EFX-with-Bounded-Charity

• EFX-with-bounded-charity is a fairness criterion for indivisible goods that permits a small, bounded pool of unallocated items to achieve near-envy-free allocations.
• It combines exact EFX on assigned bundles with quantitative limits on the charity, leveraging methodological constructs like envy and champion graphs.
• It underpins algorithms—both exact and approximate—that balance fairness with efficiency, facilitating welfare guarantees and randomized fairness support.

Searching arXiv for recent and foundational papers on EFX-with-bounded-charity to ground the article in current literature. EFX-with-bounded-charity is a relaxation of envy-freeness up to any good (EFX) for allocations of indivisible goods, in which some goods may remain unallocated provided that the allocated part satisfies EFX and the leftover set is quantitatively controlled. In its standard form, the unallocated goods form a pool or charity that no agent values above her own bundle, while a cardinality bound such as $|P|<n$, $|P|\le n-1$, or a sublinear function of $n$ or of the number of valuation types is imposed on the pool. The notion has become a principal route for obtaining exact, approximate, efficiency-aware, and randomized fairness guarantees in settings where complete EFX remains unresolved (Chaudhury et al., 2019, HV et al., 21 Aug 2025).

1. Formal model and main variants

In the bundle-allocation model, let $M$ be the set of goods, $N=[n]$ the agents, and $v_i:2^M\to\mathbb R_{\ge 0}$ their valuations. An allocation with charity is a partition $(A_1,\dots,A_n,P)$ of $M$, where each $A_i$ is assigned to agent $i$ and $|P|\le n-1$0 is the set of unallocated goods. A standard bounded-charity formulation is the condition called $|P|\le n-1$1: for every pair $|P|\le n-1$2 and every $|P|\le n-1$3, one has

$|P|\le n-1$4

for every agent $|P|\le n-1$5,

$|P|\le n-1$6

and the charity is bounded by

$|P|\le n-1$7

This combines exact EFX on the allocated bundles, non-envy of the pool, and an explicit bound on leftover goods (Kavitha et al., 22 Jul 2025).

A second major variant is approximate EFX with charity. For additive valuations and $|P|\le n-1$8, a partial allocation $|P|\le n-1$9 with charity $n$0 is $n$1-EFX with charity if for every $n$2 and every $n$3,

$n$4

and for every $n$5,

$n$6

Here the quantitative objective is to bound $n$7 as a function of $n$8, $n$9, or structural parameters such as the number of distinct valuation types $M$0 (HV et al., 21 Aug 2025).

These formulations isolate two intertwined relaxations of complete EFX: incompleteness, through the charity set, and approximation, through the factor $M$1. The literature uses both, sometimes simultaneously, to derive existence theorems, constructive algorithms, and welfare guarantees.

2. Existence guarantees and quantitative bounds

Representative guarantees are summarized below.

Setting	Guarantee	Source
Normalized, monotone valuations	EFX, no envy of $M$2, $M$3	(Chaudhury et al., 2019)
Nice cancelable valuations	EFX, no envy of $M$4, $M$5; for $M$6, $M$7	(Berger et al., 2021)
Polynomial-time approximate bound	$M$8-EFX, no strict envy of charity, $M$9 charity	(Akrami et al., 2022)
$N=[n]$0 distinct additive valuations	$N=[n]$1-EFX, no envy of charity, $N=[n]$2 charity	(HV et al., 21 Aug 2025)
Monotone and subadditive valuations, randomized	Ex-post $N=[n]$3 with ex-ante $N=[n]$4 guarantee	(Kavitha et al., 22 Jul 2025)

The foundational exact theorem shows that for any normalized, monotone valuations there exists a partition $N=[n]$5 such that $N=[n]$6 is EFX, $N=[n]$7, and $N=[n]$8 for every agent $N=[n]$9. This result established that “a little charity” always suffices to guarantee almost envy-freeness in the strong EFX sense (Chaudhury et al., 2019).

