resource allocation teams

Abstract: For a fixed $ε\ge 0$, let $ρ_{ε, n}$ be the maximum value of $ρ_n$ such that, in every allocation instance with $m$ indivisible goods and $(1 + ε)n$ agents, there is a subset of $n$ agents with the following property. The allocation instance restricted to only this subset of $n$ agents has an allocation that gives each agent at least $ρ_n$-fraction of her $n$-maximin share (MMS). When $ε> 0$ we refer to $εn$ as the {\em agent redundancy}. The case $ε= 0$ is the standard setting without agent redundancy, and we use $ρ_{0,n}$ and $ρ_n$ interchangeably. For allocation instances with additive valuations, % agents, it is known that $\frac{7}{9} \le ρ_n < 1-Ω(\frac{1}{\log^2 n})$ for every $n \ge 2$ (with stronger lower bounds for small $n$). It is an open question whether $ρ_n$ tends to~1 as $n$ grows. We show that with agent redundancy, the answer to this question is positive. That is, for every $ε> 0$, $ρ_{ε, n}$ tends to~1 as $n$ grows. A straightforward corollary of this result is that for every $c > 0$, if the number of items satisfies $m \le (2 - c)n$, then the answer is positive also for $ρ_n$ (with no agent redundancy). Adapting our techniques to a related setting, we also improve a recent result of Ezra and Garbuz, and show that without agent redundancy, there always exists an allocation that gives $n - O(\log^2 n)$ agents their full MMS.

Abstract: We study indivisible goods allocated to two prescribed nonempty groups whose members have arbitrary monotone valuations. We guarantee balanced envy-freeness up to one good (EF1) for at most five agents and up to two goods (EF2) for at most nine, for any number of goods. Bundle sizes differ by at most one. The EF1 threshold is sharp: every split of six agents can fail. EF2 improves the previous five-agent guarantee. Via the connection of Kyropoulou, Suksompong, and Voudouris, these guarantees follow from chromatic bounds for $\mathrm{KG}(2n,n,r)$, whose vertices are $n$-subsets of a $2n$-set, adjacent when their intersection has size less than $r$. We prove its chromatic number is six for $r=2$, $n\ge2$, resolving Jafari and Alipour's conjecture, and at least ten for $r=3$, $n\ge4$, improving the previous lower bound of six; the known upper bound is fourteen. We show that the standard box-complex $\mathbb{Z}_2$-index bound yields exactly five and nine for $n\ge3$ and $n\ge5$, respectively. To exceed these bounds, we exploit restrictions on maps induced by colorings: a proper five- or nine-coloring must induce a nonzero homology map, yet local constraints force all generators, supported on pentagon or rook configurations, to map to zero. The ten-color proof compares integral and fractional common centers using a sharp rounding theorem for five subset constraints. A linear-time retraction from the full allocation graph onto balanced layers adds cardinality balance to known nonlinear discrepancy guarantees. For each fixed removal allowance in the guaranteed-existence range, finding a balanced fair allocation or an explicit monotonicity violation from valuation circuits is polynomial-time equivalent to finding a monochromatic edge in a circuit-specified coloring of the corresponding graph. The reverse reduction produces normalized monotone binary valuations using ordinary circuits.

Abstract: We study the online fair division of indivisible chores among $n$ agents under upgrade recourse, where earlier assignments may be revised but each chore previously assigned to an agent must be replaced by a distinct chore that she finds no more costly. We show that a simple adaptation of round robin maintains envy-freeness up to one chore (EF1) together with a $(2-1/n)$-approximation of the maximin share (MMS), and that no deterministic algorithm without predictions can do better, even when all agents rank the chores in the same order. This stands in contrast to the allocation of goods, where the optimal MMS factor vanishes as $n$ grows and cannot be achieved together with EF1. Moreover, knowing only the number of chores modulo a period of length $O(n)$ (and nothing about the costs) suffices to improve upon this factor at the final configuration, while the same algorithm retains a factor below $3$ even if this information is wrong. On the other hand, envy-freeness up to any chore (EFX) essentially requires the exact number of chores: uncertainty between two consecutive values already rules out every finite approximation, and even exact knowledge does not allow EFX to be combined with an MMS guarantee better than the trivial factor of $n$.

