freeness (Gupta et al., 2024). Soft-conflict models allow adjacent goods to be allocated together but count each violated edge as a penalty, shifting the question from strict feasibility to a fairness-versus-violations tradeoff (Yoneda et al., 24 Feb 2026).

2. Fairness notions and efficiency criteria

Several fairness criteria coexist in this literature, and the choice of criterion materially changes both the existence theory and the algorithmics.

Notion	Formal condition	Role
Max–min fairness	$\{g,g'\}\in E$5	Welfare objective
EF1	$\{g,g'\}\in E$6, either $\{g,g'\}\in E$7 or $\{g,g'\}\in E$8 with $\{g,g'\}\in E$9	Approximate envy-freeness for goods
Maximality	No unallocated item can be feasibly added to any agent	Efficiency under hard conflicts
EF[1,1]	$g$0 such that $g$1	Mixed goods/chores or general additive valuations
SD-EF1	After removing at most one item, one bundle stochastically dominates another under a weak order	Ordinal fairness under common preferences

Under hard conflicts, maximality is especially important because unassigned goods may remain even when every agent’s bundle is feasible. An allocation is maximal exactly when every unallocated good conflicts with some good already held by every agent. This notion is orthogonal to EF1: one can have an EF1 allocation that leaves obviously assignable goods unused, or a maximal allocation that is not fair (Igarashi et al., 17 Jun 2025).

The literature also studies maximin-share guarantees and Nash welfare under conflicts. For additive valuations and feasible complete allocations, the maximin share is defined relative to the set of feasible allocations, and a maximum Nash welfare allocation maximizes the geometric mean of utilities over feasible allocations. These notions remain meaningful but no longer inherit the unconstrained existence behavior in a direct way (Hummel et al., 2021).

For common weak orders, stochastic-dominance envy-freeness up to one item provides an ordinal analogue of EF1. For common monotone additive valuations, SD-EF1 implies EF1 after ordering items by common value. For general additive valuations, especially when goods and chores coexist, EF[1,1] replaces one-sided EF1 as the natural benchmark (Haviv, 1 Jul 2026).

3. Two-agent maximal EF1: existence, algorithms, and limits

A central recent result is that for two agents and any conflict graph, a maximal EF1 allocation always exists under monotone valuations (Igarashi et al., 17 Jun 2025). One proof template reduces to identical monotone valuations and constructs a sequence of maximal allocations forming a “gapless chain.” The key local notion is ordered adjacency: consecutive allocations differ by at most one item on each side in a directional sense. The Crossing Lemma then shows that if the sign of $g$2 flips across an ordered-adjacent pair, at least one of the two allocations is EF1. By building a gapless chain from a maximal independent set assigned to agent 1 to the role-reversed situation, one obtains existence of a maximal EF1 allocation.

A complementary description uses a color-switching technique that locally modifies a maximal allocation while preserving feasibility and restoring EF1. In that formulation, the two-agent theorem is stated for arbitrary monotone valuations $g$3, not just the identical-valuation reduction, and yields a pseudopolynomial-time algorithm when valuations are integer-valued (Equbal et al., 11 May 2026). This suggests that the existence theorem is robust to the choice of constructive proof framework.

For computation, additive valuations admit a polynomial-time algorithm. In the SWAPEF1 framework, each unsuccessful iteration increases the value of a chosen maximal independent set by a multiplicative factor, yielding a total running time of

$g$4

for additive valuations, where $g$5 is the cost of the subroutine used inside the chain construction. For monotone valuations, the same framework gives

$g$6

with $g$7, hence pseudopolynomial time when valuations are integer-valued (Igarashi et al., 17 Jun 2025). Polynomial-time refinements are also known for monotone valuations on interval graphs and bipartite graphs (Equbal et al., 11 May 2026).

The two-agent theory is tight in several senses. If monotonicity is dropped, maximal EF1 need not exist even for two agents with identical additive valuations: a path with one $g$8 good and one $g$9 chore yields a maximal allocation in which envy remains after removing one item. Deciding existence becomes NP-hard for any fixed $g'$0 under identical additive valuations via a reduction from INDEPENDENT SET (Equbal et al., 11 May 2026). For three agents, the monotone case already breaks: there exists an instance with identical monotone valuations and a conflict graph on 7 goods for which no maximal EF1 allocation exists, and deciding existence is NP-hard for every fixed $g'$1 of agents; for $g'$2 the hardness holds even under identical additive valuations (Igarashi et al., 17 Jun 2025).

The goods results in this line also apply to chores by the standard transformation $g'$3, so the two-agent existence theorem, the algorithmic results, and the nonexistence and NP-hardness statements for $g'$4 all carry over to monotone non-increasing valuations (Igarashi et al., 17 Jun 2025).

