Reconstructing the evolutionary history of tumors using single-cell sequencing (SCS) data presents significant computational challenges. Existing approaches are either computationally intractable for emerging large-scale datasets or rely on heuristics that lack optimality guarantees. In this work, we propose a novel, time-efficient algorithm that constructs the phylogenetic tree of tumor evolution with a provable guarantee of optimality. Our main result is a branch-and-bound algorithm that reconstructs the most likely tumor evolutionary history up to two orders of magnitude faster than the previous best algorithm. To achieve this, we use efficient and well-known 2-approximation algorithms for the Vertex Cover problem to prune the branch-and-bound tree effectively. Unlike previous works' polynomial-time branch-and-bound bounding strategies, our bounding algorithm provides strong worst-case theoretical guarantees, leading to faster reconstruction of the tumor evolution.

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.

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.