Abstract: We give a deterministic quasi-polynomial-time algorithm for the maximum independent set problem. If an $N$-vertex graph contains an independent set of size $2^{-t}N$, where $t$ is an integer with $1\le t\le\sqrt{\log N}/12$, then our algorithm finds an independent set of size $2^t$ in time $N^{O(t)}$. This gives an approximation ratio of $N/2^{Ω(\sqrt{\log N})}$ in time $N^{O(\sqrt{\log N})}$, improving on the approximation ratio $N/2^{Ω((\log N)^{1/3})}$ obtained by Lovett [Lov26]. A limitation of this form of analysis follows from the integrality gaps of Tulsiani [Tul09]. These gaps imply that a universal guarantee of the form $\operatorname{Las}_{R(t)}(G)\ge2^{-t}N\Longrightarrowα(G)\ge2^t$, at $2^{o(\sqrt{\log N\log\log N})}$ levels, cannot hold over a range extending beyond $t=O(\sqrt{\log N\log\log N})$. Thus, the exponent $1/2$ is the limit of this form of analysis, up to a factor of $\sqrt{\log\log N}$.

Abstract: We study fair division of uncertain items, where the utility of each item is a random variable whose realization is observed only after the allocation is determined. This uncertainty creates a distinction between ex-ante fairness, evaluated before utilities are realized, and ex-post fairness, evaluated after realization. We examine to what extent these two notions are compatible. We first identify conditions under which ex-ante envy-free allocations maximize ex-post envy-freeness probability. These include two-agent instances where utilities toward the same item follow the same distribution from a convolution-power family. For an arbitrary number of agents, we establish analogous optimality results when the common generator is non-atomic or log-concave. Beyond convolution-power families, we derive approximation guarantees for ex-ante envy-free allocations relative to the optimal ex-post envy-freeness probability under log-concave or symmetric utilities. We also study the computational complexity of evaluating ex-post envy-freeness probabilities, establishing hardness results as well as pseudo-polynomial-time algorithms.

Abstract: We study online fair division of indivisible goods among $n$ agents with frequency predictions: before any good arrives, the algorithm learns the multiset of each agent's values for the goods, but neither the arrival order nor which values of different agents belong to the same good. Neoh et al. (ICML 2026) showed that such predictions allow every share-based guarantee that is achievable offline, and left open whether a positive approximation of envy-freeness up to one good (EF1) is possible. We show that deterministic online algorithms can guarantee $1/2$-EF1 for three agents and $(3-\sqrt5)/2$-EF1 for four agents, whereas for five or more agents, no online algorithm can guarantee any positive approximation of EF1, even with $n+1$ goods and identical predictions. Together with the known result for two agents, a positive approximation of EF1 is therefore possible if and only if $n\leq4$. We also rule out any positive approximation of EFX for three or more agents, even when the valuations take only two distinct positive values. Next, we use the meta-algorithm of Neoh et al. (ICML 2026) to compute an epistemic EFX (EEFX) allocation in polynomial time for any $n$, and hence a $1/(n-1)$-EF$(n-2)$ allocation for $n\geq3$, whereas algorithms that fix a picking sequence in advance can guarantee a positive approximation of EF$k$ if and only if $n\leq k+2$. Finally, for noisy predictions, we give algorithms that return EEFX allocations when the predictions are exact, and guarantee $(1-2\varepsilon+O(\varepsilon^2))$-EEFX under underestimates and, for $n\geq3$, $(1-2(n-1)\varepsilon+O(\varepsilon^2))$-EEFX under overestimates; the linear terms are the best possible, and larger overestimates rule out any positive approximation for such algorithms.

