arXiv subjects

Erel Segal-Halevi

Publications and source records attributed to Erel Segal-Halevi.

At least 19 recordsLinked to original sources

arXiv2026

An approximation notion between P and FPTAS

We present an approximation notion for NP-hard optimization problems. The notion is based on an amortized relaxation: the relaxed optimum of an input is the largest per-copy value attainable when many copies of the input are solved together. Assuming P != NP, we prove that the new notion is strictly stronger than FPTAS, but strictly weaker than having a polynomial-time algorithm. Our results therefore reveal a new computational complexity class, which is a strict superset of P and a strict subset of FPTAS.

cs.CC↗

arXiv2026

Individual Fairness in Budget Aggregation

We consider the problem of aggregating $n$ individual distributions over $m$ alternatives into a collective distribution, also known as budget aggregation. Existing fairness notions in this literature typically do not guarantee fairness to individual agents. To address this, we define two versions of individual fair share guarantees. We show that when agents' utilities are derived from $\ell_t$ metrics for any $t\geq 1$, both these guarantees can be satisfied along with Pareto efficiency, and the corresponding distributions can be computed in polynomial time. On the other hand, for $\ell_1$ utilities, we prove that Pareto efficiency, strategyproofness, and a very weak fairness notion called single-minded positive share are not always compatible for $n,m \ge 3$. For smaller parameters, we provide rules that satisfy these three axioms. We also establish similar impossibility results for $\ell_2$ utilities.

cs.GT↗

arXiv2026

Perpetually Fair Assignments Via Balanced Sequences of Permutations

There is a set of $n$ indivisible items (goods or chores), and a set of $n$ players. Each day, a single item should be assigned to each player. Assignments based on latin squares guarantee fairness after every $n$ days; our goal is to ensure fairness after every single day. We present two 'balance' conditions on latin squares. Informally, a latin square is balanced if its top rows and leftmost columns contain all $n$ labels; this ensures that all $n$ players receive one of the top items in one of the early days. One such condition can always be satisfied, but is arguably too weak; a second condition is strong, and can be satisfied for all $n\leq 12$, but cannot be satisfied for some larger values of $n$, including all $n>108$. We show that the second balance condition guarantees that the cumulative assignment is always \emph{proportional up to one item (PROP1)}, where proportionality holds in a strong ordinal sense -- for every valuations that are consistent with the item ranking. Finally, we present a weaker balance condition on a sequence, that guarantees ordinal proportionality up to two items (PROP2). Whether or not this condition can be satisfied for all $n$ remains an open question.

math.CO↗

arXiv2026

Perpetual Fully-Online Approximate Fairness

Many decision processes run for a long and unknown duration: in each round new requests arrive, an irrevocable choice must be made immediately, and the system is judged by ongoing fairness requirements. Examples include food banks allocating donations, computing systems repeatedly scheduling scarce resources across users, and institutions making repeated decisions while remaining fair over time. We propose a general approach based on \emph{deficits}, which measure how far the current outcome is from satisfying each fairness requirement. The goal is to keep all deficits small at each time step, without knowing the horizon or future agent valuations. This viewpoint also highlights a natural modeling question for long-running systems: how much of the past should be counted when fairness is evaluated? We first study the full-history model, where all past rounds count equally. We propose an efficient fully-online rule. For $n$ agents, we prove anytime guarantees: after any $t$ rounds, all requirements remain satisfied up to a slack of order $\tilde O(\sqrt{t/n})$. We instantiate the rule for online allocation of indivisible goods, yielding natural relaxations of proportionality and envy-free, and for online public decision-making. We show that this slack is tight even for weak proportionality. For unrestricted classical $\mathrm{EF}c$, the exact worst-case parameter at horizon $T$ is $\lceil T/n\rceil$. We then study discounted-memory fairness, where older deficits carry smaller weight. The same fully-online rule applies to these discounted deficits, and the resulting threshold is controlled by the discount function. In particular, the time dependence is never worse than the full-history $\sqrt t$ dependence. Overall, our results show that memory is a central part of perpetual fairness. The question is not only which requirement to impose, but also how the system should count past unfairness.

