We study the fair division of indivisible items and provide new insights into the EFX problem, which is widely regarded as the central open question in fair division, and the PMMS problem, a strictly stronger variant of EFX. Our first result constructs a three-agent instance with two monotone valuations and one additive valuation in which no PMMS allocation exists. Since EFX allocations are known to exist under these assumptions, this establishes a formal separation between EFX and PMMS. We prove existence of fair allocations for three important special cases. We show that EFX allocations exist for personalized bivalued valuations, where for each agent i there exist values aᵢ > bᵢ such that agent i assigns value vᵢ({g}) ∈ {aᵢ, bᵢ} to each good g. We establish an analogous existence result for PMMS allocations when aᵢ is divisible by bᵢ. We also prove that PMMS allocations exist for binary-valued MMS-feasible valuations, where each bundle S has value vᵢ(S) ∈ {0, 1}. Notably, this result holds even without assuming monotonicity of valuations and thus applies to the fair division of chores and mixed manna. Finally, we study a class of valuations called pair-demand valuations, which extend the well-studied unit-demand valuations to the case where each agent derives value from at most two items, and we show that PMMS allocations exist in this setting. Our proofs are constructive, and we provide polynomial-time algorithms for all three existence results.
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The fair division of indivisible goods is not only a subject of theoretical research, but also an important problem in practice, with solutions being offered on several online platforms. Little is known, however, about the characteristics of real-world allocation instances and how they compare to synthetic instances. Using dimensionality reduction, we compute a map of allocation instances: a 2-dimensional embedding such that an instance's location on the map is predictive of the instance's origin and other key instance features. Because the axes of this map closely align with the utility matrix's two largest singular values, we define a second, explicit map, which we theoretically characterize.
Picking a Representative Set of Solutions in Multiobjective Optimization: Axioms, Algorithms, and Experiments
PDF ↗Many real-world decision-making problems involve optimizing multiple objectives simultaneously, rendering the selection of the most preferred solution a non-trivial problem: All Pareto optimal solutions are viable candidates, and it is typically up to a decision maker to select one for implementation based on their subjective preferences. To reduce the cognitive load on the decision maker, previous work has introduced the Pareto pruning problem, where the goal is to compute a fixed-size subset of Pareto optimal solutions that best represent the full set, as evaluated by a given quality measure. Reframing Pareto pruning as a multiwinner voting problem, we conduct an axiomatic analysis of existing quality measures, uncovering several unintuitive behaviors. Motivated by these findings, we introduce a new measure, directed coverage. We also analyze the computational complexity of optimizing various quality measures, identifying previously unknown boundaries between tractable and intractable cases depending on the number and structure of the objectives. Finally, we present an experimental evaluation, demonstrating that the choice of quality measure has a decisive impact on the characteristics of the selected set of solutions and that our proposed measure performs competitively or even favorably across a range of settings.
Despite extensive theoretical research on proportionality in approval-based multiwinner voting, its impact on which committees and candidates can be selected in practice remains poorly understood. We address this gap by (i) analyzing the computational complexity of several natural problems related to the behavior of proportionality axioms, and (ii) conducting an extensive experimental study on both real-world and synthetic elections. Our findings reveal substantial variation in the restrictiveness of proportionality across instances, including previously unobserved high levels of restrictiveness in some real-world cases. We also introduce and evaluate new measures for quantifying a candidate’s importance for achieving proportional outcomes, which differ clearly from assessing candidate strength by approval score.