Abstract: Reinforcement learning has emerged as an effective approach for combinatorial optimization, but most existing methods focus on problems such as the Travelling Salesperson Problem (TSP) in which all elements must be served. Selective combinatorial optimization problems (SCOPs), including the Orienteering Problem (OP), Prize-Collecting TSP (PCTSP), Stochastic PCTSP and Knapsack Problem (KP), additionally require selecting which elements to include. This selection creates sampled solutions with substantially different objective values, yet standard preference optimization reduces such comparisons to binary rankings and discards the magnitude of these differences. We introduce Reward-Weighted Preference Optimization (RWPO), a pairwise training objective that incorporates reward-gap information into preference learning. For candidate trajectories sampled from the same instance, RWPO weights each preference loss according to the normalized difference in their rewards, assigning greater training emphasis to comparisons that represent larger improvements in solution quality. We further combine RWPO with scale-specific fine-tuning and reward- and likelihood-weighted tilted sampling to adapt trained policies to larger and shifted instance distributions without relying on local-search post-processing. Experiments on the OP, PCTSP, Stochastic PCTSP, and KP show that RWPO consistently improves over unweighted preference optimization and policy-gradient baselines across a wide range of instance sizes. On the OP, the resulting solver exceeds the mean reward of the Compass metaheuristic at scales from 100 to 2,000 nodes. The results demonstrate that preserving reward-gap information can substantially improve preference-based learning for selective combinatorial optimization and support generalization to large-scale instances.

Abstract: We give a bounded envy-free cake-cutting protocol for any number $n$ of agents in the Robertson--Webb model, with a short, self-contained, and elementary correctness proof. The protocol uses $2^{O(n \log n)}$ queries. Most ingredients are already known and well understood in the fair division literature; our main contribution is a simpler organization of these ideas into a protocol whose complete description and proof fit in a few pages.

Abstract: We give a randomized combinatorial \((1/2 - \varepsilon)\)-approximation for nonnegative non-monotone \(k\)-submodular maximization under a fixed number \(d\) of knapsack constraints, for \(k \ge 2 \). The algorithm selects whole elements and follows the proportional top-2 rule of Chen et al. \cite{chen2026}: it samples from the two labels with the largest marginal gains, with probabilities proportional to the positive parts of these gains. Extending their single-knapsack Bellman analysis, we construct a common resource potential and associated resource weights, with potential losses charged to greedy gains. Since these weights depend on the unknown optimum, we discretize their defining coefficients and enumerate monotone guesses over phases determined by resource consumption. This yields an algorithm using at most $\mathcal{A}(d,\varepsilon)(nk+1)^{H+2}$ value queries, where $n$ is the number of elements, $\mathcal{A}(d,\varepsilon)$ depends only on $d$ and $\varepsilon$, and $H=O(d \log(2d) / \varepsilon^2)$. The approximation ratio can be made arbitrarily close to the optimal oracle threshold for \( k=2 \) and is asymptotically tight as \(k \to \infty\).

Abstract: We prove that for additive valuations, not every pure Nash equilibrium of the Round-Robin mechanism induces an allocation that is envy-free up to one good (EF1) with respect to the agents' true valuations. This corrects an earlier claim of Amanatidis et al. (WINE 2021; Math. Oper. Res. 2024). On the positive side, every pure Nash equilibrium of every fixed recursively balanced picking sequence mechanism induces an allocation that is $1/2$-EF1 and proportional up to one good (PROP1). We prove the approximate EF1 guarantee through a potential function comparing two executions, and the proportionality guarantee via a deviation that preserves the agent's most valuable goods. We also give a family of Round Robin equilibria showing that it is not possible to have a strictly better guarantee than $2/3$-EF1 for all equilibria.

Abstract: We investigate the problem of allocating indivisible chores among agents with arbitrary positive weights and additive nonnegative cost functions. An allocation is said to be $α$-weighted-maximin-share (WMMS) fair if the cost incurred by each agent is at most $α$ times her weighted maximin share. We establish that every instance admits a $4.124234528$-WMMS allocation. Moreover, for instances in which all weights and costs are rational numbers represented in binary, we give an algorithm that computes a $4.12424$-WMMS allocation in time polynomial in the number of agents, the number of chores, and the total binary encoding length of the input. Complementing these upper bounds, we construct a family of instances with strictly positive, identically ordered costs for which the optimal WMMS approximation factors converge to $e$. Together, our results substantially narrow the previously best-known gap of $[2,12]$ established by Li et al.~\cite{LWX26} to $[e,4.12424]$.

