Agents enter barter exchanges to swap items they have for items they want. We study Barter Exchange with Asymmetric Valuations (BAV), a centralized barter exchange where each agent has an individual valuation over items. Given a reallocation of items, let an agent's profit be their received value minus their value given away, according to said agent's valuation of items. The goal of the clearinghouse (the party facilitating the exchange) is to output a reallocation of items that maximizes welfare (sum of agent profits) subject to each agent receiving non-negative profit.

Abstract We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in ([1]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms ([2, 3]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read- k families [4] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [4].

We study the canonical fair clustering problem where each cluster is constrained to have close to population-level representation of each group. Despite significant attention, the salient issue of having incomplete knowledge about the group membership of each point has been superficially addressed. In this paper, we consider a setting where the assigned group memberships are noisy. We introduce a simple noise model that requires a small number of parameters to be given by the decision maker. We then present an algorithm for fair clustering with provable \emph{robustness} guarantees. Our framework enables the decision maker to trade off between the robustness and the clustering quality. Unlike previous work, our algorithms are backed by worst-case theoretical guarantees. Finally, we empirically verify the performance of our algorithm on real world datasets and show its superior performance over existing baselines.

We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in ([Frederick Qiu and Sahil Singla, 2022]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms ([Chandra Chekuri et al., 2010; Nicholas J. A. Harvey and Neil Olver, 2014]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [Dmitry Gavinsky et al., 2015] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [Dmitry Gavinsky et al., 2015].

In barter exchanges agents enter seeking to swap their items for other items on their wishlist. We consider a centralized barter exchange with a set of agents and items where each item has a positive value. The goal is to compute a (re)allocation of items maximizing the agents' collective utility subject to each agent's total received value being comparable to their total given value. Many such centralized barter exchanges exist and serve crucial roles; e.g., kidney exchange programs, which are often formulated as variants of directed cycle packing. We show finding a reallocation where each agent's total given and total received values are equal is NP-hard. On the other hand, we develop a randomized algorithm that achieves optimal utility in expectation and where, i) for any agent, with probability 1 their received value is at least their given value minus v^* where v^* is said agent's most valuable owned and wished-for item, and ii) each agent's given and received values are equal in expectation. Our algorithm builds on the dependent rounding techniques from \citetgandhiApproximationAlgorithmsPartial2004.

Online bipartite-matching platforms are ubiquitous and find applications in important areas such as crowdsourcing and ridesharing. In the most general form, the platform consists of three entities: two sides to be matched and a platform operator that decides the matching. The design of algorithms for such platforms has traditionally focused on the operator’s (expected) profit. Since fairness has become an important consideration that was ignored in the existing algorithms a collection of online matching algorithms have been developed that give a fair treatment guarantee for one side of the market at the expense of a drop in the operator’s profit. In this paper, we generalize the existing work to offer fair treatment guarantees to both sides of the market simultaneously, at a calculated worst case drop to operator profit. We consider group and individual Rawlsian fairness criteria. Moreover, our algorithms have theoretical guarantees and have adjustable parameters that can be tuned as desired to balance the trade-off between the utilities of the three sides. We also derive hardness results that give clear upper bounds over the performance of any algorithm.

Kidney exchange programs (KEPs) typically seek to match incompatible patient-donor pairs based on a utilitarian objective where the number or overall quality of transplants is maximized---implicitly penalizing certain classes of difficult to match (e.g., highly-sensitized) patients. Prioritizing the welfare of highly-sensitized (hard-to-match) patients has been studied as a natural \textit{fairness} criterion. We formulate the KEP problem as $k$-set packing with a probabilistic group fairness notion of proportionality fairness---namely, fair $k$-set packing (\f{}). In this work we propose algorithms that take arbitrary proportionality vectors (i.e., policy-informed demands of how to prioritize different groups) and return a probabilistically fair solution with provable guarantees. Our main contributions are randomized algorithms as well as hardness results for \f{} variants. Additionally, the tools we introduce serve to audit the price of fairness involved in prioritizing different groups in realistic KEPs and other $k$-set packing applications. We conclude with experiments on synthetic and realistic kidney exchange \textsc{FairSP} instances.

