arXiv subjects

Uriel Feige

Publications and source records attributed to Uriel Feige.

At least 19 recordsLinked to original sources

arXiv2026

A lone divider allocation algorithm with subjective divisibility

In the subjective divisibility allocation model, all goods are divisible, every agent $i$ has a non-negative additive valuation function $v_i$, where for some goods $v_i$ is additive over fractions of the good, and for some goods, any fraction smaller than~1 has value~0. Previous work established the existence of $\frac{5}{9}$-MMS allocations in this setting. We show that $\frac{3}{5}$-MMS allocations exist. Our proof is based on an adaptation of the lone divider framework.

cs.GT↗

arXiv2026

Fair Allocation with Optional Selling

We consider fair allocation of indivisible goods in a setting in which agents have subjective valuation functions over the set of goods, and in addition, goods may be sold at given market prices. In this setting, a fair allocation involves {deciding which goods to sell, how to allocate the unsold goods, and how to divide the money received from the sold goods.} We adapt to this setting the definitions of share-based fairness notions, such as the maximin share (MMS) and the truncated proportional share (TPS), and comparison-based fairness notions such as EF1 and EFX (which we adapt to SEF1 and SEFX). We show the following results when the utility of each agent is additive both over goods and over money. With two agents, there are allocations that are simultaneously MMS and SEFX. With three agents, there are instances in which no allocation gives every agent more than $\frac{11}{12}$-MMS. With any number of agents, there are $\frac{2}{3}$-MMS allocations. There also are allocations that are simultaneously SEFX and $\frac{n}{2n-1}$-TPS. This latter ratio is best possible, even without the SEFX requirement.

cs.GT↗

arXiv2026

Simultaneous EF1 and approximate MMS allocations for submodular valuations

There are two common classes of fairness notions that are considered when allocating $m$ indivisible items to $n$ agents of equal entitlements. One is that of share-based fairness notions, with the maximin share (MMS) and its relaxations to $\rho$-MMS being prominent representatives of this class. The other is that of comparison-based fairness notions, with envy-freeness (EF) and its relaxations such as EF1 being prominent representatives of this class. In general, no class offers good guarantees for the other class. In this work, we design allocations that simultaneously satisfy notions from both classes, and specifically, are $\rho$-MMS for constant $\rho$ and EF1 (in fact, also EFL). Such results were previously known when agents have additive valuations, and we prove such results for the more general class of submodular valuations.

cs.GT↗

arXiv2026

On MMS, APS and XOS

We consider allocations of a set of $m$ indivisible goods to $n$ agents of equal entitlements that have valuations from the class XOS. A previous sequence of works showed allocations that obtain an $\alpha$-approximation for the maximin share (MMS), for values of $\alpha$ that gradually approach $\frac{1}{4}$ from below (the currently known ratio is $\frac{4}{17}$). In this work we attempt to obtain ratios better than $\frac{1}{4}$, and manage to do so for sufficiently large $n$. Our methodology is to first investigate the gap between the anyprice share (APS) and the MMS when all agents have the same XOS valuations, for which we design an allocation algorithm and prove that each agent receives at least $\alpha > \frac{11}{40}$ times the APS. Then, we derive inspiration from this algorithm, and modify it so that it applies also when agents have different XOS valuations. Using this modified version, we show that for some sufficiently large $n_0$, there is an $\alpha$-MMS allocation (in fact, an $\alpha$-APS allocation) for every $n \geq n_0$.