Abstract: We study the relationship between fair division with $\textit{subsidies}$ and $\textit{best-of-both-worlds}$ (BoBW) fairness for indivisible items. We prove a general reduction of BoBW existence to envy-freeability. Specifically, for any scale-free ex-post property $P$ (e.g., EF1, EFX, PO) and any scale-free valuation class $\mathcal{V}$ (e.g, additive, submodular, subadditive, monotone), if every instance in $\mathcal{V}$ admits a $P$-allocation that can be made envy-free using subsidies, then every such instance admits an ex-ante envy-free lottery supported on $P$-allocations. The proof uses linear programming duality. As applications, the reduction gives alternative proofs of BoBW existence results for additive goods, chores, and mixtures of goods and chores. The converse of our general reduction holds for two agents but fails for three: We provide a four-good counterexample with monotone submodular valuations where an ex-ante envy-free and ex-post EF1 lottery exists, but no allocation is simultaneously EF1 and envy-freeable. For lexicographic valuations, we prove simultaneous EFX, Pareto optimality, and envy-freeability (and hence BoBW guarantees for EFX+PO) for three agents, and show that even EFX and envy-freeability can be incompatible for four agents.

Abstract: Motivated by social services where consumers pay with non-transferable ordeals, we study mechanisms that maximize consumer utility for multiple unit-demand buyers and heterogeneous items whose values are drawn independently from known prior distributions. Prior work in utility maximization approximates social welfare and shows that the gap between optimal utility and social welfare is logarithmic. We resolve the question of whether simple mechanisms can guarantee a constant-factor approximation to optimal utility itself. Our mechanisms achieve a $(5.67+\varepsilon)$-approximation for general independent values, improving to $2e/(e-1)<3.164$ when values are i.i.d. Each buyer chooses their favorite option from posted item prices or free item lotteries, and contention resolution determines which buyers' requests are served; this is BIC, ex-post individually rational, and computable in polynomial time. Our main technical contribution is a general upper bound on the optimal ex-ante-constrained utility that separates the contributions captured by posted prices and free lotteries. These results establish a utility counterpart to the theory of simple, approximately revenue-optimal mechanisms.

Abstract: A random assignment of n indivisible objects to n agents is specified by its assignment matrix and implemented by drawing a deterministic assignment from a Birkhoff-von Neumann decomposition. Kawase et al. observed that the choice of decomposition matters for fairness: a matrix that is envy-free in the sense of stochastic dominance (SD-EF) can be decomposed so that some agent envies another with probability close to 1. They call a decomposition Dec-EF if every agent envies every other agent with probability at most 1/2, proved that every SD-EF matrix admits a Dec-EF decomposition when n <= 3 or when there are at most two distinct preferences, and left the general case open. We settle the first open case: every SD-EF matrix with four agents admits a Dec-EF decomposition. The worst case over the SD-EF polytope of a profile is attained at a vertex, and our computer-aided proof enumerates all 26,927 vertices for the 762 profiles up to symmetry in exact arithmetic and certifies each by a rational decomposition. The same method settles five agents with at most four distinct preferences and the probabilistic serial rule for all five-agent profiles, and adversarial search up to seven agents finds no counterexample. For general n, an envy-budget identity shows that 1/2 is the best possible threshold. We prove that every SD-EF matrix with at most two distinct rows admits a Dec-EF decomposition, and that the maximum-entropy decomposition is Dec-EF whenever all agents but two share a preference; the latter proof rests on a new monotonicity lemma for weighted least-squares rankings. In general, natural decompositions fail: greedy Birkhoff-von Neumann can come arbitrarily close to envy probability (n-1)/n, and maximum entropy fails at n = 4 when all preferences differ. Deciding whether an arbitrary random assignment, not necessarily SD-EF, admits a Dec-EF decomposition is strongly NP-complete.

Abstract: We initiate the study of $α$-fair prophet inequalities. This interpolates between utilitarian welfare $(α=0)$, Nash welfare $(α=1)$, and Rawlsian max-min fairness $(α\to\infty)$. Given the non-linearity of the objective, it matters when the expectation is applied. For instance, for the Rawlsian objective, it matters whether we aim to maximize $\min \mathbb{E}[u_i]$ or $\mathbb{E}[\min u_i]$. We refer to the former as the ex-ante model, and the latter as the ex-post model. For ex-ante fairness, full distributional knowledge yields a tight competitive ratio of exactly $1/2$ for every $α\ge 0$. Under sample access, $O(n\log n)$ samples per distribution suffice for a constant competitive ratio when $α\in(0,1]$. In contrast, for every $α>1$, no finite number of samples improves upon the trivial $1/n$ guarantee. Thus, unlike in the utilitarian setting, full-information and sample-access prophet inequalities become fundamentally separated. For ex-post fairness, under full information, we obtain a uniform constant ratio for all $α\in(0,1)$, while for every $α>1$ the competitive ratio collapses to $1/n$. In the sample-access model, one sample per distribution suffices for each fixed $α<1$, but no sample budget depending only on $n$ yields a uniform constant guarantee as $α\to 1$. Beyond these phase transitions for $α$-fairness, our results open the door to a broader theory of prophet inequalities for non-linear welfare objectives.