The widespread use of machine learning algorithms in settings that directly affect human lives has instigated significant interest in designing variants of these algorithms that are provably fair. Recent work in this direction has produced numerous algorithms for the fundamental problem of clustering under many different notions of fairness. Perhaps the most common family of notions currently studied is group fairness, in which proportional group representation is ensured in every cluster. We extend this direction by considering the downstream application of clustering and how group fairness should be ensured for such a setting. Specifically, we consider a common setting in which a decision-maker runs a clustering algorithm, inspects the center of each cluster, and decides an appropriate outcome (label) for its corresponding cluster. In hiring for example, there could be two outcomes, positive (hire) or negative (reject), and each cluster would be assigned one of these two outcomes. To ensure group fairness in such a setting, we would desire proportional group representation in every label but not necessarily in every cluster as is done in group fair clustering. We provide algorithms for such problems and show that in contrast to their NP-hard counterparts in group fair clustering, they permit efficient solutions. We also consider a well-motivated alternative setting where the decision-maker is free to assign labels to the clusters regardless of the centers' positions in the metric space. We show that this setting exhibits interesting transitions from computationally hard to easy according to additional constraints on the problem. Moreover, when the constraint parameters take on natural values we show a randomized algorithm for this setting that always achieves an optimal clustering and satisfies the fairness constraints in expectation. Finally, we run experiments on real world datasets that validate the effectiveness of our algorithms.

Load balancing of skewed data in MapReduce systems like Hadoop is a well-studied problem. Many heuristics already exist to improve the load balance of the reducers thereby reducing the overall execution time. In this paper, we propose a lightweight optimization approach for MapReduce systems to minimize the makespan for repetitive tasks involving a typical frequency distribution. Our idea is to analyze the observed frequency distribution for the given task so as to identify an optimal offset parameter c to add in the hash function to minimize makespan. For two different bucketing methods - modulo labeling and consecutive binning - we present efficient algorithms for finding the optimal value of c. Finally, we present simulation results for both bucketing methods. The results vary with the data distribution and the number of reducers, but generally reduce makespan by 20% on average for power-law distributions, Results are confirmed with experiments on well-known real-world data sets.

A determinacy race occurs if two or more logically parallel instructions access the same memory location and at least one of them tries to modify its content. Races are often undesirable as they can lead to nondeterministic and incorrect program behavior. A data race is a special case of a determinacy race which can be eliminated by associating a mutual-exclusion lock with the memory location in question or allowing atomic accesses to it. However, such solutions can reduce parallelism by serializing all accesses to that location. For associative and commutative updates to a memory cell, one can instead use a reducer, which allows parallel race-free updates at the expense of using some extra space. More extra space usually leads to more parallel updates, which in turn contributes to potentially lowering the overall execution time of the program. We start by asking the following question. Given a fixed budget of extra space for mitigating the cost of races in a parallel program, which memory locations should be assigned reducers and how should the space be distributed among those reducers in order to minimize the overall running time? We argue that under reasonable conditions the races of a program can be captured by a directed acyclic graph (DAG), with nodes representing memory cells and arcs representing read-write dependencies between cells. We then formulate our original question as an optimization problem on this DAG. We concentrate on a variation of this problem where space reuse among reducers is allowed by routing every unit of extra space along a (possibly different) source to sink path of the DAG and using it in the construction of multiple (possibly zero) reducers along the path. We consider two different ways of constructing a reducer and the corresponding duration functions (i.e., reduction time as a function of space budget). We generalize our race-avoiding space-time tradeoff problem to a discrete resource-time tradeoff problem with general non-increasing duration functions and resource reuse over paths of the given DAG. For general DAGs, we show that even if the entire DAG is available offline the problem is strongly NP-hard under all three duration functions, and we give approximation algorithms for solving the corresponding optimization problems. We also prove hardness of approximation for the general resource-time tradeoff problem and give a pseudo-polynomial time algorithm for series-parallel DAGs.