cs.GT↗

arXiv2026

The Min Max Average Cycle Weight Problem

When an old apartment building is demolished and rebuilt, how can we fairly redistribute the new apartments to minimize envy among residents? We reduce this question to a combinatorial optimization problem called the *Min Max Average Cycle Weight* problem. In that problem we seek to assign objects to agents in a way that minimizes the maximum average weight of directed cycles in an associated envy graph. While this problem reduces to maximum-weight matching when starting from a clean slate (achieving polynomial-time solvability), we show that this is not the case when we account for preexisting conditions, such as residents' satisfaction with their original apartments. Whether the problem is polynomial-time solvable in the general case remains an intriguing open problem.

cs.DS↗

arXiv2026

Time and Supply Fairness in Electricity Distribution using $k$-times bin packing

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into the minimum number of bins such that the sum of the item sizes in each bin does not exceed the capacity. We define a new variant, k-times bin-packing (kBP), in which the goal is to pack the items so that each item appears exactly k times in k different bins. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. The fair electricity division problem motivates the study of kBP. The goal is to allocate the available supply among households using some fairness criteria, such as the egalitarian principle. We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve kBP and apply them to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time. We study another variant of the egalitarian allocation problem, in which the goal is to maximize the minimum number of watts allocated to a household. For this variant, we prove an impossibility result: there does not exist such a k that depends only on the number of agents. This impossibility result motivates us to develop four different heuristic algorithms to solve the egalitarian allocation of watts problem. We evaluate the heuristics by summing the minimum watts allocated to any household in each hour, yielding a fairness metric that reflects the lowest watt allocation across all hours. A higher total minimum of watts indicates a more equitable distribution. Thus, we establish new benchmarks for fair allocation of watts.

cs.DS↗

arXiv2026

What Are People's Actual Utility Functions in Budget Aggregation?

Budget aggregation is a process in which citizens vote by declaring their individual ideal budget allocation, and a pre-determined rule aggregates all votes into a single outcome. Recent theoretical work has proposed various aggregation rules, along with impossibility results for satisfying desirable axioms simultaneously. These analyses rely on assumptions about how voters evaluate non-ideal allocations, yet such assumptions have not been empirically validated on human subjects. We present a framework for empirically testing hypotheses about human utility functions using simple pairwise comparisons. We introduce a modular, open-source polling system that, after eliciting a subject's ideal allocation, presents carefully generated pairs of non-ideal alternatives. Different pair-generation algorithms allow testing various properties of utility functions. Using this framework, we conduct polls with hundreds of participants. The results show that standard utility models, including $\ell_1$, $\ell_2$, and Leontief, fail to capture human preferences, as very few participants behave consistently with any single model. In contrast, we find strong empirical support for more general properties, such as star-shaped, multi-dimensional single-peaked, and peak-linear preferences. We also find that most participants exhibit asymmetries both with respect to sign (gains vs. losses) and issue, contradicting any utility model based on an $\ell_p$ metric. These findings suggest that developing practical budget-aggregation mechanisms requires more flexible models of human utility functions.

cs.GT↗

arXiv2026

Efficiently Computing Equilibria in Budget-Aggregation Games

Budget aggregation deals with the social choice problem of distributing an exogenously given budget among a set of public projects, given agents' preferences. Taking a game-theoretic perspective, we study budget-aggregation games where each agent has virtual decision power over some fraction of the budget. We investigate the structure and show efficient computability of Nash equilibria for various common preference models in this setting. In particular, we show that equilibria for Leontief utilities can be found in polynomial time, solving an open problem from Brandt et al. [2023], and give an explicit polynomial-time algorithm for computing equilibria for $\ell_1$ preferences.

cs.GT↗

arXiv2026

Fair Allocation of Improvements: When Old Endowments Shape New Assignments

This work is motivated by a common urban renewal process called Reconstruct and Divide. It involves the demolition of old buildings and the construction of new ones. Original homeowners are compensated with upgraded apartments, while surplus units are sold for profit, so theoretically it is a win-win project for all parties involved. However, many Reconstruct and Divide projects are withheld or delayed due to disagreements over the assignment of new apartments, claiming they are not fair. The goal of this research is to develop algorithms for envy-free assignment of the new apartments, possibly using monetary payments to reduce envy. In contrast to previous works on envy-free assignment, in our setting the envy depends also on the value of the old apartments, as people with more valuable old apartments expect to get more valuable new apartments. This presents two challenges. First, in some cases, no assignment and payment-vector satisfy the common fairness notions of envy-freeness and proportionality. Hence, we focus on minimizing the envy and the disproportionality (the distance between an agent's value and their proportional share). We present a strongly polynomial-time algorithm that, for a given assignment, finds a payment vector that minimizes the maximum pairwise-envy. We also present a strongly polynomial-time algorithm that computes an assignment and payment-vector that together minimize the maximum disproportionality. Second, directly asking the agents for their subjective valuations for their old apartments is infeasible, as it is a dominant strategy for them to report very high values for their old apartments. We introduce a novel method to elicit agents' valuations indirectly. Using this method, we identify conditions under which our Minimum Disproportionality algorithm is risk-averse truthful.

