resource allocation engineers

Abstract: When allocating indivisible goods, the maximum Nash welfare rule is known to be the unique welfarist rule satisfying envy-freeness up to one good. We strengthen this characterization by showing that envy-freeness up to one good can be replaced by substantially weaker fairness requirements: envy-freeness up to $k$ goods or proportionality up to $k$ goods, for any fixed $k$. Our characterization holds even for profiles with identical goods.

Abstract: We study the order-selection prophet inequality. A gambler knows the distributions of independent nonnegative random values, chooses the order in which to inspect them, and may irrevocably accept at most one value, at the time it is revealed. Since the order is irrelevant when the values are i.i.d., no algorithm can guarantee more than the optimal i.i.d.\ ratio $Γ_\star\approx0.74544$ of the expected maximum. We give an algorithm with competitive ratio $0.74500001$, less than $4.5\times10^{-4}$ below this bound. The algorithm samples independent arrival times and uses item-dependent thresholds. It is designed so that, conditional on its value, every item is selected with probability equal to a fixed fraction of its probability of being the maximum. We show that the constructions of this form that apply one local rule to every item are exactly those given by an allocation kernel, that is, by a coupling of two uniform random variables. Requiring every rarely winning item to follow the optimal i.i.d.\ threshold path singles out a baseline kernel, for which we prove analytically a ratio $Γ_0\approx0.7389$, defined through a one-dimensional control problem. Our final kernel perturbs this baseline, and its guarantee combines analytic estimates with a rigorous computer-assisted certificate.

Abstract: As constellations of satellite grow in size, and become more costly, they are increasingly funded by multiple stakeholders who expect access to the constellation's resources that is commensurate with their investment. Such cooperation raises significant challenges for constellation operation, in particular regarding the fairness of the computed schedules among stakeholders. While several existing works deal with fairness in the Earth Observation Satellite Scheduling Problem (EOSSP), most of them assume that users (e.g. space agencies, research labs, countries) have the same weights, i.e. contributed equally to the project and therefore have equal access to the constellation. However, in practice, users' contributions often differ significantly. Hence, in this paper, we study a weighted variant of the EOSSP where users are assigned weights reflecting their respective contributions to the constellation. We propose several solution concepts with varying focus on efficiency and fairness, namely weighted utilitarianism, weighted egalitarianism, weighted leximin, and weighted ordered weighed averaging (WOWA). We formulate the related optimization problems as Mixed-Integer Linear Programs (MILPs), and compare them against two natural heuristics: weighted greedy and weighted round robin. All methods are evaluated on scenarios drawn from the literature, under strict runtime constraints, in order to assess the performance of each approach with respect to both efficiency and fairness metrics.

Abstract: We consider the problem of allocating indivisible goods to agents with additive valuations, under two classic notions of fairness and efficiency: envy-freeness up to any good (EFX) and Pareto optimality. While some of the earliest work on EFX showed that Pareto optimal EFX allocations may not exist if agents can have zero value for some goods, it has remained open whether the same is true when all values are positive. We present a simple instance in which every good has positive value to every agent, but no Pareto optimal EFX allocation exists. We then generalize to a family of instances showing that Pareto optimal $α$-EFX allocations may not exist for any $α> 2/3$.

Abstract: We revisit the online submodular welfare maximization problem (OSW), a vast generalization of numerous well-studied online allocation problems, including online bipartite matching, display ads and AdWords. A natural greedy OSW algorithm achieves a competitive ratio of $1/2$ [LLN06], and $1-1/e$ under i.i.d. arrivals [KPV13]. Both bounds are tight for polynomial-time algorithms unless NP=RP [KPV13]. The information-theoretic limits for OSW, in contrast, remained open. This prompted researchers to ask for tighter upper bounds [Mehta13] or conjecture the existence of better-competitive (slow) algorithms [Huang23, LTX+23, CHS24, Huang2024, HTW24}. We resolve Mehta's question and refute the subsequent conjectures, proving that the greedy algorithm is in fact tight for both adversarial and i.i.d. arrivals, regardless of computation time. Seemingly contradicting recent $(1-1/e)$-competitive fractional algorithms for OSW [HJP+24,Patton26], we show that this impossibility even holds for fractional algorithms, provided their outputs correspond to marginals that are roundable in hindsight. Conceptually, our proofs can be seen as identifying that a bottleneck for OSW, even under (deterministic) adversarial arrivals, is the inherently random notion of correlation gap [ADSY12].

