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This work examines the Conditional Approval Framework for elections involving multiple interdependent issues, specifically focusing on the Conditional Minisum Approval Voting Rule. We first conduct a detailed analysis of the computational complexity of this rule, demonstrating that no approach can significantly outperform the brute-force algorithm under common computational complexity assumptions and various natural input restrictions. In response, we propose two practical restrictions (the first in the literature) that make the problem computationally tractable and show that these restrictions are essentially tight. Overall, this work provides a clear picture of the tractability landscape of the problem, contributing to a comprehensive understanding of the complications introduced by conditional ballots and indicating that conditional approval voting can be applied in practice, albeit under specific conditions.

The Pandora's Box problem models the search for the best alternative when evaluation is costly. In its simplest variant, a decision maker is presented with $n$ boxes, each associated with a cost of inspection and a distribution over the reward hidden within. The decision maker inspects a subset of these boxes one after the other, in a possibly adaptive ordering, and obtains as utility the difference between the largest reward uncovered and the sum of the inspection costs. While this classic version of the problem is well understood (Weitzman 1979), recent years have seen a flourishing of the literature on variants of the problem. In this paper, we introduce a general framework -- the Pandora's Box Over Time problem -- that captures a wide range of variants where time plays a role, e.g., as it might constrain the schedules of exploration and influence both costs and rewards. In the Pandora's Box Over Time problem, each box is characterized by time-dependent rewards and costs, and inspecting it might require a box-specific processing time. Moreover, once a box is inspected, its reward may deteriorate over time, possibly differently for each box. Our main result is an efficient $21.3$-approximation to the optimal strategy, which is NP-hard to compute in general. We further obtain improved results for the natural special cases where boxes have no processing time, or when costs and reward distributions do not depend on time (but rewards may deteriorate after inspecting).

We study the problem of allocating a set of indivisible goods to a set of agents with additive valuation functions, aiming to achieve approximate envy-freeness up to any good ($\alpha$-EFX). The state-of-the-art results on the problem include that (exact) EFX allocations exist when (a) there are at most three agents, or (b) the agents' valuation functions can take at most two values, or (c) the agents' valuation functions can be represented via a graph. For $\alpha$-EFX, it is known that a $0.618$-EFX allocation exists for any number of agents with additive valuation functions. In this paper, we show that $2/3$-EFX allocations exist when (a) there are at most \emph{seven agents}, (b) the agents' valuation functions can take at most \emph{three values}, or (c) the agents' valuation functions can be represented via a \emph{multigraph}. Our results can be interpreted in two ways. First, by relaxing the notion of EFX to $2/3$-EFX, we obtain existence results for strict generalizations of the settings for which exact EFX allocations are known to exist. Secondly, by imposing restrictions on the setting, we manage to beat the barrier of $0.618$ and achieve an approximation guarantee of $2/3$. Therefore, our results push the \emph{frontier} of existence and computation of approximate EFX allocations, and provide insights into the challenges of settling the existence of exact EFX allocations., Comment: The conference version of this work has been accepted to the Twenty-Fifth ACM Conference on Economics and Computation (EC 2024)

We consider a voting problem in which a set of agents have metric preferences over a set of alternatives, and are also partitioned into disjoint groups. Given information about the preferences of the agents and their groups, our goal is to decide an alternative to approximately minimize an objective function that takes the groups of agents into account. We consider two natural group-fair objectives known as Max-of-Avg and Avg-of-Max which are different combinations of the max and the average cost in and out of the groups. We show tight bounds on the best possible distortion that can be achieved by various classes of mechanisms depending on the amount of information they have access to. In particular, we consider group-oblivious full-information mechanisms that do not know the groups but have access to the exact distances between agents and alternatives in the metric space, group-oblivious ordinal-information mechanisms that again do not know the groups but are given the ordinal preferences of the agents, and group-aware mechanisms that have full knowledge of the structure of the agent groups and also ordinal information about the metric space.