Subsequent work sharpened the exact bound under additional valuation structure. For nice cancelable valuations, there exists an EFX allocation with at most $v_i:2^M\to\mathbb R_{\ge 0}$0 unallocated items and no envying of the unallocated set, and for four agents the charity can be reduced to at most one item (Berger et al., 2021). On the approximate side, the charity bound was pushed from $v_i:2^M\to\mathbb R_{\ge 0}$1 to $v_i:2^M\to\mathbb R_{\ge 0}$2 through improved combinatorial bounds on rainbow cycles (Akrami et al., 2022). More recently, when there are only $v_i:2^M\to\mathbb R_{\ge 0}$3 distinct additive valuations among $v_i:2^M\to\mathbb R_{\ge 0}$4 agents, the dependence on $v_i:2^M\to\mathbb R_{\ge 0}$5 can be replaced by dependence on $v_i:2^M\to\mathbb R_{\ge 0}$6, yielding $v_i:2^M\to\mathbb R_{\ge 0}$7 charity (HV et al., 21 Aug 2025).

3. Constructive methods for exact bounded charity

The original constructive framework maintains an EFX partial allocation and an acyclic envy graph $v_i:2^M\to\mathbb R_{\ge 0}$8, where $v_i:2^M\to\mathbb R_{\ge 0}$9 iff $(A_1,\dots,A_n,P)$0. The algorithm repeatedly applies one of three update rules. Rule $(A_1,\dots,A_n,P)$1 is a safe-add step: if there exists a source agent $(A_1,\dots,A_n,P)$2 in $(A_1,\dots,A_n,P)$3 and a good $(A_1,\dots,A_n,P)$4 such that adding $(A_1,\dots,A_n,P)$5 to $(A_1,\dots,A_n,P)$6 preserves EFX, then $(A_1,\dots,A_n,P)$7 is allocated to $(A_1,\dots,A_n,P)$8. Rule $(A_1,\dots,A_n,P)$9, labeled “charity-land grab,” applies when some agent $M$0 satisfies $M$1; an inclusion-wise minimal subset $M$2 with $M$3 is moved to $M$4, while $M$5’s old bundle returns to the pool. Rule $M$6, a reallocation-cycle step, uses a cycle of sources and goods from the pool together with “most-envious agents” to generate a welfare-improving reassignment. Every application of $M$7 reduces $M$8, while each application of $M$9 or $A_i$0 raises total welfare by at least the smallest nonzero valuation-difference $A_i$1, so the process terminates in at most

$A_i$2

steps. When no rule applies, one derives $A_i$3 (Chaudhury et al., 2019).

A different exact paradigm is based on champion graphs and Pareto-improvable cycles. For a partial EFX allocation $A_i$4 with unallocated set $A_i$5, one builds a champion graph $A_i$6 containing ordinary envy edges, $A_i$7-champion edges $A_i$8, and generalized edges $A_i$9. A directed cycle of such edges that frees precisely the required goods yields a new EFX allocation that Pareto-dominates the old one. This framework implies that for four agents one can iteratively reduce charity until $i$0, and for general $i$1 until $i$2 (Berger et al., 2021).

These two lines of work differ in their combinatorial scaffolding—envy-graph updates versus champion-graph cycles—but they share a common template: maintain EFX, preserve or improve agents’ valuations, and convert large charity into a contradiction unless a bounded-charity allocation has been reached.

4. Approximate EFX and the rainbow-cycle program

Approximate EFX with sublinear charity was developed through a reduction to a multipartite digraph problem. For additive valuations and $i$3, one maintains a partial $i$4-EFX allocation $i$5 and uses three update types: $i$6 allocates a good from the pool to a source if the addition is not strongly envied, $i$7 handles agents that heavily envy the entire pool, and $i$8 eliminates champion cycles and strictly improves some bundle by a factor $i$9. The combinatorial core is the rainbow-cycle number $|P|\le n-1$00, defined as the largest $|P|\le n-1$01 for which a $|P|\le n-1$02-partite digraph with parts of size at most $|P|\le n-1$03 can satisfy a full-incoming condition while containing no directed cycle that uses at most one vertex from each part. Any upper bound on $|P|\le n-1$04 translates into a charity bound. Using $|P|\le n-1$05, one obtains

$|P|\le n-1$06

and in particular

$|P|\le n-1$07

The same framework can be initialized from a partial EFX allocation with high Nash welfare, yielding final Nash welfare at least $|P|\le n-1$08 of optimum (Chaudhury et al., 2021).