Abstract: We consider the problem of allocating $m$ indivisible items to $n$ agents. For $n \ge 2$, we define $μ(n)$ as the largest $m$ for which every allocation instance with $n$ agents and $m$ items has an allocation that gives each agent at least her maximin share (MMS). We prove that for general monotone valuations, $μ(n) = n + Θ(\log n)$. This holds both for goods and for chores. For arbitrary valuations (not monotone), we prove that $μ(n) = \log n + O(1)$.

Abstract: We consider allocation of indivisible goods to agents with equal entitlements and subadditive valuations. As an ex-post fairness notion we consider the maximin share (MMS), and as an ex-ante fairness notion we consider the maximum expectation share (MES), which is always at least as large as the MMS, and sometimes much larger. We present a simple transformation that for every $0 < ρ\le 1$, given any algorithm that produces $ρ$-MMS allocations, transforms it into a randomized allocation algorithm that offers $ρ$-MMS ex-post simultaneously with $η$-MES ex-ante. We prove several new properties of MES, and use them to show that $η\ge \min[\fracρ{2 + ρ}, \frac{1}{4}]$. We also present cases in which the transformation results in a higher value of $η$. Applying our transformation to currently known allocation algorithms shows for subadditive valuations the existence of randomized allocations that are simultaneously $Ω(\frac{1}{\log\log n})$-MES ex-ante and $Ω(\frac{1}{\log\log n})$-MMS ex-post, and for XOS valuations the existence of randomized allocations that are simultaneously $\frac{4}{27}$-MES ex-ante and $\frac{4}{17}$-MMS ex-post.

Abstract: We study fair allocation of indivisible goods among agents with additive valuations. Our main result guarantees every agent her exact AnyPrice Share (APS) whenever the total entitlement is at most $1-\varepsilon$ and each individual entitlement is sufficiently small. For equal entitlements, this yields a $1$-out-of-$(n+O(\sqrt{n\log n}))$ maximin share (MMS) allocation. We also establish two best-of-both-worlds guarantees that preserve proportionality in expectation: each agent receives her full MMS with probability at least $1-O((\log n/n)^{1/3})$, or every agent receives her $1$-out-of-$(n+O(n^{2/3}(\log n)^{1/3}))$ MMS in every outcome. Both guarantees extend to sufficiently small unequal entitlements, with APS replacing MMS. Quantitatively, our deterministic result allows an entitlement cap of order $\varepsilon^2/\log(1/\varepsilon)$, while an impossibility construction shows that any universally sufficient cap must be $O(\varepsilon)$.

Abstract: We study the online allocation of indivisible goods among $n$ agents, where each good must be allocated immediately and irrevocably upon arrival. Against an adaptive adversary, Neoh and Teh [2026] proved that no algorithm can guarantee a positive approximation to proportionality up to one good (PROP1) that is independent of the number of goods, and the same holds for proportionality up to $k$ goods (PROP$k$) for any fixed $k$. We instead consider an oblivious adversary, which fixes the input in advance. Choo et al. [2026] showed that the uniformly random allocation returns a $Θ(1/\log(n/δ))$-PROP1 allocation with probability at least $1-δ$. We improve this to $Ω(1/\log\log(n/δ))$; our algorithm does not take $δ$ as input, so the same algorithm achieves this bound for every $δ\in(0,1)$. Moreover, with the same probability, a variant of our algorithm gives every agent almost her bundle, and even without adding any good when no single good is too valuable relative to this share. In contrast, for envy-freeness up to one good (EF1), we show that, for every $α\in(0,1]$, every randomized algorithm has an input on which its probability of returning an $α$-EF1 allocation is at most $e^{-Ω(n)}$. For envy-freeness up to any good (EFX), this probability is at most $1/n!$ with only $n+1$ goods, a bound that is optimal within a factor of $(n+1)/2$. For the maximin share (MMS), this probability is at most $5/6$, however small $α$ is. Allowing more removals gives a positive envy-freeness guarantee: allocating each good to a uniformly random agent among those with positive values achieves, with high probability, an approximation factor arbitrarily close to one for envy-freeness up to logarithmically many goods, and logarithmically many goods are necessary for this rule.