Abstract: In the classical prophet inequality, an algorithm observes a sequence of random variables with known distributions in an online fashion, and it must select one of the random variables with the goal of maximizing the expected value of its selection. The performance of the algorithm is compared to an \textit{omniscient prophet} who observes all of the random variables before having to make its selection. Combinatorial extensions of the classical prophet inequality in which the algorithm gets to pick a subset of the random variables (constrained to belong to some family of feasible sets) have been studied extensively. We study prophet inequalities with a $k$-matchoid constraint (a common generalization of a $k$-matroid intersection constraint and a $k$-bounded hypergraph matching constraint) in two common online arrival models. We give guarantees with respect to the \textit{ex-ante} fractional relaxation of the omniscient prophet, obtaining an ex-ante competitive ratio of $\frac{1}{k+1}$ in the adversarial order case, and $\frac{1-e^{-k}}{k}$ in the random order case. Using the duality framework of Lee and Singla \cite{Lee2018}, this also yields online contention resolution schemes in these settings. Our adversarial-order prophet inequality can be viewed as a generalization of a recent $\frac12$-competitive matroid prophet inequality by Kalantarzadeh and Pashkovich, 2026. This generalization introduces a new framework: coordinated weighted principal partitions across multiple matroids. Our random-order prophet inequality is a generalization of the $k=1$ matroid case of Lee and Singla, 2018. The two results improve previously known competitive ratios for $k$-matroid intersection, which were $\frac{1}{(e+o(1))k}$ and $\frac{1}{k+1}$, respectively.

Abstract: We study the problem of fairly allocating m indivisible items among n agents with possibly unequal entitlements in the mixed manna setting, where each item may be perceived as a good or a chore by different agents. We focus on the fundamental fairness notion of proportionality. Since proportional allocations need not exist in this setting, we allow monetary subsidies to restore proportionality while minimizing the total subsidy. When each item's (dis)utility is bounded by 1, a total subsidy of at least τ(n) \approx n/4 may be necessary. For goods-only or chores-only instances, the best previously known upper bound was n/3-1/6 due to Wu and Zhou~(2024). We close this gap by proving that a total subsidy of at most τ(n) always suffices, thereby establishing the tight subsidy bound. Our results hold even in the more general setting of weighted mixed manna, resolving an open question posed by~Wu et al. (2023) and Garg et al. (2026). The allocation also satisfies weighted proportionality up to one item (WPROP1). Our proof develops a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) fixed-point theorem, extending the KKM framework to share-based fairness notions. Finally, we design a polynomial-time algorithm to compute such allocations for any fixed number of agents.

Abstract: This work considers multi-agent coordination with arbitrary information networks among the agents using a game-theoretic approach. A system designer aims to assign local utility functions to the agents to guide their actions toward a desired system objective. The performance of the assigned local utilities is measured by the well known pure price of anarchy (pPoA) metric that equals the ratio of the system objective at the worst pure Nash equilibrium of the corresponding game to the optimal system objective. Our aim is to derive the utility functions which optimize the pPoA-based performance guarantees for any given information network and system objective. We develop a linear program that derives the optimal pPoA for any arbitrary information network and arbitrary system objective. Our work is the first to solve optimal utility design for arbitrary networks; our techniques generalize previous approaches which considered only the full-information setting. For supermodular objective functions, we prove that counterintuitively, a fully communication-denied utility design is optimal irrespective of the original information network. For submodular system objectives, an exhaustive numerical analysis suggests that the optimal utility design is robust to communication failures even for this case. When the system objective is weighted maximum coverage, the marginal contribution utility design provably optimizes the pPoA for a wide variety of information networks of interest.