Approximately Envy-free and Equitable Allocations of Indivisible Items for Non-monotone Valuations
PDF ↗We revisit the setting of fair allocation of indivisible items among agents with heterogeneous, non-monotone valuations. We explore the existence and efficient computation of allocations that approximately satisfy either envy-freeness or equity constraints. Approximate envy-freeness ensures that each agent values her bundle at least as much as those given to the others, after some (or any) item removal, while approximate equity guarantees roughly equal valuations among agents, under similar adjustments. As a key technical contribution of this work, by leveraging fixed-point theorems (such as Sperner's Lemma and its variants), we establish the existence of envy-free-up-to-one-good-and-one-chore (EF1_g^c) and equitable-up-to-one-good-and-one-chore (EQ1_g^c) allocations, for non-monotone valuations that are always either non-negative or non-positive. These notions represent slight relaxations of the well-studied envy-free-up-to-one-item (EF1) and equitable-up-to-one-item (EQ1) guarantees, respectively. Our existential results hold even when items are arranged in a path and bundles must form connected sub-paths. The case of non-positive valuations, in particular, has been solved by proving a novel multi-coloring variant of Sperner's Lemma that constitutes a combinatorial result of independent interest. In addition, we also design a polynomial-time dynamic programming algorithm that computes an EQ1_g^c allocation. For monotone non-increasing valuations and path-connected bundles, all the above results can be extended to EF1 and EQ1 guarantees as well. Finally, we provide existential and computational results for certain stronger up-to-any-item equity notions under objective valuations, where items are partitioned into goods and chores.
We introduce a new notion of deterministic stable solution for non-cooperative games, termed subsidized equilibrium. It assumes that an amount of money can be used as a pool of subsidies to stabilize a strategy profile that otherwise would not be accepted by (some of) the players. Roughly speaking, for a given amount of money, a strategy profile is a subsidized equilibrium if the total payoff loss incurred by players not playing best-responses does not exceed that amount, i.e., there is enough money to refund all players experiencing a regret. With respect to many other solution concepts in the literature, the notion of subsidized equilibrium has important advantages. Specifically, for a sufficiently high value of money, a subsidized equilibrium always exists and can even be computed in polynomial time; also, existence of an efficient subsidized equilibrium can be guaranteed. Thus, determining for which amounts of money existence, polynomial time computability and efficiency can or cannot be achieved becomes an intriguing question. We provide initial results towards this direction for some widely studied classes of games.
Equitability is a well-studied fairness notion in fair division, where an allocation is equitable if all agents receive equal utility from their allocation. For indivisible items, an exactly equitable allocation may not exist, hence, a natural relaxation is EQ1, which stipulates that any inequitability should be resolved by the removal of a single item. In this paper, we study equitability in the context of randomized allocations. Specifically, we aim to achieve equitability in expectation (ex ante EQ) and require that each deterministic outcome in the support satisfies ex post EQ1. Such an allocation is commonly known as a `Best of Both Worlds' allocation, and has been studied, e.g., for envy-freeness and MMS. We characterize the existence of such allocations using a geometric condition on convex combinations of allocations, and use this to give comprehensive results on both existence and computation. For two agents, we show that ex ante EQ and ex post EQ1 allocations always exist and can be computed in polynomial time. For three or more agents, however, such allocations may not exist. We prove that deciding existence of such allocations is strongly NP-complete in general, and weakly NP-complete even for three agents. We also present a pseudo-polynomial time algorithm for a constant number of agents. Additionally, we show that when agents have binary valuations, best of both worlds allocations that additionally satisfy welfare guarantees exist and are efficiently computable.
On the Edge of Core (Non-)Emptiness: An Automated Reasoning Approach to Approval-Based Multi-Winner Voting
PDF ↗Core stability is a natural and well-studied notion for group fairness in multi-winner voting, where the task is to select a committee from a pool of candidates. We study the setting where voters either approve or disapprove of each candidate; here, it remains a major open problem whether a core-stable committee always exists. In this work, we develop an approach based on mixed-integer linear programming for deciding whether and when core-stable committees are guaranteed to exist. In contrast to SAT-based approaches popular in computational social choice, our method can produce proofs for a specific number of candidates independent of the number of voters. In addition to these computational gains, our program lends itself to a novel duality-based reformulation of the core stability problem, from which we obtain new existence results in special cases. Further, we use our framework to reveal previously unknown relationships between core stability and other desirable properties, such as notions of priceability.
