Abstract We consider the fundamental problem of fairly allocating a set of indivisible items among agents having valuations that are represented by a multi-graph – here, agents appear as vertices and items as edges between them and each vertex (agent) only values the set of its incident edges (items). The goal is to find a fair, i.e., envy-free up to any item ( $$\textsf {EFX}$$ ) allocation. This model has recently been introduced by [22] where they show that $$\textsf {EFX}$$ allocations always exist on simple graphs for monotone valuations, i.e., where any two agents can share at most one edge (item). A natural question arises as to what happens when we go beyond simple graphs and study various classes of multi-graphs? We answer the above question affirmatively for the valuation class of bipartite multi-graphs and multi-cycles . The main contribution of this work is to establish the existence of $$\textsf {EFX}$$ allocations on bipartite multi-graphs for monotone valuations and on multi-cycles for $$\textsf {MMS}$$ -feasible valuations. We also present pseudo-polynomial time algorithms to compute $$\textsf {EFX}$$ allocations for the above settings. Furthermore, we show that for bipartite multi-graphs with cancelable valuations, $$\textsf {EFX}$$ allocations can be computed in polynomial time. We thus deepen the understanding of $$\textsf {EFX}$$ allocations by expanding the spectrum of settings in which they are guaranteed to exist for an arbitrary number of agents. Next, we study $$\textsf {EFX}$$ orientations (allocations where every item is assigned to one of its two endpoint agents) and provide a complete characterization of their existence on bipartite multi-graphs in terms of two key parameters—the number of edges shared between any two agents and the diameter of the graph. Finally, we prove that it is $$\textsf {NP}$$ -complete to determine whether a given fair division instance on a bipartite multi-graph admits an $$\textsf {EFX}$$ orientation, even with a constant number of agents.

We study the fundamental problem of fairly dividing a set of indivisible items among agents with (general) monotone valuations. The notion of envy-freeness up to any item (EFX) is considered to be one of the most fascinating fairness concepts in this line of work. Unfortunately, despite significant efforts, existence of EFX allocations is a major open problem in fair division, thereby making the study of approximations and relaxations of EFX a natural line of research. Recently, Caragiannis et al. [2023] introduced a promising relaxation of EFX, called epistemic EFX (EEFX). An allocation is EEFX, if for every agent, it is possible to shuffle the items in the remaining bundles so that she becomes ``EFX-satisfied''. Caragiannis et al. [2023] prove existence and polynomial-time computability of EEFX allocations for additive valuations. A natural question asks what happens when we consider valuations more general than additive? We address this important open question and answer it affirmatively by establishing the existence of EEFX allocations for an arbitrary number of agents with general monotone valuations. To the best of our knowledge, besides EF1, EEFX is the only known relaxation of EFX to have such strong existential guarantees. Furthermore, we complement our existential result by proving computational and information-theoretic lower bounds. We prove that even for an arbitrary number of (more than one) agents with identical submodular valuations, it is PLS-hard to compute EEFX allocations and it requires exponentially-many value queries to do so.

We study the problem of computing fair divisions of a set of indivisible goods among agents with additive valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having envy-based or share-based lens. For the discrete setting of resource-allocation problems, envy-free up to any good (EFX) and maximin share (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors by Chaudhury et al. and Amanatidis et al. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both 2/3-MMS and EFX, and (ii) a complete allocation that is both 2/3-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to 2/3 - e for any constant e>0 and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to (1-d)-EFX (or (1-d)-EF1) for any constant d>0. In particular, we improve from the best approximation factor known prior to our work by Chaudhury et al., which computes partial allocations that are 1/2-MMS and EFX in pseudo-polynomial time.

Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi (2023), we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (specifically, O(n^5) of queries). This solves the main open problem stated by Caragiannis and Rathi (2023). Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.