Constrained maximization of submodular functions poses a central problem in combinatorial optimization. In many realistic scenarios, a number of agents need to maximize multiple submodular objectives over the same ground set. We study such a setting, where the different solutions must be disjoint, and thus, questions of algorithmic fairness arise. Inspired from the fair division literature, we suggest a simple round-robin protocol, where agents are allowed to build their solutions one item at a time by taking turns. Unlike what is typical in fair division, however, the prime goal here is to provide a fair algorithmic environment; each agent is allowed to use any algorithm for constructing their respective solutions. We show that just by following simple greedy policies, agents have solid guarantees for both monotone and non-monotone objectives, and for combinatorial constraints as general as $p$-systems (which capture cardinality and matroid intersection constraints). In the monotone case, our results include approximate EF1-type guarantees and their implications in fair division may be of independent interest. Further, although following a greedy policy may not be optimal in general, we show that consistently performing better than that is computationally hard.

We study elections where voters are faced with the challenge of expressing preferences over an extreme number of issues under consideration. This is largely motivated by emerging blockchain governance systems, which include voters with different weights and a massive number of community generated proposals. In such scenarios, it is natural to expect that voters will have incomplete preferences, as they may only be able to evaluate or be confident about a very small proportion of the alternatives. As a result, the election outcome may be significantly affected, leading to suboptimal decisions. Our central inquiry revolves around whether delegation of ballots to proxies possessing greater expertise or a more comprehensive understanding of the voters' preferences can lead to outcomes with higher legitimacy and enhanced voters' satisfaction in elections where voters submit incomplete preferences. To explore its aspects, we introduce the following model: potential proxies advertise their ballots over multiple issues, and each voter either delegates to a seemingly attractive proxy or casts a ballot directly. We identify necessary and sufficient conditions that could lead to a socially better outcome by leveraging the participation of proxies. We accompany our theoretical findings with experiments on instances derived from real datasets. Overall, our results enhance the understanding of the power of delegation towards improving election outcomes.

Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer (2010). An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, they might offer multiple levels of a service, or they may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings where each agent's services are either fully acquired or not at all. The main reason for this, is that in settings with partial allocations like the ones mentioned, there are strong inapproximability results. Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results in this work. We design a polynomial-time, deterministic, truthful, budget-feasible $(2+\sqrt{3})$-approximation mechanism for the setting where each agent offers multiple levels of service and the auctioneer has a discrete separable concave valuation function. We then use this result to design a deterministic, truthful and budget-feasible mechanism for the setting where any fraction of a service can be acquired and the auctioneer's valuation function is separable concave (i.e., the sum of concave functions). The approximation ratio of this mechanism depends on how `nice' the concave functions are, and is $O(1)$ for valuation functions that are sums of $O(1)$-regular functions (e.g., functions like $\log(1+x)$). For the special case of a linear valuation function, we improve the best known approximation ratio for the problem from $1+\phi$ (by Klumper & Sch\"afer (2022)) to $2$. This establishes a separation between this setting and its indivisible counterpart.

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide non-trivial fairness guarantees. Recently, Amanatidis et al. [2021] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents' true valuation functions. Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents' true valuation functions. Further, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist!

Allocating resources to individuals in a fair manner has been a topic of interest since ancient times, with most of the early mathematical work on the problem focusing on resources that are infinitely divisible. Over the last decade, there has been a surge of papers studying computational questions regarding the indivisible case, for which exact fairness notions such as envy-freeness and proportionality are hard to satisfy. One main theme in the recent research agenda is to investigate the extent to which their relaxations, like maximin share fairness (MMS) and envy-freeness up to any good (EFX), can be achieved. In this survey, we present a comprehensive review of the recent progress made in the related literature by highlighting different ways to relax fairness notions, common algorithm design techniques, and the most interesting questions for future research., Comment: This survey unifies and extends preliminary versions that appeared in IJCAI 2022 (arXiv:2202.07551) and SIGecom Exchanges (arXiv:2202.08713). It has been accepted for publication to the journal of Artificial Intelligence (AIJ)

