arXiv subjects

Haris Aziz

Publications and source records attributed to Haris Aziz.

At least 19 recordsLinked to original sources

arXiv2026

Pairwise Maximin Share Allocations Need Not Exist

We study the fair allocation of indivisible goods among agents with strictly positive additive valuations. Pairwise maximin share fairness (PMMS) asks that, for every ordered pair of agents, the first agent value her own bundle at least as highly as the best worst-case share she could secure by repartitioning the two agents' combined bundles into two parts. Whether a PMMS allocation always exists for positive additive valuations has remained open since the notion was introduced. We resolve this question in the negative. We construct an instance with four agents and strictly positive additive valuations that admits no complete PMMS allocation.

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arXiv2026

On the Incompatibility of Weighted PROPX and Pareto Optimality for Indivisible Chores

Proportionality (PROP) is one of the simplest fairness criteria for allocating items among agents with additive preferences. With indivisible chores, however, PROP is not always satisfiable. We study proportionality up to any item (PROPX), which requires every agent to satisfy proportionality after any chore is removed from her bundle. Under strictly positive costs, we settle the weighted compatibility question negatively: weighted PROPX and Pareto optimality are incompatible already for two agents and four chores. Moreover, for every $n\geq3$, we give an $n$-agent, $(n+1)$-chore counterexample whose shares can be arbitrarily close to equal. These counterexamples are item-minimal: under strictly positive costs, weighted PROPX and Pareto optimality are always compatible when the number of chores is at most the number of agents, and they are compatible for two agents with at most three chores. Our impossibility result contrasts with the compatibility theorem of Mahara (2026) for weighted envy-freeness up to one item (EF1) and Pareto optimality .

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arXiv2026

Anchoring for Truthfulness: The Random-Anchor Volume Mechanism for Multi-Facility Location

We study the strategyproof placement of $k$ facilities on the real line for $n$ agents who privately report their locations, without monetary transfers. For two facilities, the Proportional Mechanism of Lu, Sun, Wang, and Zhu (2010) is strategyproof in expectation and achieves a constant-factor approximation to the optimal social cost. Whether such a guarantee is possible for three facilities in the standard model, where each agent is served by her nearest open facility, has remained open. We resolve this question affirmatively by introducing the \emph{Random-Anchor Volume} mechanism. The mechanism first opens a facility at the report of a uniformly random agent, called the \emph{anchor}, and then jointly selects two additional reports, assigning each pair probability proportional to the product of the two consecutive gaps formed by the pair and the anchor. We prove that the mechanism is strategyproof in expectation and has expected social cost at most $8 OPT_3$, where $OPT_k$ denotes the minimum social cost achievable using at most $k$ facilities. The mechanism naturally extends to every $k\geq 2$ by selecting $k-1$ additional reports with probability proportional to the product of the consecutive gaps among them and the anchor. Under truthful reporting, this generalization has expected social cost at most $4(k-1)OPT_k$. Its incentive guarantee, however, has a sharp boundary: the mechanism is strategyproof in expectation for $k\in\{1,2,3\}$, but is manipulable for every $k\geq 4$.

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arXiv2026

Best-of-Both-Worlds Fairness and Pareto Optimality

We consider fair allocation of indivisible items among agents with non-negative and additive valuations. The goal is to construct a lottery over deterministic allocations whose induced fractional allocation is envy-free, while every realised allocation is envy-free up to one item and Pareto optimal. We show that this is always possible for two agents. We then prove a stronger result that there always exists a lottery over deterministic allocations whose induced fractional allocation is envy-free, while every realised allocation is envy-free up each one item (EFX) and Pareto optimal. For non-negative integral additive valuations, such a lottery can be computed in pseudo-polynomial time. We also prove that for three agents and four items, there may be no ex-ante envy-free lottery that can be supported on allocations that are simultaneously EFX and Pareto optimal.