4. Max–min welfare under hard conflicts

A parallel research line formulates fairness as max–min welfare rather than envy minimization. Here the goal is to maximize the minimum total profit allocated to any agent, with each agent receiving an independent set. This model was systematized by Chiarelli et al., who emphasized its interpretation as partial graph coloring with weighted color classes (Chiarelli et al., 2020).

The hardness picture is broad. The problem is strongly NP-hard for bipartite graphs and for line graphs of bipartite graphs, even with identical profit functions. The inapproximability theory is correspondingly severe: by transferring hardness from Independent Set, no $g'$5-approximation is obtainable even for unit profits. These results show that perfect-graph structure alone does not guarantee tractability (Chiarelli et al., 2020).

Positive results arise on more structured graph classes. For every fixed number $g'$6 of agents, pseudo-polynomial dynamic programs are known for cocomparability graphs, chordal graphs, biconvex bipartite graphs, and graphs of bounded treewidth. The cocomparability algorithm uses a transitive orientation of the complement and interprets independent sets as directed paths; the chordal and bounded-treewidth algorithms are bag-based dynamic programs over tree decompositions; the biconvex case combines a decomposition into a cocomparability “middle” part with guessed endpoints and a convolution over connected components. Each of these pseudo-polynomial algorithms can be turned into an FPTAS by standard scaling and rounding of profit coordinates (Chiarelli et al., 2020).

This tractability boundary was extended further to convex bipartite graphs, graphs of bounded clique-width, and graphs of bounded tree-independence number. The convex bipartite algorithm grows the graph along ordered interval endpoints and maintains profit-profile sets indexed by the highest chosen vertices; the clique-width algorithm performs dynamic programming over an $g'$7-expression with state indexed by label sets used by each agent; the tree-independence-number algorithm generalizes tree-decomposition DP from bounded treewidth to bags of bounded independence number (Chiarelli et al., 2023).

5. Parameterized and structural tractability

Parameterized analysis has refined the max–min picture beyond classical NP-hardness. In the “conflict-free fair allocation” formulation, the natural decision question asks whether there exists a conflict-free allocation $g'$8 with $g'$9. A trivial enumeration over assignments gives $N=\{1,\dots,n\}$0, but FFT-based subset-convolution methods reduce this to a single-exponential

$N=\{1,\dots,n\}$1

algorithm in the number of items $N=\{1,\dots,n\}$2 (Gupta et al., 2023). A closely related formulation also gives an $N=\{1,\dots,n\}$3-time algorithm and an ETH lower bound ruling out $N=\{1,\dots,n\}$4 (Bandopadhyay et al., 2024).

The parameterized complexity is mixed. The number of agents $N=\{1,\dots,n\}$5 alone is para-NP-hard, and the bundle-size bound $N=\{1,\dots,n\}$6 alone is also para-NP-hard once $N=\{1,\dots,n\}$7. By contrast, the combined parameter $N=\{1,\dots,n\}$8 admits an exact characterization on hereditary graph classes: size-bounded CFFA is FPT in parameter $N=\{1,\dots,n\}$9 if and only if size-bounded MWIS is FPT in parameter $A=(A_1,\dots,A_n)$0 on the same graph class. This yields FPT results for interval, chordal, perfect, bipartite, planar, bounded-degeneracy, and related classes (Gupta et al., 2023).

Other structural parameters support kernelization or FPT algorithms. Neighborhood diversity yields a polynomial kernel, and the number $A=(A_1,\dots,A_n)$1 of non-edges yields an FPT algorithm by enumerating cliques in the complement using degeneracy. For treewidth, current results are more asymmetric: there are lower bounds excluding polynomial kernels for $A=(A_1,\dots,A_n)$2, but the FPT status of $A=(A_1,\dots,A_n)$3 remains open in one formulation (Bandopadhyay et al., 2024).

These results underscore a persistent theme. Conflict graphs simultaneously encode packing and coloring difficulty, so tractability tends to require either strong graph structure, small item sets, or carefully combined parameters rather than any single obvious quantity.

6. Degree bounds, strong colorability, and relaxed conflicts

More recent work has identified graph-theoretic thresholds for envy-based fairness under common preferences. A particularly clean framework uses variants of the strong chromatic number. If $A=(A_1,\dots,A_n)$4, then every graph $A=(A_1,\dots,A_n)$5 admits a feasible SD-EF1 allocation for every common weak order and a feasible EF1 allocation for every common monotone additive valuation. If $A=(A_1,\dots,A_n)$6, then every common additive valuation admits a feasible EF[1,1] allocation. By Haxell’s bound, every graph of maximum degree $A=(A_1,\dots,A_n)$7 therefore admits feasible SD-EF1, EF1, and EF[1,1] allocations whenever the number of agents is at least $A=(A_1,\dots,A_n)$8; moreover, for any $A=(A_1,\dots,A_n)$9, deterministic polynomial-time algorithms are available when the number of agents is at least $A_i$0 (Haviv, 1 Jul 2026).