cs.GT↗

arXiv2026

Truthful-in-Expectation Mechanisms for MMS Approximation

We study fair allocation of indivisible goods among strategic agents with additive valuations. Motivated by impossibility results for deterministic truthful mechanisms, we focus on randomized mechanisms that are \emph{Truthful-in-Expectation (TIE)}. From a fairness perspective, we seek to guarantee every agent a large fraction of their \emph{Maximin Share (MMS)} ex-post. Among other results, Bu~and~Tao~[FOCS 2025] presented a TIE mechanism that guarantees $\frac{1}{n}$-MMS ex-post. First, we present an ordinal TIE mechanism that guarantees $\frac{1}{H_{n-1} + 2}$-MMS ex-post, where $H_k$ is the $k$-th harmonic number ($H_k \simeq \ln k$). This is nearly best possible for ordinal mechanisms, as even non-truthful ordinal allocation algorithms cannot obtain an approximation better than $\frac{1}{H_n}$. We then show that with just a small amount of additional cardinal information, the ex-post guarantee can be improved to $\Omega(\frac{1}{\log\log n})$-MMS, at the cost of relaxing the incentive requirement to $(1-\varepsilon(n))$-TIE for negligible $\varepsilon(n)$. Finally, for two agents, we present a TIE mechanism that is $\frac{2}{3}$-MMS ex-post. All our mechanisms are ex-ante proportional (thus also providing ``Best-of-Both-Worlds'' results) and run in polynomial time. Moreover, all our results extend to the truncated proportional share (TPS), which is at least as large as the MMS. Our two-agent $\frac{2}{3}$-TPS result is best possible for the TPS.

cs.GT↗

arXiv2026

The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees

We study the problem of fairly allocating indivisible goods among agents with monotone valuations. We introduce a new share-based fairness notion, the residual maximin share (RMMS), and show that it provides a unified framework for several existing lone-divider style techniques in fair division. RMMS satisfies two key properties: feasibility and self-maximization. Using RMMS, we give simple proofs of the existence of partial allocations that are both RMMS and envy-free up to any good (EFX), and complete allocations that are both RMMS and envy-free up to one good (EF1), in fact satisfying the stronger notion of EFL. This unifies and strengthens several previously known results. We further demonstrate the power of the share-based approach by studying the compatibility of fairness notions related to the long-standing EFX problem. While allocations satisfying either epistemic EFX (EEFX) or EF1 are known to exist for general monotone valuations, whether they can always be achieved simultaneously has remained open in every setting where EFX existence itself is unresolved. For additive valuations, we resolve this question affirmatively by proving the existence of allocations that satisfy both EEFX and EFL. Our proof introduces the strong EEFX share, a new share notion implying EEFX feasibility of bundles. We show that the strong EEFX share is upper bounded by RMMS, enabling us to derive EEFX+EFL allocations via the RMMS framework. This answers the main open question of Akrami and Rathi (2025). Finally, although our algorithm for computing EEFX and EF1 allocations may take exponential time in general, we develop a polynomial-time algorithm for restricted additive valuations. Unlike the lone-divider approach, our algorithm exploits the structural properties of restricted additive valuations to compute allocations satisfying both EEFX and EF1.

cs.GT↗

arXiv2025

Large cliques and large independent sets: can they coexist?

For a graph $G$ and a parameter $k$, we call a vertex $k$-enabling if it belongs both to a clique of size $k$ and to an independent set of size $k$, and we call it $k$-excluding otherwise. Motivated by issues that arise in secret sharing schemes, we study the complexity of detecting vertices that are $k$-excluding. We show that for every $\epsilon$, for sufficiently large $n$, if $k > (\frac{1}{4} + \epsilon)n$, then every graph on $n$ vertices must have a $k$-excluding vertex, and moreover, such a vertex can be found in polynomial time. In contrast, if $k < (\frac{1}{4} - \epsilon)n$, a regime in which it might be that all vertices are $k$-enabling, deciding whether a graph has no $k$-excluding vertex is NP-hard.

cs.DS↗

arXiv2025

From multi-allocations to allocations, with subadditive valuations

We consider the problem of fair allocation of $m$ indivisible items to $n$ agents with monotone subadditive valuations. For integer $d \ge 2$, a $d$-multi-allocation is an allocation in which each item is allocated to at most $d$ different agents. We show that $d$-multi-allocations can be transformed into allocations, while not losing much more than a factor of $d$ in the value that each agent receives. One consequence of this result is that for allocation instances with equal entitlements and subadditive valuations, if $\rho$-MMS $d$-multi-allocations exist, then so do $\frac{\rho}{4d}$-MMS allocations. Combined with recent results of Seddighin and Seddighin [EC 2025], this implies the existence of $\Omega(\frac{1}{\log\log n})$-MMS allocations.