The later improvement replaces the $|P|\le n-1$09 bound on $|P|\le n-1$10 by

$|P|\le n-1$11

via a probabilistic argument together with derandomization by conditional expectations. When this is inserted into the reduction, one obtains charity

$|P|\le n-1$12

while preserving the $|P|\le n-1$13-EFX condition and the requirement that no agent strictly envies the set of unallocated goods (Akrami et al., 2022).

This program made bounded charity depend on an extremal graph parameter rather than solely on allocation dynamics. A plausible implication is that further asymptotic improvements in charity bounds are tightly coupled to new upper bounds on rainbow-cycle numbers.

5. Type-parameterized charity: dependence on $|P|\le n-1$14 instead of $|P|\le n-1$15

When there are $|P|\le n-1$16 agents but only $|P|\le n-1$17 distinct additive valuations, the charity bound can be parameterized by $|P|\le n-1$18 rather than by $|P|\le n-1$19. For every $|P|\le n-1$20, one can efficiently compute a partial allocation $|P|\le n-1$21 that is $|P|\le n-1$22-EFX and satisfies

$|P|\le n-1$23

The logarithmic term is inherited from the best known bound $|P|\le n-1$24 on the rainbow-cycle number. This strictly strengthens the earlier $|P|\le n-1$25 guarantee whenever $|P|\le n-1$26 (HV et al., 21 Aug 2025).

The proof works only on $|P|\le n-1$27 “leading” agents, one per valuation type. It maintains a partial $|P|\le n-1$28-EFX allocation and iteratively applies one of four improvement steps. If some agent heavily envies the charity, her bundle is swapped with a suitable subset of the pool. If there is an unallocated good that nobody values up to factor $|P|\le n-1$29, the good is given to any source in the envy graph. Otherwise the pool is partitioned into high-demand and low-demand goods. High-demand goods are few because each leading agent values at most $|P|\le n-1$30 items, implying at most $|P|\le n-1$31 such goods. On the low-demand side, one builds a $|P|\le n-1$32-partite champion graph with size at most $|P|\le n-1$33 per part. By the definition of $|P|\le n-1$34, either a rainbow cycle exists, producing a Pareto-improving reallocation, or there are at most $|P|\le n-1$35 low-demand goods. Choosing

$|P|\le n-1$36

balances the two terms.

Two lemmas drive the argument. The valuable-goods bound states that if $|P|\le n-1$37 is already $|P|\le n-1$38-EFX and no agent heavily envies the charity, then each agent finds at most $|P|\le n-1$39 goods in the charity valuable in the sense $|P|\le n-1$40. The rainbow-cycle lemma states that the low-demand champion graph either has at most $|P|\le n-1$41 parts or contains a rainbow cycle yielding a $|P|\le n-1$42-EFX Pareto-improvement. The same paper also proves that a $|P|\le n-1$43-EFX allocation exists for any number of agents when there are at most four distinct valuations, which situates the charity theorem within a broader few-types agenda (HV et al., 21 Aug 2025).

6. Welfare, ex-ante guarantees, and randomized support

Bounded charity has also been used to preserve efficiency. For additive valuations, there always exists a subset $|P|\le n-1$44 and an EFX allocation $|P|\le n-1$45 of $|P|\le n-1$46 such that

$|P|\le n-1$47

where $|P|\le n-1$48 and $|P|\le n-1$49 is the maximum Nash welfare over all allocations of $|P|\le n-1$50. The construction starts from a maximum-Nash-welfare allocation, repeatedly removes items from carefully chosen bundles, and then finds a perfect matching in an EFX-feasibility graph. The factor $|P|\le n-1$51 is best possible. Under the $|P|\le n-1$52-large market assumption $|P|\le n-1$53 for all $|P|\le n-1$54, the resulting EFX allocation on a subset satisfies

$|P|\le n-1$55

and a polynomial-time variant based on a $|P|\le n-1$56-approximate Nash-welfare allocation yields

$|P|\le n-1$57

This established bounded charity as a mechanism for combining strong fairness with controlled welfare loss (Caragiannis et al., 2019).