Motivated by governance models adopted in blockchain applications, we study the problem of selecting appropriate system updates in a decentralised way. Contrary to most existing voting approaches, we use the input of a set of motivated experts of varying levels of expertise. In particular, we develop an approval voting inspired selection mechanism through which the experts approve or disapprove the different updates according to their perception of the quality of each alternative. Given their opinions, and weighted by their expertise level, a single update is then implemented and evaluated, and the experts receive rewards based on their choices. We show that this mechanism always has approximate pure Nash equilibria and that these achieve a constant factor approximation with respect to the quality benchmark of the optimal alternative. Finally, we study the repeated version of the problem, where the weights of the experts are adjusted after each update, according to their performance. Under mild assumptions about the weights, the extension of our mechanism still has approximate pure Nash equilibria in this setting.

In most social choice settings, the participating agents express their preferences over the different alternatives in the form of linear orderings. While this clearly simplifies preference elicitation, it inevitably leads to poor performance with respect to optimizing a cardinal objective, such as the social welfare, since the values of the agents remain virtually unknown. This loss in performance because of lack of information is measured by the notion of distortion. A recent array of works put forward the agenda of designing mechanisms that learn the values of the agents for a small number of alternatives via queries, and use this limited extra information to make better-informed decisions, thus improving distortion. Following this agenda, in this work we focus on a class of combinatorial problems that includes most well-known matching problems and several of their generalizations, such as One-Sided Matching, Two-Sided Matching, General Graph Matching, and $k$-Constrained Resource Allocation. We design two-query mechanisms that achieve the best-possible worst-case distortion in terms of social welfare, and outperform the best-possible expected distortion achieved by randomized ordinal mechanisms., Comment: A conference version of this work was accepted in the 36th Conference on Neural Information Processing Systems (NeurIPS 2022)

Allocating resources to individuals in a fair manner has been a topic of interest since the ancient times, with most of the early rigorous mathematical work on the problem focusing on infinitely divisible resources. Recently, there has been a surge of papers studying computational questions regarding various different notions of fairness for the indivisible case, like maximin share fairness (MMS) and envy-freeness up to any good (EFX). We survey the most important results in the discrete fair division literature, focusing on the case of additive valuation functions and paying particular attention to the progress made in the last 10 years.

The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, in which every node $i$ has a degree from an interval not too far away from a given $d \in \N$. In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, we provide the first non-trivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given near-regular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly log-concave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\"uller-Hannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of near-regular degree intervals of size at most one., Comment: Accepted at STACS 2023

We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers and, therefore, a mechanism in our setting is an algorithm that takes as input the reported -- rather than the true -- values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX), and we positively answer the above question. In particular, we study two algorithms that are known to produce such allocations in the non-strategic setting: Round-Robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden [SIAM Journal of Discrete Mathematics, 2020] (EFX allocations for two agents). For Round-Robin we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, while for the algorithm of Plaut and Roughgarden we show that the corresponding allocations not only are EFX but also satisfy maximin share fairness, something that is not true for this algorithm in the non-strategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria which all induce EFX allocations., Comment: The conference version of this work was presented at the 17th Conference on Web and Internet Economics (WINE 2021). The journal version has been accepted to Mathematics of Operations Research

Submodular maximization is a classic algorithmic problem with multiple applications in data mining and machine learning; there, the growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the adaptive complexity, which captures the number of sequential rounds of parallel computation needed by an algorithm to terminate. In this work we obtain the first constant factor approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with near-optimal $O(\log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: a crucial feature to account for is represented by the total number of function evaluations (or value queries). Our algorithm asks $\tilde{O}(n^2)$ value queries, but can be modified to run with only $\tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(\log^2n)$. Besides the above improvement in adaptivity, this is also the first combinatorial approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives., Comment: This version addresses a gap in the probabilistic analysis of the approximation guarantees in the previous version of this work. We provide a simple fix via a standard sampling routine while maintaining the same approximation guarantees and complexity bounds. (formerly appeared as arXiv:2007.05014v2 in error)