cs.GT↗

arXiv2025

Upper bounds on the theta function of random graphs

The theta function of Lovasz is a graph parameter that can be computed up to arbitrary precision in polynomial time. It plays a key role in algorithms that approximate graph parameters such as maximum independent set, maximum clique and chromatic number, or even compute them exactly in some models of random and semi-random graphs. For Erdos-Renyi random $G_{n,1/2}$ graphs, the expected value of the theta function is known to be at most $2\sqrt{n}$ and at least $\sqrt{n}$. These bounds have not been improved in over 40 years. In this work, we introduce a new class of polynomial time computable graph parameters, where every parameter in this class is an upper bound on the theta function. We also present heuristic arguments for determining the expected values of parameters from this class in random graphs. The values suggested by these heuristic arguments are in agreement with results that we obtain experimentally, by sampling graphs at random and computing the value of the respective parameter. Based on parameters from this new class, we feel safe in conjecturing that for $G_{n,1/2}$, the expected value of the theta function is below $1.55 \sqrt{n}$. Our paper falls short of rigorously proving such an upper bound, because our analysis makes use of unproven assumptions.

cs.DS↗

arXiv2025

The residual maximin share

We consider fair allocations of indivisible goods to agents with general monotone valuations. We observe that it is useful to introduce a new share-based fairness notion, the {\em residual maximin share} (RMMS). This share is {\em feasible} and {\em self maximizing}. Its value is at least as large as the MXS for monotone valuations, and at least as large as $\frac{2}{3}$-MMS for additive valuations. Known techniques easily imply the existence of partial allocations that are both RMMS and EFX, and complete allocations that are both RMMS and EFL. This unifies and somewhat improves upon several different results from previous papers.

cs.GT↗

arXiv2025

Fair allocations with subadditive and XOS valuations

We consider the problem of fair allocation of $m$ indivisible goods to $n$ agents with either subadditive or XOS valuations, in the arbitrary entitlement case. As fairness notions, we consider the anyprice share (APS) ex-post, and the maximum expectation share (MES) ex-ante. We observe that there are randomized allocations that ex-ante are at least $\frac{1}{2}$-MES in the subadditive case and $(1-\frac{1}{e})$-MES in the XOS case. Our more difficult results concern ex-post guarantees. We show that $(1 - o(1))\frac{\log\log m}{\log m}$-APS allocations exist in the subadditive case, and $\frac{1}{6}$-APS allocations exist in the XOS case. For the special case of equal entitlements, we show $\frac{4}{17}$-APS allocations for XOS. Our results are the first for subadditive and XOS valuations in the arbitrary entitlement case, and also improve over the previous best results for the equal entitlement case.

cs.GT↗

arXiv2025

Concentration and maximin fair allocations for subadditive valuations

We consider fair allocation of $m$ indivisible items to $n$ agents of equal entitlements, with submodular valuation functions. Previously, Seddighin and Seddighin [{\em Artificial Intelligence} 2024] proved the existence of allocations that offer each agent at least a $\frac{1}{c \log n \log\log n}$ fraction of her maximin share (MMS), where $c$ is some large constant (over 1000, in their work). We modify their algorithm and improve its analysis, improving the ratio to $\frac{1}{14 \log n}$. Some of our improvement stems from tighter analysis of concentration properties for the value of any subadditive valuation function $v$, when considering a set $S' \subseteq S$ of items, where each item of $S$ is included in $S'$ independently at random (with possibly different probabilities). In particular, we prove that up to less than the value of one item, the median value of $v(S')$, denoted by $M$, is at least two-thirds of the expected value, $M \geq \frac{2}{3}\E[v(S')] - \frac{11}{12}\max_{e \in S} v(e)$.