Time-inconsistent behavior, such as procrastination or abandonment of long-term goals, arises when agents evaluate immediate outcomes disproportionately higher than future ones. This leads to globally suboptimal behavior, where plans are frequently revised or abandoned entirely. In the influential model of Kleinberg and Oren (2014) such behavior is modeled by a present-biased agent navigating a task graph toward a goal, making locally optimal decisions at each step based on discounted future costs. As a result, the agent may repeatedly deviate from initially intended plans. Recent work by Belova et al. (2024) introduced a two-agent extension of this model, where a fully-aware principal attempts to guide the present-biased agent through a specific set of critical tasks without causing abandonment. This captures a rich class of principal–agent dynamics in behavioral settings. In this paper, we provide a comprehensive algorithmic characterization of this problem. We analyze its computational complexity through the framework of parameterized algorithms, focusing on graph parameters that naturally emerge in this setting, such as treewidth, vertex cover, and feedback vertex set. Our main result is a fixed-parameter tractable algorithm when parameterized by the treewidth of the task graph and the number of distinct (v,t)-path costs. Our algorithm encaptures several input settings, such as bounded edge costs and restricted task graph structure. We demonstrate that our main result yields efficient algorithms for a number of such configurations. We complement this with tight hardness results, that highlight the extreme difficulty of the problem even on simplest graphs with bounded number of nodes and constant parameter values, and motivate our choice of parameters. We delineate tractable and intractable regions of the problem landscape, which include answers to open questions of Belova et al. (2024).
In recent years, research in Participatory Budgeting (PB) has put a greater emphasis on rules satisfying notions of fairness and proportionality, with the Method of Equal Shares (MES) being a prominent example. However, proportionality can come at a cost to the total utilitarian welfare. Our work formalizes this relationship, by deriving minimum utilitarian welfare guarantees for MES for a subclass of satisfaction functions called DNS functions, which includes two of the most popular ways of measuring a voter’s utility in the PB setting: considering (1) the total cost of approved projects or (2) the total number of those projects. Our results are parameterized in terms of minimum and maximum project costs, which allows us to improve on the mostly negative results found in prior studies, and reduce to the existing multiwinner guarantee when project costs are equal. We show that our guarantees are asymptotically tight for rules satisfying Extended Justified Representation up to one project, showing that no proportional rule can achieve a better utilitarian guarantee than MES.
Designing Optimal Mechanisms to Locate Facilities with Insufficient Capacity for Bayesian Agents
PDF ↗In this paper, we study the Facility Location Problem with Scarce Resources (FLPSR) under the assumption that agents' type follows a probability distribution on [0,1]. In the FLPSR, the goal is to identify the optimal locations for one or more capacitated facilities to maximize Social Welfare (SW), defined as the sum of the utilities of all agents. Since the total capacity of the facilities is insufficient to serve all agents, they compete in a First-Come-First-Served game to get accommodated. The main contribution of the paper ties Optimal Transport theory to the problem of selecting a truthful mechanism tailored to the agents' distributions. For the case of a single facility, we show that an optimal mechanism always exists. We examine three classes of probability distributions and characterize the optimal mechanism analytically or provide a routine to numerically compute it. We extend our results to the case in which we have two capacitated facilities to place. Initially, we assume that agents are independent and identically distributed, but our techniques generalize to scenarios where agents are not identically distributed. Finally, we validate our findings through several numerical experiments, including: (i) deriving optimal mechanisms for the class of beta distributions, (ii) assessing the Bayesian approximation ratio of these mechanisms for small numbers of agents, and (iii)assessing how quickly the expected mechanism SW converges to its limit.
We study the computational complexity of explaining preference data through Boolean attribute models (BAMs), motivated by extensive research involving attribute models and their promise in understanding preference structure and enabling more efficient decision-making processes. In a BAM, each alternative possesses a subset of binary attributes, each voter cares about a subset of attributes, and voters prefer alternatives with more of their desired attributes. In the BAM problem, we are given a preference profile and want to know whether there is a k-attribute model explaining the profile. We establish a complexity dichotomy for the number of attributes k: BAM is linear-time solvable for k≤2 but NP-complete for k≥3. The problem remains hard even when preference orders have length two. On the positive side, BAM becomes fixed-parameter tractable when parameterized by the number of alternatives m. For the special case of two voters, we provide a linear-time algorithm. We also analyze variants where partial information is given: When voter preferences over attributes are known (BAM With Cares) or when alternative attributes are specified (BAM With Has), showing that for most parameters BAM With Cares is more difficult whereas BAM With Has is more tractable except for being NP-hard even for one voter.