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arXiv2026

Fair Allocation of Divisible Goods under Non-Linear Valuations

We study the problem of dividing homogeneous divisible goods among agents with non-linear valuations. Specifically, the value that an agent gains from a given good depends only on the amount of the good they receive, and is not necessarily linear with respect to the amount. For instance, under one-breakpoint piecewise-constant valuations, each agent specifies a threshold for each good such that this agent receives utility zero (resp., full utility of the good) when getting an amount below (resp., at least) the threshold. Given non-linear valuations that are additive across the goods, we focus on designing fair allocation algorithms and consider two well-known fairness properties: the maximin share (MMS) guarantee and envy-freeness (EF). For MMS, we devise an algorithm which always produces a $\frac{1}{2n-1}$-MMS allocation for $n$ agents with arbitrary non-decreasing valuations. It is worth noting that this algorithmic result is almost tight as we give an impossibility of guaranteeing more than $1/n$ approximation to MMS, even when agents have one-breakpoint piecewise-constant valuations. For $n \leq 3$ agents, we show the ratio $1/n$ is tight. Regarding envy-freeness, we show it is NP-hard to check the existence of an EF and Pareto optimal (PO) allocation for $n$ agents and at least three goods, even when agents have one-breakpoint piecewise-constant valuations. We complement the hardness result by considering the case with a single divisible good, and devising a polynomial-time algorithm to check whether an EF and PO allocation exists or not for agents with piecewise-linear valuations.

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arXiv2026

Best-of-Both-Worlds Fairness for Mixed Goods and Chores

We study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-of-both-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envy-freeness up to one item (EF1). Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality -- via Sion's minimax theorem -- with carefully designed biased flow networks.

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arXiv2026

Menu Selection: A Computational Approach to Minimizing Food Waste

We introduce a novel collective decision making problem that captures the ubiquitous issue of ordering food to cater for varied dietary preferences and requirements. Our settings involve agents with diverse dietary requirements over menu options with varied serving sizes. The goal is to select a menu where everyone has enough food they can consume and wastage of food is minimized. We introduce two different consumption models: optimistic and pessimistic. Optimistic consumption assumes a situation when a central planner can optimally allocate the food ordered among the agents to maximize the number of people who get enough to eat. Pessimistic considers the worst case guarantee on consumption when agents fill their own plates in an arbitrary order. Under either consumption model, we seek valid menus (under which all agents are sufficiently fed) of minimum size. Our work provides two sets of characterizations: (1) we characterize valid menus under either consumption model and (2) we characterize the space of instances that admit polynomial-time algorithms to find minimum sized menus. Our results also help us design Integer Linear Programs to find minimum sized menus in general settings. Furthermore, we present polynomial-time algorithms for important special cases. We then consider the worst case discrepancy between the size of minimum sized optimistic and pessimistic menus. We call this the waste of pessimism, captured by the ratio of the minimum sized pessimistic menu to that of the minimum sized optimistic menu. We show tight upper bounds on this ratio. Our results also provide additional insights on the problem of finding a minimum sized maximal matching, which may be of independent interest.

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arXiv2026

The Team Order Problem: Maximizing the Probability of Matching Being Large Enough

We consider a matching problem, which is meaningful in team competitions, as well as in information theory, recommender systems, and assignment problems. In the competitions which we study, each competitor in a team order plays a match with the corresponding opposing player. The team that wins more matches wins. We consider a problem where the input is the graph of probabilities that a team 1 player can win against the team 2 player, and the output is the optimal ordering of team 1 players given the fixed ordering of team 2. Our central result is a polynomial-time approximation scheme (PTAS) to compute a matching whose winning probability is at most epsilon less than the winning probability of the optimal matching. We also provide tractability results for several special cases of the problem, as well as an analytical bound on how far the winning probability of a maximum weight matching of the underlying graph is from the best achievable winning probability.

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arXiv2026

Approximate Strategyproofness in Approval-based Budget Division

In approval-based budget division, the task is to allocate a divisible resource to the candidates based on the voters' approval preferences over the candidates. For this setting, Brandl et al. [2021] have shown that no distribution rule can be strategyproof, efficient, and fair at the same time. In this paper, we aim to circumvent this impossibility theorem by focusing on approximate strategyproofness. To this end, we analyze the incentive ratio of distribution rules, which quantifies the maximum multiplicative utility gain of a voter by manipulating. While it turns out that several classical rules have a large incentive ratio, we prove that the Nash product rule ($\mathsf{NASH}$) has an incentive ratio of $2$, thereby demonstrating that we can bypass the impossibility of Brandl et al. by relaxing strategyproofness. Moreover, we show that an incentive ratio of $2$ is optimal subject to some of the fairness and efficiency properties of $\mathsf{NASH}$, and that the positive result for the Nash product rule even holds when voters may report arbitrary concave utility functions. Finally, we complement our results with an experimental analysis.