cs.GT↗

arXiv2024

Low communication protocols for fair allocation of indivisible goods

We study the multi-party randomized communication complexity of computing a fair allocation of $m$ indivisible goods to $n < m$ equally entitled agents. We first consider MMS allocations, allocations that give every agent at least her maximin share. Such allocations are guaranteed to exist for simple classes of valuation functions. We consider the expected number of bits that each agent needs to transmit, on average over all agents. For unit demand valuations, we show that this number is only $O(1)$ (but $\Theta(\log n)$, if one seeks EF1 allocations instead of MMS allocations), for binary additive valuations we show that it is $\Theta(\log \frac{m}{n})$, and for 2-valued additive valuations we show a lower bound of $\Omega(\frac{m}{n})$. For general additive valuations, MMS allocations need not exist. We consider a notion of {\em approximately proportional} (Aprop) allocations, that approximates proportional allocations in two different senses, being both Prop1 (proportional up to one item), and $\frac{n}{2n-1}$-TPS (getting at least a $\frac{n}{2n-1}$ fraction of the {\em truncated proportional share}, and hence also at least a $\frac{n}{2n-1}$ fraction of the MMS). We design randomized protocols that output Aprop allocations, in which the expected average number of bits transmitted per agent is $O(\log m)$. For the stronger notion of MXS ({\em minimum EFX share}) we show a lower bound of $\Omega(\frac{m}{n})$.

cs.GT↗

arXiv2024

Share-Based Fairness for Arbitrary Entitlements

We consider the problem of fair allocation of indivisible items to agents that have arbitrary entitlements to the items. Every agent $i$ has a valuation function $v_i$ and an entitlement $b_i$, where entitlements sum up to~1. Which allocation should one choose in situations in which agents fail to agree on one acceptable fairness notion? We study this problem in the case in which each agent focuses on the value she gets, and fairness notions are restricted to be {\em share based}. A {\em share} $s$ is an function that maps every $(v_i,b_i)$ to a value $s(v_i,b_i)$, representing the minimal value $i$ should get, and $s$ is {\em feasible} if it is always possible to give every agent $i$ value of at least $s(v_i,b_i)$. Our main result is that for additive valuations over goods there is an allocation that gives every agent at least half her share value, regardless of which feasible share-based fairness notion the agent wishes to use. Moreover, the ratio of half is best possible. More generally, we provide tight characterizations of what can be achieved, both ex-post (as single allocations) and ex-ante (as expected values of distributions of allocations), both for goods and for chores. We also show that for chores one can achieve the ex-ante and ex-post guarantees simultaneously (a ``best of both world" result), whereas for goods one cannot.

cs.GT↗

arXiv2023

The inversion paradox, and classification of fairness notions

Several different fairness notions have been introduced in the context of fair allocation of goods. In this manuscript, we compare between some fairness notions that are used in settings in which agents have arbitrary (perhaps unequal) entitlements to the goods. This includes the proportional share, the anyprice share, the weighted maximin share, weighted envy freeness, maximum weight Nash social welfare and competitive equilibrium. We perform this comparison in two settings, that of a divisible homogeneous good and arbitrary valuations, and that of indivisible goods and additive valuations. Different fairness notions are not always compatible with each other, and might dictate selecting different allocations. The purpose of our work is to clarify various properties of fairness notions, so as to allow, when needed, to make an educated choice among them. Also, such a study may motivate introducing new fairness notions, or modifications to existing fairness notions. Among other properties, we introduce definitions for monotonicity that postulate that having higher entitlement should be better to the agent than having lower entitlement. Some monotonicity notions, such as population monotonicity and weight monotonicity, appeared in previous work, but we prefer to consider other monotonicity properties that we refer to as global monotonicity and individual monotonicity. We find that some of the fairness notions (but not all) violate our monotonicity properties in a strong sense, that we refer to as the inversion paradox. Under this paradox, a fairness notion enforces that the value received by an agent decreases when the entitlement of the agent increases.