A separate direction studies randomized allocations supported on bounded-charity outcomes. For monotone valuations, ex-post EFX-with-charity can be achieved alongside ex-ante $|P|\le n-1$58-EF, and for monotone subadditive valuations there is a pseudopolynomial-time randomized algorithm returning a distribution over integral allocations $|P|\le n-1$59 such that each realized allocation is $|P|\le n-1$60. In expectation, the algorithm guarantees for every $|P|\le n-1$61,

$|P|\le n-1$62

The algorithm, called Random-Swap-with-Little-Charity, first performs randomized envy-elimination by repeatedly selecting an inclusion-wise minimal envied subset $|P|\le n-1$63, choosing a random envier $|P|\le n-1$64 uniformly from the set of agents that prefer $|P|\le n-1$65 to their current bundles, and swapping $|P|\le n-1$66 with $|P|\le n-1$67. A second deterministic “little-charity” stage then reduces the charity to at most $|P|\le n-1$68 goods without harming the ex-ante guarantee (Kavitha et al., 22 Jul 2025).

These results show that bounded charity is compatible not only with EFX itself, but also with high Nash welfare, large-market near-optimality, and ex-ante fairness guarantees in randomized mechanisms.

7. Limitations, adjacent models, and open directions

Several assumptions recur across the literature. The type-parameterized $|P|\le n-1$69-EFX-with-charity result works under additive valuations and assumes $|P|\le n-1$70, so that the heavy-envy and value-swap steps can be carried out cleanly. It also uses a non-degeneracy assumption to avoid tie-breaking issues, although this can be removed by standard perturbation arguments. The resulting $|P|\le n-1$71 bound is stated to be tight up to logarithmic factors under the current understanding $|P|\le n-1$72; improving the $|P|\le n-1$73 factor would require new combinatorial bounds on rainbow cycles (HV et al., 21 Aug 2025).

The exact bounded-charity program is also explicitly framed as an intermediate route toward full EFX. Reducing the number of unallocated goods for arbitrary numbers of agents is presented as a systematic way to settle the general existence question. Open problems recorded in the exact literature include whether full EFX always exists for additive valuations, whether the last unallocated item can be removed for four agents, whether the general $|P|\le n-1$74 bound can be improved to $|P|\le n-1$75 or to a constant independent of $|P|\le n-1$76, what happens for valuation classes such as submodular or subadditive, and whether strongly polynomial-time algorithms exist for these bounded-charity allocations (Berger et al., 2021).

An adjacent model replaces bundles of goods by graph orientations. In this fair-orientation setting, agents are vertices, goods are edges, and charity means leaving at most $|P|\le n-1$77 edges unoriented. The bounded-charity decision problem asks whether there exists $|P|\le n-1$78 with $|P|\le n-1$79 such that the remaining graph admits an EFX orientation. This variant is fixed-parameter tractable by $|P|\le n-1$80, with running time $|P|\le n-1$81, and for polynomially bounded valuations it is decidable in time $|P|\le n-1$82. At the same time, it is NP-complete on simple symmetric instances with vertex-cover number $|P|\le n-1$83, NP-complete on symmetric multigraphs with only $|P|\le n-1$84 vertices, W[1]-hard parameterized by vertex-cover number, XNLP-hard by pathwidth, and XALP-hard by treewidth, even for polynomial weights. In simple graphs there is also a trivial upper bound: removing at most $|P|\le n-1$85 edges makes the graph bipartite, and every bipartite graph admits an EFX orientation (Kanellopoulos et al., 31 Dec 2025).

Taken together, these results position EFX-with-bounded-charity as both a relaxation and a methodological framework. It supports exact existence theorems under broad monotonicity assumptions, sharper quantitative bounds under structural restrictions such as few valuation types, combinatorial reductions via rainbow cycles, welfare-preserving constructions, and randomized mechanisms with ex-ante guarantees, while also exposing a clear frontier of unresolved questions around full EFX and the ultimate necessity of charity.