cs.GT↗

arXiv2026

It's Not All Black and White: Degree of Truthfulness for Risk-Avoiding Agents

The classic notion of \emph{truthfulness} requires that no agent has a profitable manipulation -- an untruthful report that, for \emph{some} combination of reports of the other agents, increases her utility. This strong notion implicitly assumes that the manipulating agent either knows what all other agents are going to report, or is willing to take the risk and act as-if she knows their reports. Without knowledge of the others' reports, most manipulations are \emph{risky} -- they might decrease the manipulator's utility for some other combinations of reports by the other agents. Accordingly, a recent paper (Bu, Song and Tao, ``On the existence of truthful fair cake cutting mechanisms'', Artificial Intelligence 319 (2023), 103904) suggests a relaxed notion, which we refer to as \emph{risk-avoiding truthfulness (RAT)}, which requires only that no agent can gain from a \emph{safe} manipulation -- one that is sometimes beneficial and never harmful. Truthfulness and RAT are two extremes: the former considers manipulators with complete knowledge of others, whereas the latter considers manipulators with no knowledge at all. In reality, agents often know about some -- but not all -- of the other agents. This paper introduces the \emph{RAT-degree} of a mechanism, defined as the smallest number of agents whose reports, if known, may allow another agent to safely manipulate, or $n$ if there is no such number. This notion interpolates between classic truthfulness (degree $n$) and RAT (degree at least $1$): a mechanism with a higher RAT-degree is harder to manipulate safely. To illustrate the generality and applicability of this concept, we analyze the RAT-degree of prominent mechanisms across various social choice settings, including auctions, indivisible goods allocations, cake-cutting, voting, and two-sided matching.

cs.GT↗

arXiv2025

Optimal Budget Aggregation with Star-Shaped Preference Domains

We study the problem of aggregating distributions, such as budget proposals, into a collective distribution. An ideal aggregation mechanism would be Pareto efficient, strategyproof, and fair. Most previous work assumes that agents evaluate budgets according to the $\ell_1$ distance to their ideal budget. We investigate and compare different models from the larger class of star-shaped utility functions - a multi-dimensional generalization of single-peaked preferences. For the case of two alternatives, we extend existing results by proving that under very general assumptions, the uniform phantom mechanism is the only strategyproof mechanism that satisfies proportionality - a minimal notion of fairness introduced by Freeman et al. (2021). Moving to the case of more than two alternatives, we establish sweeping impossibilities for $\ell_1$ and $\ell_\infty$ disutilities: no mechanism satisfies efficiency, strategyproofness, and proportionality. We then propose a new kind of star-shaped utilities based on evaluating budgets by the ratios of shares between a given budget and an ideal budget. For these utilities, efficiency, strategyproofness, and fairness become compatible. In particular, we prove that the mechanism that maximizes the Nash product of individual utilities is characterized by group-strategyproofness and a core-based fairness condition.

econ.TH↗

arXiv2025

Coordinating Charitable Donations with Leontief Preferences

We consider the problem of funding public goods that are complementary in nature. Examples include charities handling different needs (e.g., protecting animals vs. providing healthcare), charitable donations to different individuals, or municipal units handling different issues (e.g., security vs. transportation). We model these complementarities by assuming Leontief preferences; that is, each donor seeks to maximize an individually weighted minimum of all contributions across the charities. Decentralized funding may be inefficient due to a lack of coordination among the donors; centralized funding may be undesirable as it ignores the preferences of individual donors. We present a mechanism that combines the advantages of both methods. The mechanism efficiently distributes each donor's contribution so that no subset of donors has an incentive to redistribute their donations. Moreover, it is group-strategyproof, satisfies desirable monotonicity properties, maximizes Nash welfare, returns a unique Lindahl equilibrium, and can be implemented via natural best-response spending dynamics.