Computing Approximately Proportional Allocations of Indivisible Goods: Beyond Additive and Monotone Valuations
PDF ↗Although approximate notions of envy-freeness—such as envy-freeness up to one good (EF1)—have been extensively studied for indivisible goods, the seemingly simpler fairness concept of proportionality up to one good (PROP1) has received far less attention. For additive valuations, every EF1 allocation is PROP1, and well-known algorithms such as round-robin and envy-cycle elimination compute such allocations in polynomial time. PROP1 is also compatible with Pareto efficiency, as maximum Nash welfare allocations are EF1 and hence PROP1. We ask whether these favorable properties extend to non-additive valuations. We study a broad class of allocation instances with satiating goods, where agents have non-negative valuation functions that need not be monotone, allowing for negative marginal values. We present the following results: --EF1 implies PROP1 for submodular valuations over satiating goods, ensuring existence and efficient computation via envy-cycle elimination for monotone submodular valuations; --Round-robin computes a partial PROP1 allocation after the second-to-last round for satiating submodular goods and a complete PROP1 for submodular monotone valuations; --PROP1 allocations for satiating subadditive goods can be computed in polynomial-time; --Maximum Nash welfare allocations are PROP1 for monotone submodular goods, revealing yet another facet of their ``unreasonable fairness.''
Humans exhibit time-inconsistent behavior, in which planned actions diverge from executed actions. Understanding time inconsistency and designing appropriate interventions is a key research challenge in computer science and behavioral economics. Previous work focuses on progress-based tasks and derives a closed-form description of agent behavior, from which they obtain optimal intervention strategies. They model time-inconsistency using the β–δ discounting (quasi-hyperbolic discounting), but the analysis is limited to the case δ = 1. In this paper, we relax that constraint and show that a closed-form description of agent behavior remains possible for the general case 0 < δ ≤ 1. Based on this result, we derive the conditions under which agents abandon tasks and develop efficient methods for computing optimal interventions. Our analysis reveals that agent behavior and optimal interventions depend critically on the value of δ, suggesting that fixing δ = 1 in many prior studies may unduly simplify real-world decision-making processes.
We study a class of two-player zero-sum Colonel Blotto games in which, after allocating soldiers across battlefields, players engage in (possibly distinct) normal-form games on each battlefield. Per-battlefield payoffs are parameterized by the soldier allocations. This generalizes the classical Blotto setting, where outcomes depend only on relative soldier allocations. We consider both discrete and continuous allocation models and examine two types of aggregate objectives: linear aggregation and worst-case battlefield value. For each setting, we analyze the existence and computability of Nash equilibrium. The general problem is not convex-concave, which limits the applicability of standard convex optimization techniques. However, we show that in several settings it is possible to reformulate the strategy space in a way where convex-concave structure is recovered. We evaluate the proposed methods on synthetic and real-world instances inspired by security applications, suggesting that our approaches scale well in practice.
In voting with ranked ballots each agent submits a strict ranking of the form a > b > c > d over the alternatives, and the voting rule decides on the winner based on these rankings. Although this ballot format has desirable characteristics, there is a question of whether it is expressive enough for the agents. Kahng et. al. address this issue by adding intensities to the rankings. They introduce ranking with intensities ballot format, where agents can use both >> and > in their rankings to express intensive and normal preferences between consecutive alternatives in their rankings. While Kahng et. al. focus on analyzing this ballot format in the utilitarian distortion framework, in this work, we look at the potentials of using this ballot format from the metric distortion view point. We design a class of voting rules coined Positional Scoring Rules, which can be used for different problems in the metric setting, and show that by solving a zero-sum game we can find the optimal member of this class for our problem. This rule takes intensities into account and achieves a lower distortion. In addition, by proving a bound on the price of ignoring intensities, we show that we might lose a great deal in terms of distortion by not taking the intensities into account.