We consider the one-sided matching problem, where n agents have preferences over n items, and these preferences are induced by underlying cardinal valuation functions. The goal is to match every agent to a single item so as to maximize the social welfare. Most of the related literature, however, assumes that the values of the agents are not a priori known, and only access to the ordinal preferences of the agents over the items is provided. Consequently, this incomplete information leads to loss of efficiency, which is measured by the notion of distortion. In this paper, we further assume that the agents can answer a small number of queries, allowing us partial access to their values. We study the interplay between elicited cardinal information (measured by the number of queries per agent) and distortion for one-sided matching, as well as a wide range of well-studied related problems. Qualitatively, our results show that with a limited number of queries, it is possible to obtain significant improvements over the classic setting, where only access to ordinal information is given.

Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n \log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a 9-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data., Comment: Same as v1. Version 2 was a replacement intended for arXiv:2102.08327 and erroneously updated here

We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations.

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a $(\phi-1)$-approximation of envy-freeness up to any good (EFX) and a $\frac{2}{\phi +2}$-approximation of groupwise maximin share fairness (GMMS), where $\phi$ is the golden ratio ($\phi \approx 1.618$). The best known approximation factor for either one of these fairness notions prior to this work was $1/2$. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a $2/3$-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve the guarantees for GMMS or PMMS. Finally, we show that GMMS -- and thus PMMS and EFX -- allocations always exist when the number of goods does not exceed the number of agents by more than two.

Aggregating the preferences of individuals into a collective decision is the core subject of study of social choice theory. In 2006, Procaccia and Rosenschein considered a utilitarian social choice setting, where the agents have explicit numerical values for the alternatives, yet they only report their linear orderings over them. To compare different aggregation mechanisms, Procaccia and Rosenschein introduced the notion of distortion, which quantifies the inefficiency of using only ordinal information when trying to maximize the social welfare, i.e., the sum of the underlying values of the agents for the chosen outcome. Since then, this research area has flourished and bounds on the distortion have been obtained for a wide variety of fundamental scenarios. However, the vast majority of the existing literature is focused on the case where nothing is known beyond the ordinal preferences of the agents over the alternatives. In this paper, we take a more expressive approach, and consider mechanisms that are allowed to further ask a few cardinal queries in order to gain partial access to the underlying values that the agents have for the alternatives. With this extra power, we design new deterministic mechanisms that achieve significantly improved distortion bounds and, in many cases, outperform the best-known randomized ordinal mechanisms. We paint an almost complete picture of the number of queries required by deterministic mechanisms to achieve specific distortion bounds.

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible and $O(1)$-approximate mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only $O(1)$-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). Ours is the first mechanism for the problem where---crucially---the agents are not ordered with respect to their marginal value per cost. This allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present. We obtain $O(p)$-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a $p$-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about non-trivial approximation guarantees in polynomial time, our results are asymptotically best possible., Comment: Accepted to EC 2019

We consider the classic cake-cutting problem of producing envy-free allocations, restricted to the case of four agents. The problem asks for a partition of the cake to four agents, so that every agent finds her piece at least as valuable as every other agent's piece. The problem has had an interesting history so far. Although the case of three agents is solvable with less than 15 queries, for four agents no bounded procedure was known until the recent breakthroughs of Aziz and Mackenzie (STOC 2016, FOCS 2016). The main drawback of these new algorithms, however, is that they are quite complicated and with a very high query complexity. With four agents, the number of queries required is close to 600. In this work we provide an improved algorithm for four agents, which reduces the current complexity by a factor of 3.4. Our algorithm builds on the approach of Aziz and Mackenzie (STOC 2016) by incorporating new insights and simplifying several steps. Overall, this yields an easier to grasp procedure with lower complexity.