A different degree-based line studies complete feasible EF1 allocations. For tiered valuations of size $A_i$1, including ordered valuations, if

$A_i$2

then there always exists a complete, conflict-feasible EF1 allocation. The algorithm processes tiers one by one and uses a matching in a feasibility graph between agents and current-tier items. When $A_i$3, this bound is tight even for identical valuations. For general additive valuations with $A_i$4, a Round Robin plus matching method yields complete EF1 allocations under several explicit degree-versus-size conditions, including $A_i$5, or $A_i$6 with $A_i$7, and more generally $A_i$8 with $A_i$9 (Markakis et al., 11 Jun 2026).

Soft-conflict models relax feasibility rather than fairness. Here all goods are assigned, EF1 is required, and the objective is to keep the number of violated conflict edges small. For identical additive valuations and fixed $G$0, a cyclic-shift round-robin algorithm returns a balanced EF1 allocation in $G$1 time with at most $G$2 violations, and a star lower bound shows that the leading term $G$3 is unavoidable in general. For general additive valuations and fixed $G$4, a linear-time algorithm returns a balanced EF1 allocation with at most

$G$5

violations, using a combination of the Biswas–Barman cardinality-constraint method and a geometric “closest points” argument on conflict profiles (Yoneda et al., 24 Feb 2026).

Taken together, these results show that the graph structure can enter the fairness theory in qualitatively different ways: as a hard feasibility constraint, as a degree-controlled sufficient condition for complete EF1, or as a source of bounded violation under soft conflicts.

7. Variants, empirical observations, and open problems

The subject has broadened beyond EF1 and max–min welfare. Budget-feasible egalitarian allocation of conflicting jobs adds per-agent costs and budgets to the conflict graph. This problem is NP-hard even for two agents and no conflicts, admits an FPTAS on bounded-treewidth graphs when the number of agents is constant, and has a parameterized landscape that includes W[1]-hardness for the number of agents, an exact $G$6-style algorithm in the number of jobs, and polynomial-time solvability for bundle size $G$7 via matching (Gupta et al., 2024).

Approximate maximin-share fairness and Nash welfare under conflicts form another line. For additive valuations and conflict graph maximum degree $G$8, a $G$9-approximate MMS allocation can be found in polynomial time whenever $\bigcup_i A_i\subseteq M$0, and a $\bigcup_i A_i\subseteq M$1-approximate MMS allocation always exists when $\bigcup_i A_i\subseteq M$2. Experimental evidence on 15,000 random instances with $\bigcup_i A_i\subseteq M$3 found that all sampled instances admitted both EF1 and MMS, and that in 99.9% of cases the maximum-Nash-welfare allocation was EF1 (Hummel et al., 2021). A plausible implication is that worst-case nonexistence phenomena coexist with a much milder empirical regime on random sparse instances.

Several open problems recur across the literature. For maximal EF1, it remains open whether a three-agent counterexample exists under additive valuations, and closing the gap between pseudopolynomial and truly polynomial time for monotone two-agent valuations remains unresolved (Igarashi et al., 17 Jun 2025). In the parameterized max–min line, the FPT status of $\bigcup_i A_i\subseteq M$4 remains open in one formulation, as do several intermediate bipartite classes and the case of cluster graphs with two cliques and arbitrary utilities (Bandopadhyay et al., 2024, Gupta et al., 2023). In the strong-colorability framework, the conjectured $\bigcup_i A_i\subseteq M$5 threshold would sharpen the current $\bigcup_i A_i\subseteq M$6 existence bound (Haviv, 1 Jul 2026). In the soft-conflict setting, the natural conjecture is that the lower-order term can be removed and every instance should admit EF1 with at most $\bigcup_i A_i\subseteq M$7 violations (Yoneda et al., 24 Feb 2026).

The resulting picture is technically diverse but conceptually coherent. Conflict constraints transform fair allocation into a graph-constrained packing problem whose difficulty depends sharply on the chosen fairness notion, the graph class, and whether one insists on completeness or maximality. Two-agent maximal EF1 under monotone valuations is now essentially settled; multi-agent hard-conflict EF1 is governed by nonexistence and hardness unless additional graph structure or common-preference assumptions are imposed; and max–min welfare remains broadly hard but highly amenable to dynamic programming on structured graphs.