We study the fair division of indivisible chores among agents with additive disutility functions. We investigate the existence of allocations satisfying the popular fairness notion of envy-freeness up to any chore (EFX), and its multiplicative approximations. The existence of 4-EFX allocations was recently established by Garg, Murhekar, and Qin (2025). We improve this guarantee by proving the existence of 2-EFX allocations for all instances with additive disutilities. This approximation was previously known only for restricted instances such as bivalued disutilities (Lin, Wu, and Zhou (2025)) or three agents (Afshinmehr, Ansaripour, Danaei, and Mehlhorn (2024)). We obtain our result by providing a general framework for achieving approximate-EFX allocations. The approach begins with a suitable initial allocation and performs a sequence of local swaps between the bundles of envious and envied agents. For our main result, we begin with an initial allocation that satisfies envy-freeness up to one chore (EF1) and Pareto-optimality (PO); the existence of such an allocation was recently established in a major breakthrough by Mahara (2025). We further demonstrate the strength and generality of our framework by giving simple and unified proofs of existing results, namely (i) 2-EFX for bivalued instances, (ii) 2-EFX for three agents, (iii) EFX when the number of chores is at most twice the number of agents, and (iv) 4-EFX for all instances. We expect this framework to have broader applications in approximate-EFX due to its simplicity and generality.

Data powers almost everything we experience on the web today---from the recommendations and ads we see to the AI systems and online marketplaces that shape our digital interactions. The increasing demand for high-quality data has given rise to platforms that facilitate the buying and selling of data. A key practical challenge in such markets is determining how to price data. Competitive equilibrium (CE), a foundational concept in classical market economics, determines prices for rivalrous goods by matching their supply and demand. In this work, we initiate the study of CE in data markets, explicitly incorporating the role of data in improving predictive performance in buyers' utility functions, and the non-rival nature of data by adapting the standard market-clearing condition to allow the simultaneous allocation of data records to multiple buyers. We analyze the existence, structure, and computation of CE in such data markets. We establish that CE always exists, and almost all instances admit a unique and rational equilibrium price vector. In general, however, there could be a non-convex set of prices, which rules out convex-programming approaches for finding a CE. Despite these challenges, we design an FPTAS for computing approximate equilibria using a Walrasian-style price adjustment algorithm. Our framework opens avenues for studying richer buyer utilities under correlated data sellers, and deeper structural and algorithmic aspects of data markets.

Federated learning (FL) is a popular collaborative learning paradigm, whereby agents with individual datasets can jointly train an ML model. While higher data sharing improves model accuracy and leads to higher payoffs, it also raises costs associated with data acquisition or loss of privacy, causing agents to be strategic about their data contribution. This leads to undesirable behavior at a Nash equilibrium (NE) such as *free-riding*, resulting in sub-optimal fairness, data sharing, and welfare. To address this, we design $\mathcal{M}^{Shap}$, a budget-balanced payment mechanism for FL, that admits Nash equilibria under mild conditions, and achieves *reciprocal fairness*: where each agent's payoff equals her contribution to the collaboration, as measured by the Shapley share. In addition to fairness, we show that the NE under $\mathcal{M}^{Shap}$ has desirable guarantees in terms of accuracy, welfare, and total data collected. We validate our theoretical results through experiments, demonstrating that $\mathcal{M}^{Shap}$ outperforms baselines in terms of fairness and efficiency.

Organizations increasingly seek to share and access datasets to improve their ML models and derive insights. Despite the immense demand for quality data, data exchange and collaboration have not reached their full potential. One of the key reasons is the lack of reciprocity, where some participants perceive their contribution to others to be of higher value than what they receive in return.

We study the problem of scheduling jobs on a machine with time-varying capacity. The capacity at each time slot represents the maximum number of jobs that can be scheduled in parallel at that time slot. Each job is associated with a release time, processing time, deadline, and a profit. The objective is to find a non-preemptive schedule that respects all capacity constraints and maximizes the throughput, defined as the total profit of jobs that complete within their deadline. We consider different variants of the problem depending on job profits (identical / arbitrary), capacities (small / large) and environment (offline / online).

Previous work on fairness in federated learning introduced the notion of core stability, which provides utility-based fairness guarantees to any subset of participating agents. However, these guarantees require strong assumptions on agent utilities that render them impractical. To address this shortcoming, we measure the quality of output models in terms of their ordinal rank instead of their cardinal utility, and use this insight to adapt the classical notion of proportional veto core (PVC) from social choice theory to the federated learning setting. We prove that models that are PVC-stable exist in very general learning paradigms, even allowing non-convex model sets, as well as non-convex and non-concave loss functions. We also design Rank-Core-Fed, a distributed federated learning algorithm, to train a PVC-stable model. Finally, we demonstrate that Rank-Core-Fed outperforms baselines in terms of fairness on different datasets.

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation and a non-constructive proof of the existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO), which is a stronger notion than PO. We present a pseudopolynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive and that it can be computed in pseudo-polynomial time. We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. For such instances, we show that an EF1+fPO allocation can be computed in strongly polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in strongly polynomial time. Next, we consider instances where the number of agents is constant and show that an EF1+PO (likewise, an EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. We also design a polynomial-time algorithm that computes a Nash welfare maximizing allocation when there are constantly many agents with constant many different values for the goods. Finally, on the complexity side, we show that the problem of computing an EF1+fPO allocation lies in the complexity class PLS.