In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight, or almost tight, results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions., Comment: accepted in IJCAI 2018

The switch Markov chain has been extensively studied as the most natural Markov Chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We use comparison arguments with other, less natural but simpler to analyze, Markov chains, to show that the switch chain mixes rapidly in two different settings. We first study the classic problem of uniformly sampling simple undirected, as well as bipartite, graphs with a given degree sequence. We apply an embedding argument, involving a Markov chain defined by Jerrum and Sinclair (TCS, 1990) for sampling graphs that almost have a given degree sequence, to show rapid mixing for degree sequences satisfying strong stability, a notion closely related to $P$-stability. This results in a much shorter proof that unifies the currently known rapid mixing results of the switch chain and extends them up to sharp characterizations of $P$-stability. In particular, our work resolves an open problem posed by Greenhill (SODA, 2015). Secondly, in order to illustrate the power of our approach, we study the problem of uniformly sampling graphs for which, in addition to the degree sequence, a joint degree distribution is given. Although the problem was formalized over a decade ago, and despite its practical significance in generating synthetic network topologies, small progress has been made on the random sampling of such graphs. The case of a single degree class reduces to sampling of regular graphs, but beyond this almost nothing is known. We fully resolve the case of two degree classes, by showing that the switch Markov chain is always rapidly mixing. Again, we first analyze an auxiliary chain for strongly stable instances on an augmented state space and then use an embedding argument., Comment: Accepted to SODA 2019

We study the mechanism design problem of allocating a set of indivisible items without monetary transfers. Despite the vast literature on this very standard model, it still remains unclear how do truthful mechanisms look like. We focus on the case of two players with additive valuation functions and our purpose is twofold. First, our main result provides a complete characterization of truthful mechanisms that allocate all the items to the players. Our characterization reveals an interesting structure underlying all truthful mechanisms, showing that they can be decomposed into two components: a selection part where players pick their best subset among prespecified choices determined by the mechanism, and an exchange part where players are offered the chance to exchange certain subsets if it is favorable to do so. In the remaining paper, we apply our main result and derive several consequences on the design of mechanisms with fairness guarantees. We consider various notions of fairness, (indicatively, maximin share guarantees and envy-freeness up to one item) and provide tight bounds for their approximability. Our work settles some of the open problems in this agenda, and we conclude by discussing possible extensions to more players., Comment: To appear in the 18th ACM Conference on Economics and Computation (ACM EC '17)

We study a class of procurement auctions with a budget constraint, where an auctioneer is interested in buying resources or services from a set of agents. Ideally, the auctioneer would like to select a subset of the resources so as to maximize his valuation function, without exceeding a given budget. As the resources are owned by strategic agents however, our overall goal is to design mechanisms that are truthful, budget-feasible, and obtain a good approximation to the optimal value. Budget-feasibility creates additional challenges, making several approaches inapplicable in this setting. Previous results on budget-feasible mechanisms have considered mostly monotone valuation functions. In this work, we mainly focus on symmetric submodular valuations, a prominent class of non-monotone submodular functions that includes cut functions. We begin first with a purely algorithmic result, obtaining a $\frac{2e}{e-1}$-approximation for maximizing symmetric submodular functions under a budget constraint. We view this as a standalone result of independent interest, as it is the best known factor achieved by a deterministic algorithm. We then proceed to propose truthful, budget feasible mechanisms (both deterministic and randomized), paying particular attention on the Budgeted Max Cut problem. Our results significantly improve the known approximation ratios for these objectives, while establishing polynomial running time for cases where only exponential mechanisms were known. At the heart of our approach lies an appropriate combination of local search algorithms with results for monotone submodular valuations, applied to the derived local optima., Comment: A conference version appears in WINE 2017