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arXiv2026

Smart Lotteries in School Choice: Ex-ante Pareto-Improvement with Ex-post Stability

In a typical school choice application, the students have strict preferences over the schools while the schools have coarse priorities over the students based on their distance and their enrolled siblings. The outcome of a centralized admission mechanism is then usually obtained by the Deferred Acceptance (DA) algorithm with random tie-breaking. Therefore, every possible outcome of this mechanism is a stable solution for the coarse priorities that will arise with certain probability. This implies a probabilistic assignment, where the admission probability for each student-school pair is specified. In this paper, we propose a new efficiency-improving stable `smart lottery' mechanism. We aim to improve the probabilistic assignment ex-ante in a stochastic dominance sense, while ensuring that the improved random matching is still ex-post stable, meaning that it can be decomposed into stable matchings regarding the original coarse priorities. Therefore, this smart lottery mechanism can provide a clear Pareto-improvement in expectation for any cardinal utilities compared to the standard DA with lottery solution, without sacrificing the stability of the final outcome. We show that although the underlying computational problem is NP-hard, we can solve the problem by using advanced optimization techniques such as integer programming with column generation. We conduct computational experiments on generated and real instances. Our results show that the welfare gains by our mechanism are substantially larger than the expected gains by standard methods that realize efficiency improvements after ties have already been broken.

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arXiv2026

Fair Transit Stop Placement: A Clustering Perspective and Beyond

We study the transit stop placement (TrSP) problem in general metric spaces, where agents travel between source-destination pairs and may either walk directly or utilize a shuttle service via selected transit stops. We investigate fairness in TrSP through the lens of justified representation (JR) and the core, and uncover a structural correspondence with fair clustering. Specifically, we show that a constant-factor approximation to proportional fairness in clustering can be used to guarantee a constant-factor biparameterized approximation to core. We establish a lower bound of 1.366 on the approximability of JR, and moreover show that no clustering algorithm can approximate JR within a factor better than 3. Going beyond clustering, we propose the Expanding Cost Algorithm, which achieves a tight 2.414-approximation for JR, but does not give any bounded core guarantee. In light of this, we introduce a parameterized algorithm that interpolates between these approaches, and enables a tunable trade-off between JR and core. Finally, we complement our results with an experimental analysis using small-market public carpooling data.

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arXiv2025

Learning-Augmented Facility Location Mechanisms for Envy Ratio

The augmentation of algorithms with predictions of the optimal solution, such as from a machine-learning algorithm, has garnered significant attention in recent years, particularly in facility location problems. Moving beyond the traditional focus on utilitarian and egalitarian objectives, we design learning-augmented facility location mechanisms on a line for the envy ratio objective, a fairness metric defined as the maximum ratio between the utilities of any two agents. For the deterministic setting, we propose the $\alpha$-Bounding Interval Mechanism ($\alpha$-BIM), which utilizes predictions to achieve $\alpha$-consistency and $\frac{\alpha}{\alpha - 1}$-robustness for a selected parameter $\alpha \in [1,2]$, and prove its optimality. We also resolve open questions raised by Ding et al. [10], devising a randomized mechanism without predictions to improve upon the best-known approximation ratio from $2$ to approximately $1.8944$. Building upon these advancements, we construct a novel randomized mechanism, the Bias-Aware Mechanism (BAM), which incorporates predictions to achieve improved consistency and robustness guarantees.