Abstract Motivated by the success of the serial dictatorship mechanism in social choice settings, we explore its usefulness in tackling various combinatorial optimization problems. We do so by considering an abstract model, in which a set of agents are asked to act in a particular ordering, called the action sequence. Each agent acts in a way that gives her the maximum possible value, given the actions of the agents who preceded her in the action sequence. Our goal is to compute action sequences that yield approximately optimal total value to the agents (a.k.a., social welfare). We assume query access to the value $$v_i(S)$$ v i ( S ) that the agent i gets when she acts after the agents in the ordered set S. We establish tight bounds on the social welfare that can be achieved using polynomially many queries. Even though these bounds show a marginally sublinear approximation of optimal social welfare in general, excellent approximations can be obtained when the valuations stem from an underlying combinatorial domain. Indicatively, when the valuations are defined using bipartite matchings, arborescences in directed graphs, and satisfiability of Boolean expressions, simple query-efficient algorithms yield 2-approximations. We discuss issues related to truthfulness and show how some of our algorithms can be implemented truthfully using VCG-like payments. Finally, we introduce and study the price of serial dictatorship, a notion that provides an optimistic measure of the quality of combinatorial optimization solutions generated by action sequences.

For the fundamental problem of fairly dividing a set of indivisible items among agents, envy-freeness up to any item (EFX) and maximin fairness (MMS) are arguably the most compelling fairness concepts proposed till now. Unfortunately, despite significant efforts over the past few years, whether EFX allocations always exist is still an enigmatic open problem, let alone their efficient computation. Furthermore, today we know that MMS allocations are not always guaranteed to exist. These facts weaken the usefulness of both EFX and MMS, albeit their appealing conceptual characteristics. We propose two alternative fairness concepts—called epistemic EFX (EEFX) and minimum EFX value fairness (MXS)---inspired by EFX and MMS. For both, we explore their relationships to well-studied fairness notions and, more importantly, prove that EEFX and MXS allocations always exist and can be computed efficiently for additive valuations. Our results justify that the new fairness concepts are excellent alternatives to EFX and MMS.

This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation of cake cutting wherein each agent must receive a contiguous piece of the cake. Although multiple hardness results exist in this setup for finding fair/efficient cake divisions, we show that, if the value densities of the agents satisfy the monotone likelihood ratio property (MLRP), then strong algorithmic results hold for various notions of fairness and economic efficiency. Addressing cake-cutting instances with MLRP, first we develop an algorithm that finds cake divisions (with connected pieces) that are envy free, up to an arbitrary precision. The time complexity of our algorithm is polynomial in the number of agents and the bit complexity of an underlying Lipschitz constant. We obtain similar positive results for maximizing social, egalitarian, and Nash social welfare. Many distribution families bear MLRP. In particular, this property holds if all the value densities belong to any one of the following families: Gaussian (with the same variance), linear, Poisson, and exponential distributions, linear translations of any log-concave function. Hence, through MLRP, the current work obtains novel cake-cutting algorithms for multiple distribution families.

We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that well-behaved utilities of n − 1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm for the case of quasilinear utilities. In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively.One of the main technical contributions of this paper is the development of novel connections between different fairdivision paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm.

We study the problem of fair rent division that entails splitting the rent and allocating the rooms of an apartment among roommates (agents) in a fair manner. In this setup, a distribution of the rent and an accompanying allocation is said to be fair if it is envy free, i.e., under the imposed rents, no agent has a strictly stronger preference for any other agent's room. The cardinal preferences of the agents are expressed via functions which specify the utilities of the agents for the rooms for every possible room rent/price. While envy-free solutions are guaranteed to exist under reasonably general utility functions, efficient algorithms for finding them were known only for quasilinear utilities. This work addresses this notable gap and develops approximation algorithms for fair rent division with minimal assumptions on the utility functions. Specifically, we show that if the agents have continuous, monotone decreasing, and piecewise-linear utilities, then the fair rent-division problem admits a fully polynomial-time approximation scheme (FPTAS). That is, we develop algorithms that find allocations and prices of the rooms such that for each agent a the utility of the room assigned to it is within a factor of (1 + ε) of the utility of the room most preferred by a. Here, ε > 0 is an approximation parameter, and the running time of the algorithms is polynomial in 1/ε and the input size. In addition, we show that the methods developed in this work provide efficient, truthful mechanisms for special cases of the rent-division problem. Envy-free solutions correspond to equilibria of a two-sided matching market with monetary transfers; hence, this work also provides efficient algorithms for finding approximate equilibria in such markets. We complement the algorithmic results by proving that the fair rent division problem (under continuous, monotone decreasing, and piecewise-linear utilities) lies in the intersection of the complexity classes PPAD and PLS.