cs.GT↗

arXiv2023

On Fair Allocation of Indivisible Goods to Submodular Agents

We consider the problem of fair allocation of indivisible goods to agents with submodular valuation functions, where agents may have either equal entitlements or arbitrary (possibly unequal) entitlements. We focus on share-based fairness notions, specifically, the maximin share (MMS) for equal entitlements and the anyprice share (APS) for arbitrary entitlements, and design allocation algorithms that give each agent a bundle of value at least some constant fraction of her share value. For the equal entitlement case (and submodular valuations), Ghodsi, Hajiaghayi, Seddighin, Seddighin, and Yami [EC 2018] designed a polynomial-time algorithm for $\frac{1}{3}$-maximin-fair allocation. We improve this result in two different ways. We consider the general case of arbitrary entitlements, and present a polynomial time algorithm that guarantees submodular agents $\frac{1}{3}$ of their APS. For the equal entitlement case, we improve the approximation ratio and obtain $\frac{10}{27}$-maximin-fair allocations. Our algorithms are based on designing strategies for a certain bidding game that was previously introduced by Babaioff, Ezra and Feige [EC 2021].

cs.GT↗

arXiv2022

On picking sequences for chores

We consider the problem of allocating $m$ indivisible chores to $n$ agents with additive disvaluation (cost) functions. It is easy to show that there are picking sequences that give every agent (that uses the greedy picking strategy) a bundle of chores of disvalue at most twice her share value (maximin share, MMS, for agents of equal entitlement, and anyprice share, APS, for agents of arbitrary entitlement). Aziz, Li and Wu (2022) designed picking sequences that improve this ratio to $\frac{5}{3}$ for the case of equal entitlement. We design picking sequences that improve the ratio to~1.733 for the case of arbitrary entitlement, and to $\frac{8}{5}$ for the case of equal entitlement. (In fact, computer assisted analysis suggests that the ratio is smaller than $1.543$ in the equal entitlement case.) We also prove a lower bound of $\frac{3}{2}$ on the obtainable ratio when $n$ is sufficiently large. Additional contributions of our work include improved guarantees in the equal entitlement case when $n$ is small; introduction of the chore share as a convenient proxy to other share notions for chores; introduction of ex-ante notions of envy for risk averse agents; enhancements to our picking sequences that eliminate such envy; showing that a known allocation algorithm (not based on picking sequences) for the equal entitlement case gives each agent a bundle of disvalue at most $\frac{4n-1}{3n}$ times her APS (previously, this ratio was shown for this algorithm with respect to the easier benchmark of the MMS).

cs.GT↗

arXiv2022

Fair Shares: Feasibility, Domination and Incentives

We consider fair allocation of a set $M$ of indivisible goods to $n$ equally-entitled agents, with no monetary transfers. Every agent $i$ has a valuation $v_i$ from some given class of valuation functions. A share $s$ is a function that maps a pair $(v_i,n)$ to a value, with the interpretation that if an allocation of $M$ to $n$ agents fails to give agent $i$ a bundle of value at least equal to $s(v_i,n)$, this serves as evidence that the allocation is not fair towards $i$. For such an interpretation to make sense, we would like the share to be feasible, meaning that for any valuations in the class, there is an allocation that gives every agent at least her share. The maximin share was a natural candidate for a feasible share for additive valuations. However, Kurokawa, Procaccia and Wang [2018] show that it is not feasible. We initiate a systematic study of the family of feasible shares. We say that a share is \emph{self maximizing} if truth-telling maximizes the implied guarantee. We show that every feasible share is dominated by some self-maximizing and feasible share. We seek to identify those self-maximizing feasible shares that are polynomial time computable, and offer the highest share values. We show that a SM-dominating feasible share -- one that dominates every self-maximizing (SM) feasible share -- does not exist for additive valuations (and beyond). Consequently, we relax the domination property to that of domination up to a multiplicative factor of $\rho$ (called $\rho$-dominating). For additive valuations we present shares that are feasible, self-maximizing and polynomial-time computable. For $n$ agents we present such a share that is $\frac{2n}{3n-1}$-dominating. For two agents we present such a share that is $(1 - \epsilon)$-dominating. Moreover, for these shares we present poly-time algorithms that compute allocations that give every agent at least her share.

econ.TH↗