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arXiv2025

Fair Division with Indivisible Goods, Chores, and Cake

We study the problem of fairly allocating indivisible items and a desirable heterogeneous divisible good (i.e., cake) to agents with additive utilities. In our paper, each indivisible item can be a good that yields non-negative utilities to some agents and a chore that yields negative utilities to the other agents. Given a fixed set of divisible and indivisible resources, we investigate almost envy-free allocations, captured by the natural fairness concept of envy-freeness for mixed resources (EFM). It requires that an agent $i$ does not envy another agent $j$ if agent $j$'s bundle contains any piece of cake yielding positive utility to agent $i$ (i.e., envy-freeness), and agent $i$ is envy-free up to one item (EF1) towards agent $j$ otherwise. We prove that with indivisible items and a cake, an EFM allocation always exists for any number of agents with additive utilities.

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arXiv2025

Distance Preservation Games

We introduce and analyze distance preservation games (DPGs). In DPGs, agents express ideal distances to other agents and need to choose locations in the unit interval while preserving their ideal distances as closely as possible. We analyze the existence and computation of location profiles that are jump stable (i.e., no agent can benefit by moving to another location) or welfare optimal for DPGs, respectively. Specifically, we prove that there are DPGs without jump stable location profiles and identify important cases where such outcomes always exist and can be computed efficiently. Similarly, we show that finding welfare optimal location profiles is NP-complete and present approximation algorithms for finding solutions with social welfare close to optimal. Finally, we prove that DPGs have a price of anarchy of at most $2$.

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arXiv2025

Social Welfare Maximization in Approval-Based Committee Voting under Uncertainty

Approval voting is widely used for making multi-winner voting decisions. The canonical rule (also called Approval Voting) used in the setting aims to maximize social welfare by selecting candidates with the highest number of approvals. We revisit approval-based multi-winner voting in scenarios where the information regarding the voters' preferences is uncertain. We present several algorithmic results for problems related to social welfare maximization under uncertainty, including computing the social welfare probability distribution of a given outcome, computing the probability that a given outcome is social welfare maximizing, computing an outcome that is social welfare maximizing with the highest probability, and understanding how robust an outcome is with respect to social welfare maximization.

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arXiv2025

Participation Incentives in Online Cooperative Games

This paper studies cooperative games where coalitions are formed online and the value generated by the grand coalition must be irrevocably distributed among the players at each timestep. We investigate the fundamental issue of strategic pariticipation incentives and address these concerns by formalizing natural participation incentive axioms. Our analysis reveals that existing value-sharing mechanisms fail to meet these criteria. Consequently, we propose several new mechanisms that not only fulfill these desirable participation incentive axioms but also satisfy the early arrival incentive for general valuation functions. Additionally, we refine our mechanisms under superadditive valuations to ensure individual rationality while preserving the previously established axioms.

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arXiv2025

Maximum Welfare Allocations under Quantile Valuations

We propose a new model for aggregating preferences over a set of indivisible items based on a quantile value. In this model, each agent is endowed with a specific quantile, and the value of a given bundle is defined by the corresponding quantile of the individual values of the items within it. Our model captures the diverse ways in which agents may perceive a bundle, even when they agree on the values of individual items. It enables richer behavioral modeling that cannot be easily captured by additive valuation functions. We study the problem of maximizing utilitarian and egalitarian welfare within the quantile-based valuation setting. For each of the welfare functions, we analyze the complexity of the objectives. Interestingly, our results show that the complexity of both objectives varies significantly depending on whether the allocation is required to be balanced. We provide near-optimal approximation algorithms for utilitarian welfare, and for egalitarian welfare, we present exact algorithms whenever possible.

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arXiv2025

Strategyproof Maximum Matching under Dichotomous Agent Preferences

We consider a two-sided matching problem in which the agents on one side have dichotomous preferences and the other side representing institutions has strict preferences (priorities). It captures several important applications in matching market design in which the agents are only interested in getting matched to an acceptable institution. These include centralized daycare assignment and healthcare rationing. We present a compelling new mechanism that satisfies many prominent and desirable properties including individual rationality, maximum size, fairness, Pareto-efficiency on both sides, strategyproofness on both sides, non-bossiness and having polynomial time running time. As a result, we answer an open problem whether there exists a mechanism that is agent-strategyproof, maximum, fair and non-bossy.

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