econ.TH↗

arXiv2025

Reducing Leximin Fairness to Utilitarian Optimization

Two prominent objectives in social choice are utilitarian - maximizing the sum of agents' utilities, and leximin - maximizing the smallest agent's utility, then the second-smallest, etc. Utilitarianism is typically computationally easier to attain but is generally viewed as less fair. This paper presents a general reduction scheme that, given a utilitarian solver, produces a distribution over states (deterministic outcomes) that is leximin in expectation. Importantly, the scheme is robust in the sense that, given an approximate utilitarian solver, it produces a lottery that is approximately-leximin (in expectation) - with the same approximation factor. We apply our scheme to several social choice problems: stochastic allocations of indivisible goods, giveaway lotteries, and fair lotteries for participatory budgeting.

cs.GT↗

arXiv2025

Fair Division with Bounded Sharing: Binary and Non-Degenerate Valuations

A set of objects is to be divided fairly among agents with different tastes, modeled by additive utility-functions. If we consider the objects as indivisible, many instances of the decision problem: ``Is there a fair division of the objects among the agents'' are negative. In addition, this question is hard to solve even for most of the special cases. The latter reasons give us a good motivation to relax the problem for which the running time complexity is better, and the number of positive instances (admitting a fair division) will significantly grow. Whereas many works relax the fairness criteria, this paper introduces another relaxation: an agent is allowed to share a \emph{bounded} number of objects between two or more agents in order to attain fairness. The paper studies various notions of fairness, such as proportionality, envy-freeness, equitability, and consensus. We analyze the run-time complexity of finding a fair allocation with a given number of sharings under several restrictions on the agents' valuations, such as: binary, generalized-binary, and non-degenerate.

cs.GT↗

arXiv2025

Fair Allocation with Money: What is Your Objective?

When allocating indivisible items, there are various ways to use monetary transfers for eliminating envy. Particularly, one can apply a balanced vector of transfer payments, or charge each agent a positive amount, or -- contrarily -- give each agent a positive amount as a ``subsidy''. In each model, one can aim to minimize the amount of payments used; this aim translates into different optimization objectives in each setting. This note compares the various models, and the relations between upper and lower bounds for these objectives.

cs.GT↗

arXiv2025

Fairly Wired: Towards Leximin-Optimal Division of Electricity

In many parts of the world - particularly in developing countries - the demand for electricity exceeds the available supply. In such cases, it is impossible to provide electricity to all households simultaneously. This raises a fundamental question: how should electricity be allocated fairly? In this paper, we explore this question through the lens of egalitarianism - a principle that emphasizes equality by prioritizing the welfare of the worst-off households. One natural rule that aligns with this principle is to maximize the egalitarian welfare - the smallest utility across all households. We show that computing such an allocation is NP-hard, even under strong simplifying assumptions. Leximin is a stronger fairness notion that generalizes the egalitarian welfare: it also requires to maximize the smallest utility, but then, subject to that, the second-smallest, then the third, and so on. The hardness results extends directly to leximin as well. Despite this, we present a Fully Polynomial-Time Approximation Scheme (FPTAS) for leximin in the special case where the network connectivity graph is a tree. This means that we can efficiently approximate leximin - and, in particular, the egalitarian welfare - to any desired level of accuracy.

cs.GT↗

arXiv2025

Asymmetric Number Partitioning with Splitting and Interval Targets

The n-way number partitioning problem, a fundamental challenge in combinatorial optimization, has significant implications for applications such as fair division and machine scheduling. Despite these problems being NP-hard, many approximation techniques exist. We consider three closely related kinds of approximations, and various objectives such as decision, min-max, max-min, and even a generalized objective, in which the bins are not considered identical anymore, but rather asymmetric (used to solve fair division to asymmetric agents or uniform machine scheduling problems). The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n=2, the running time is easy to identify. Our main results consider any fixed n>=3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s>=n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t>=n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u>=(n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both asymmetric min-max and asymmetric max-min versions.

cs.DS↗

arXiv2025

Whoever Said Money Won't Solve All Your Problems? Weighted Envy-free Allocation with Subsidy

We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any others relative to their own. Often, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work relied on characterizations of unweighted envy-freeness (EF), that fail in the weighted setting. This makes our new setting challenging. We present polynomial-time algorithms that compute WEF allocations with a guaranteed upper bound on total subsidy for monotone valuations and various subclasses thereof. We also present an efficient algorithm to compute a fair allocation of items and money, when the budget is not enough to make the allocation WEF. This algorithm is new even for the unweighted setting.

cs.GT↗