Abstract: We investigate the asymptotic behavior of max-min fair allocations for indivisible goods under i.i.d. random utilities. For $N$ agents and $K$ goods with utilities ${\mU_{i,j}}$ drawn independently from a common distribution $F$, we derive asymptotic characterizations of the max-min value in the balanced case $K=N$ (and in $K=LN$ extensions) via distributional quantiles. We then study the efficiency impact of max-min fairness by comparing the resulting total welfare with the optimal sum welfare. For distributions with sufficiently light tails, we prove that the relative efficiency loss converges to zero as the market grows, implying that max-min fairness incurs negligible welfare loss in large random instances for a broad class of distributions.

Abstract: We study a fractional assignment setting where $n$ objects are to be assigned to $n$ agents with unit capacity, and each agent specifies an ideal distribution over the objects. Unlike in classic random assignment, these ideal distributions are not necessarily degenerate, as agents may prefer a mixture of objects rather than any single object. We assume that agents seek to minimize the $\ell_1$ distance between their ideal distribution and the distribution they receive, which is equivalent to maximizing the overlap between the two distributions. We propose two mechanisms, one based on water filling (WF) and the other on quadratic programming (QP), and show that both mechanisms are utilitarian-optimal (and hence Pareto efficient), envy-free, strategyproof, and satisfy equal treatment of equals. Moreover, we highlight a distinct advantage of each mechanism: while the WF mechanism satisfies the stronger property of group-strategyproofness, the QP mechanism is more robust in terms of egalitarian overlap welfare.

Abstract: The maximin share (MMS) is a central fairness benchmark for allocating indivisible items, but it need not be simultaneously attainable even under additive preferences. While extensive work has developed approximation guarantees, quantitative impossibility bounds have received comparatively little attention. We establish improved asymptotic and constant impossibility bounds for both goods and chores. For every sufficiently large number $n$ of agents, we construct additive goods instances in which every allocation gives some agent at most a $1-Ω((\log n)^{-2})$ fraction of her MMS. This strengthens the $1/n^4$ shortfall of Feige, Sapir, and Tauber (2021) to an inverse-polylogarithmic shortfall, an exponential improvement on the logarithmic scale of $n$. For chores, we construct instances in which every allocation gives some agent cost at least a $1+Ω((\log n)^{-2})$ factor of her MMS. Consequently, for every fixed $\varepsilon>0$, guarantees of $1-O(n^{-\varepsilon})$ for goods and $1+O(n^{-\varepsilon})$ for chores are impossible. We also give four-agent, eleven-item instances that improve the universal impossibility bounds from $39/40$ to $20/21$ for goods and from $44/43$ to $31/30$ for chores.

Abstract: We study pairwise maximin share (PMMS) fairness for indivisible items with additive preferences. We give a polynomial-time reduction from chores to goods that preserves the existence of a PMMS allocation. Together with known nonexistence results for chores, this yields nonexistence for additive goods. In addition, we show that deciding if a given instance admits a PMMS allocation is NP-hard. We also give explicit instances whose PMMS factors are $226/227$ for goods and $1.102065$ for chores, certified by exact enumeration. Complementing these impossibility results, we prove that every additive-chore instance admits a $4/3$-PMMS allocation.