We study the problem of fairly and efficiently allocating indivisible chores among agents with additive disutility functions. We consider the widely used envy-based fairness properties of EF1 and EFX in conjunction with the efficiency property of fractional Pareto-optimality (fPO). Existence (and computation) of an allocation that is simultaneously EF1/EFX and fPO are challenging open problems, and we make progress on both of them. We show the existence of an allocation that is - EF1 + fPO, when there are three agents, - EF1 + fPO, when there are at most two disutility functions, - EFX + fPO, for three agents with bivalued disutility functions. These results are constructive, based on strongly polynomial-time algorithms. We also investigate non-existence and show that an allocation that is EFX+fPO need not exist, even for two agents.

Federated learning (FL) has emerged as a powerful scheme to facilitate the collaborative learning of models amongst a set of agents holding their own private data. Although the agents benefit from the global model trained on shared data, by participating in federated learning, they may also incur costs (related to privacy and communication) due to data sharing. In this paper, we model a collaborative FL framework, where every agent attempts to achieve an optimal trade-off between her learning payoff and data sharing cost. We show the existence of Nash equilibrium (NE) under mild assumptions on agents' payoff and costs. Furthermore, we show that agents can discover the NE via best response dynamics. However, some of the NE may be bad in terms of overall welfare for the agents, implying little incentive for some fraction of the agents to participate in the learning. To remedy this, we design a budget-balanced mechanism involving payments to the agents, that ensures that any $p$-mean welfare function of the agents' utilities is maximized at NE. In addition, we introduce a FL protocol FedBR-BG that incorporates our budget-balanced mechanism, utilizing best response dynamics. Our empirical validation on MNIST and CIFAR-10 substantiates our theoretical analysis. We show that FedBR-BG outperforms the basic best-response-based protocol without additional incentivization, the standard federated learning protocol FedAvg, as well as a recent baseline MWFed in terms of achieving superior $p$-mean welfare.

Several cloud-based applications, such as cloud gaming, rent servers to execute jobs which arrive in an online fashion. Each job has a resource demand, such as GPU requirement, and must be dispatched to a cloud server which has enough resources to execute the job, which departs after its completion. Under the "pay-as-you-go'' billing model, the server rental cost is proportional to the total time that servers are actively running jobs. The problem of efficiently allocating a sequence of online jobs to servers without exceeding the resource capacity of any server while minimizing total server usage time can be modelled as a variant of the dynamic bin packing problem (DBP), called MinUsageTime DBP [10].

We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that EF1+PO allocations exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of chores to admit an EF1+PO allocation and an efficient algorithm for its computation. We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time.

We study the problem of fair and efficient allocation of a set of indivisible goods to agents with additive valuations using the popular fairness notions of envy-freeness up to one good (EF1) and equitability up to one good (EQ1) in conjunction with Pareto-optimality (PO). There exists a pseudo-polynomial time algorithm to compute an EF1+PO allocation, and a non-constructive proof of existence of allocations that are both EF1 and fractionally Pareto-optimal (fPO). We present a pseudo-polynomial time algorithm to compute an EF1+fPO allocation, thereby improving the earlier results. Our techniques also enable us to show that an EQ1+fPO allocation always exists when the values are positive, and that it can be computed in pseudo-polynomial time. We also consider the class of k-ary instances where k is a constant, i.e., each agent has at most k different values for the goods. We show that for such instances an EF1+fPO allocation can be computed in polynomial time. When all values are positive, we show that an EQ1+fPO allocation for such instances can be computed in polynomial time. Next, we consider instances where the number of agents is constant, and show that an EF1+PO (also EQ1+PO) allocation can be computed in polynomial time. These results significantly extend the polynomial-time computability beyond the known cases of binary or identical valuations. Further, we show that the problem of computing an EF1+PO allocation polynomial-time reduces to a problem in the complexity class PLS. We also design a polynomial-time algorithm that computes Nash welfare maximizing allocations when there are constantly many agents with constant many different values for the goods.

We study fair allocation of indivisible public goods subject to cardinality (budget) constraints. In this model, we have n agents and m available public goods, and we want to select k ≤ m goods in a fair and efficient manner. We first establish fundamental connections between the models of private goods, public goods, and public decision making by presenting polynomial-time reductions for the popular solution concepts of maximum Nash welfare (MNW) and leximin. These mechanisms are known to provide remarkable fairness and efficiency guarantees in private goods and public decision making settings. We show that they retain these desirable properties even in the public goods case. We prove that MNW allocations provide fairness guarantees of Proportionality up to one good (Prop1), 1/n approximation to Round Robin Share (RRS), and the efficiency guarantee of Pareto Optimality (PO). Further, we show that the problems of finding MNW or leximin-optimal allocations are NP-hard, even in the case of constantly many agents, or binary valuations. This is in sharp contrast to the private goods setting that admits polynomial-time algorithms under binary valuations. We also design pseudo-polynomial time algorithms for computing an exact MNW or leximin-optimal allocation for the cases of (i) constantly many agents, and (ii) constantly many goods with additive valuations. We also present an O(n)-factor approximation algorithm for MNW which also satisfies RRS, Prop1, and 1/2-Prop.

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds.

Neural sequence-to-sequence models have been successfully extended for summary generation. However, existing frameworks generate a single summary for a given input and do not tune the summaries towards any additional constraints/preferences. Such a tunable framework is desirable to account for linguistic preferences of the specific audience who will consume the summary. In this paper, we propose a neural framework to generate summaries constrained to a vocabulary-defined linguistic preferences of a target audience. The proposed method accounts for the generation context by tuning the summary words at the time of generation. Our evaluations indicate that the proposed approach tunes summaries to the target vocabulary while still maintaining a superior summary quality against a state-of-the-art word embedding based lexical substitution algorithm, suggesting the feasibility of the proposed approach. We demonstrate two applications of the proposed approach - to generate understandable summaries with simpler words, and readable summaries with shorter words.