Multi-Agent Pointer Transformer: Seq-to-Seq Reinforcement Learning for Multi-Vehicle Dynamic Pickup-Delivery Problems
PDF ↗This paper addresses the cooperative Multi-Vehicle Dynamic Pickup and Delivery Problem with Stochastic Requests (MVDPDPSR) and proposes an end-to-end centralized decision-making framework based on sequence-to-sequence, named Multi-Agent Pointer Transformer (MAPT). MVDPDPSR is an extension of the vehicle routing problem and a spatio-temporal system optimization problem, widely applied in scenarios such as on-demand delivery. Classical operations research methods face bottlenecks in computational complexity and time efficiency when handling large-scale dynamic problems. Although existing reinforcement learning methods have achieved some progress, they still encounter several challenges: 1) Independent decoding across multiple vehicles fails to model joint action distributions; 2) The feature extraction network struggles to capture inter-entity relationships; 3) The joint action space is exponentially large. To address these issues, we designed the MAPT framework, which employs a Transformer Encoder to extract entity representations, combines a Transformer Decoder with a Pointer Network to generate joint action sequences in an AutoRegressive manner, and introduces a Relation-Aware Attention module to capture inter-entity relationships. Additionally, we guide the model's decision-making using informative priors to facilitate effective exploration. Experiments on 8 datasets demonstrate that MAPT significantly outperforms existing baseline methods in terms of performance and exhibits substantial computational time advantages compared to classical operations research methods.
Traffic flow prediction is a typical spatio-temporal prediction problem and has a wide range of applications. The core challenge lies in modeling the underlying complex spatio-temporal dependencies. Various methods have been proposed, and recent studies show that the modeling of dynamics is useful to meet the core challenge. While handling spatial dependencies and temporal dependencies using separate base model structures may hinder the modeling of spatio-temporal correlations, the modeling of dynamics can bridge this gap. Incorporating spatio-temporal heterogeneity also advances the main goal, since it can extend the parameter space and allow more flexibility. Despite these advances, two limitations persist: 1) the modeling of dynamics is often limited to the dynamics of spatial topology (e.g., adjacency matrix changes), which, however, can be extended to a broader scope; 2) the modeling of heterogeneity is often separated for spatial and temporal dimensions, but this gap can also be bridged by the modeling of dynamics. To address the above limitations, we propose a novel framework for traffic prediction, called Meta Dynamic Graph (MetaDG). MetaDG leverages dynamic graph structures of node representations to explicitly model spatio-temporal dynamics. This generates both dynamic adjacency matrices and meta-parameters, extending dynamic modeling beyond topology while unifying the capture of spatio-temporal heterogeneity into a single dimension. Extensive experiments on four real-world datasets validate the effectiveness of MetaDG.
Dynamic graph learning plays a pivotal role in modeling evolving relationships over time, especially for temporal link prediction tasks in domains such as traffic systems, social networks, and recommendation platforms. While Transformer-based models have demonstrated strong performance by capturing long-range temporal dependencies, their reliance on self-attention results in quadratic complexity with respect to sequence length, limiting scalability on high-frequency or large-scale graphs. In this work, we revisit the necessity of self-attention in dynamic graph modeling. Inspired by recent findings that attribute the success of Transformers more to their architectural design than attention itself, we propose GLFormer, a novel attention-free Transformer-style framework for dynamic graphs. GLFormer introduces an adaptive token mixer that performs context-aware local aggregation based on interaction order and time intervals. To capture long-term dependencies, we further design a hierarchical aggregation module that expands the temporal receptive field by stacking local token mixers across layers. Experiments on six widely used dynamic graph benchmarks show that GLFormer achieves competitive or superior performance, which reveals that attention-free architectures can match or surpass Transformer baselines in dynamic graph settings with significantly improved efficiency.
Structured data question answering (QA), including table QA, Knowledge Graph (KG) QA, and temporal KG QA, is a pivotal research area. Advances in large language models (LLMs) have driven significant progress in unified structural QA frameworks like TrustUQA. However, these frameworks face challenges when applied to small-scale LLMs since small-scale LLMs are prone to errors in generating structured queries. To improve the structured data QA ability of small-scale LLMs, we propose a self-correction distillation (SCD) method. In SCD, an error prompt mechanism (EPM) is designed to detect errors and provide customized error messages during inference, and a two-stage distillation strategy is designed to transfer large-scale LLMs' query-generation and error-correction capabilities to small-scale LLM. Experiments across 5 benchmarks with 3 structured data types demonstrate that our SCD achieves the best performance and superior generalization on small-scale LLM (8B) compared to other distillation methods, and closely approaches the performance of GPT4 on some datasets. Furthermore, large-scale LLMs equipped with EPM surpass the state-of-the-art results on most datasets.