We study a type of reverse (procurement) auction problems in the presence of budget constraints. The general algorithmic problem is to purchase a set of resources, which come at a cost, so as not to exceed a given budget and at the same time maximize a given valuation function. This framework captures the budgeted version of several well known optimization problems, and when the resources are owned by strategic agents the goal is to design truthful and budget feasible mechanisms, i.e. elicit the true cost of the resources and ensure the payments of the mechanism do not exceed the budget. Budget feasibility introduces more challenges in mechanism design, and we study instantiations of this problem for certain classes of submodular and XOS valuation functions. We first obtain mechanisms with an improved approximation ratio for weighted coverage valuations, a special class of submodular functions that has already attracted attention in previous works. We then provide a general scheme for designing randomized and deterministic polynomial time mechanisms for a class of XOS problems. This class contains problems whose feasible set forms an independence system (a more general structure than matroids), and some representative problems include, among others, finding maximum weighted matchings, maximum weighted matroid members, and maximum weighted 3D-matchings. For most of these problems, only randomized mechanisms with very high approximation ratios were known prior to our results.

We study a revenue maximization problem in the context of social networks. Namely, we consider a model introduced by Alon, Mansour, and Tennenholtz (EC 2013) that captures inequity aversion, i.e., prices offered to neighboring vertices should not be significantly different. We first provide approximation algorithms for a natural class of instances, referred to as the class of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for example, to settings where the seller will only consider a fixed number of discount types or special offers. We then resolve one of the open questions posed in Alon et al., by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be large, or even when the number of allowed distinct prices is as small as three. Finally, we provide some extensions of the model, regarding either the allowed set of prices or the demand type of the clients., Comment: A preliminary conference version of this work appeared in MFCS 2016

We study a fair division problem with indivisible items, namely the computation of maximin share allocations. Given a set of $n$ players, the maximin share of a single player is the best she can guarantee to herself, if she would partition the items in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then is to find an allocation, so that each player is guaranteed her maximin share. Previous works have studied this problem mostly algorithmically, providing constant factor approximation algorithms. In this work we embark on a mechanism design approach and investigate the existence of truthful mechanisms. We propose three models regarding the information that the mechanism attempts to elicit from the players, based on the cardinal and ordinal representation of preferences. We establish positive and negative (impossibility) results for each model and highlight the limitations imposed by truthfulness on the approximability of the problem. Finally, we pay particular attention to the case of two players, which already leads to challenging questions.

In this paper we introduce extensions and modifications of the classical degree sequence graphic realization problem studied by Erd\H{o}s-Gallai and Havel-Hakimi, as well as of the corresponding connected graphic realization version. We define the joint-degree matrix graphic (resp. connected graphic) realization problem, where in addition to the degree sequence, the exact number of desired edges between vertices of different degree classes is also specified. We give necessary and sufficient conditions, and polynomial time decision and construction algorithms for the graphic and connected graphic realization problems. These problems arise naturally in the current topic of graph modeling for complex networks. From the technical point of view, the joint-degree matrix realization algorithm is straightforward. However, the connected joint-degree matrix realization algorithm involves a novel recursive search of suitable local graph modifications. Also, we outline directions for further work of both theoretical and practical interest. In particular, we give a Markov chain which converges to the uniform distribution over all realizations. We show that the underline state space is connected, and we leave the question of the mixing rate open., Comment: Unpublished manuscript of 2009

We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of $n$ agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into $n$ bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, so we resort to approximation algorithms. Our main result is a $2/3$-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang, which also produces a $2/3$-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of their algorithm. Furthermore, motivated by the apparent difficulty, both theoretically and experimentally, in finding lower bounds on the existence of approximate solutions, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in relevant works. Finally, we provide further positive results for two special cases that arise from previous works. The first one is the intriguing case of $3$ agents, for which it is already known that exact maximin share allocations do not always exist (contrary to the case of $2$ agents). We provide a $7/8$-approximation algorithm, improving the previously known result of $3/4$. The second case is when all item values belong to $\{0, 1, 2\}$, extending the $\{0, 1\}$ setting studied in Bouveret and Lema\^itre. We obtain an exact algorithm for any number of agents in this case.