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We study the problem of assigning items to agents so as to maximize the \emph{weighted} Nash Social Welfare (NSW) under submodular valuations. The best-known result for the problem is an $O(nw_{\max})$-approximation due to Garg, Husic, Li, Vega, and Vondrak~\cite{GHL23}, where $w_{\max}$ is the maximum weight over all agents. Obtaining a constant approximation algorithm is an open problem in the field that has recently attracted considerable attention. We give the first such algorithm for the problem, thus solving the open problem in the affirmative. Our algorithm is based on the natural Configuration LP for the problem, which was introduced recently by Feng and Li~\cite{FL24} for the additive valuation case. Our rounding algorithm is similar to that of Li \cite{Li25} developed for the unrelated machine scheduling problem to minimize weighted completion time. Roughly speaking, we designate the largest item in each configuration as a large item and the remaining items as small items. So, every agent gets precisely 1 fractional large item in the configuration LP solution. With the rounding algorithm in \cite{Li25}, we can ensure that in the obtained solution, every agent gets precisely 1 large item, and the assignments of small items are negatively correlated.

We study a public event scheduling problem, where multiple public events are scheduled to coordinate the availability of multiple agents. The availability of each agent is determined by solving a separate flexible interval job scheduling problem, where the jobs are required to be preemptively processed. The agents want to attend as many events as possible, and their agreements are considered to be the total length of time during which they can attend these events. The goal is to find a schedule for events as well as the job schedule for each agent such that the total agreement is maximized. We first show that the problem is NP-hard, and then prove that a simple greedy algorithm achieves $\frac{1}{2}$-approximation when the whole timeline is polynomially bounded. Our method also implies a $(1-\frac{1}{e})$-approximate algorithm for this case. Subsequently, for the general timeline case, we present an algorithmic framework that extends a $\frac{1}{\alpha}$-approximate algorithm for the one-event instance to the general case that achieves $\frac{1}{\alpha+1}$-approximation. Finally, we give a polynomial time algorithm that solves the one-event instance, and this implies a $\frac{1}{2}$-approximate algorithm for the general case., Comment: To appear in IJCAI 2024

The paper designs revenue-maximizing auction mechanisms for agents who aim to maximize their total obtained values rather than the classical quasi-linear utilities. Several models have been proposed to capture the behaviors of such agents in the literature. In the paper, we consider the model where agents are subject to budget and return-on-spend constraints. The budget constraint of an agent limits the maximum payment she can afford, while the return-on-spend constraint means that the ratio of the total obtained value (return) to the total payment (spend) cannot be lower than the targeted bar set by the agent. The problem was first coined by [Balseiro et al., EC 2022]. In their work, only Bayesian mechanisms were considered. We initiate the study of the problem in the worst-case model and compare the revenue of our mechanisms to an offline optimal solution, the most ambitious benchmark. The paper distinguishes two main auction settings based on the accessibility of agents' information: fully private and partially private. In the fully private setting, an agent's valuation, budget, and target bar are all private. We show that if agents are unit-demand, constant approximation mechanisms can be obtained; while for additive agents, there exists a mechanism that achieves a constant approximation ratio under a large market assumption. The partially private setting is the setting considered in the previous work [Balseiro et al., EC 2022] where only the agents' target bars are private. We show that in this setting, the approximation ratio of the single-item auction can be further improved, and a $\Omega(1/\sqrt{n})$-approximation mechanism can be derived for additive agents., Comment: 29 pages

Scheduling with testing falls under the umbrella of the research on optimization with explorable uncertainty. In this model, each job has an upper limit on its processing time that can be decreased to a lower limit (possibly unknown) by some preliminary action (testing). Recently, D{\"{u}}rr et al. \cite{DBLP:journals/algorithmica/DurrEMM20} has studied a setting where testing a job takes a unit time, and the goal is to minimize total completion time or makespan on a single machine. In this paper, we extend their problem to the budget setting in which each test consumes a job-specific cost, and we require that the total testing cost cannot exceed a given budget. We consider the offline variant (the lower processing time is known) and the oblivious variant (the lower processing time is unknown) and aim to minimize the total completion time or makespan on a single machine. For the total completion time objective, we show NP-hardness and derive a PTAS for the offline variant based on a novel LP rounding scheme. We give a $(4+\epsilon)$-competitive algorithm for the oblivious variant based on a framework inspired by the worst-case lower-bound instance. For the makespan objective, we give an FPTAS for the offline variant and a $(2+\epsilon)$-competitive algorithm for the oblivious variant. Our algorithms for the oblivious variants under both objectives run in time $O(poly(n/\epsilon))$. Lastly, we show that our results are essentially optimal by providing matching lower bounds., Comment: To appear in ESA 2023

We revisit the online dynamic acknowledgment problem. In the problem, a sequence of requests arrive over time to be acknowledged, and all outstanding requests can be satisfied simultaneously by one acknowledgement. The goal of the problem is to minimize the total request delay plus acknowledgement cost. This elegant model studies the trade-off between acknowledgement cost and waiting experienced by requests. The problem has been well studied and the tight competitive ratios have been determined. For this well-studied problem, we focus on how to effectively use machine-learned predictions to have better performance. We develop algorithms that perform arbitrarily close to the optimum with accurate predictions while concurrently having the guarantees arbitrarily close to what the best online algorithms can offer without access to predictions, thereby achieving simultaneous optimum consistency and robustness. This new result is enabled by our novel prediction error measure. No error measure was defined for the problem prior to our work, and natural measures failed due to the challenge that requests with different arrival times have different effects on the objective. We hope our ideas can be used for other online problems with temporal aspects that have been resisting proper error measures., Comment: To appear in INFOCOM 2023

The paper revisits the robust $s$-$t$ path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with $n$ vertices and $k$ distinct cost functions (scenarios) defined over edges, and aim to choose an $s$-$t$ path such that the total cost of the path is always provable no matter which scenario is realized. With the view of each cost function being associated with an agent, our goal is to find a common $s$-$t$ path minimizing the maximum objective among all agents, and thus create a fair solution for them. The problem is hard to approximate within $o(\log k)$ by any quasi-polynomial time algorithm unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly}\log n})$, and the best approximation ratio known to date is $\widetilde{O}(\sqrt{n})$ which is based on the natural flow linear program. A longstanding open question is whether we can achieve a polylogarithmic approximation even when a quasi-polynomial running time is allowed. We give the first polylogarithmic approximation for robust $s$-$t$ path since the problem was proposed more than two decades ago. In particular, we introduce a $O(\log n \log k)$-approximate algorithm running in quasi-polynomial time. The algorithm is built on a novel linear program formulation for a decision-tree-type structure which enables us to get rid of the $\Omega(\max\{k,\sqrt{n}\})$ integrality gap of the natural flow LP. Further, we also consider some well-known graph classes, e.g., graphs with bounded treewidth, and show that the polylogarithmic approximation can be achieved polynomially on these graphs. We hope the new proposed techniques in the paper can offer new insights into the robust $s$-$t$ path problem and related problems in robust optimization.

In the submodular ranking (SR) problem, the input consists of a set of submodular functions defined on a ground set of elements. The goal is to order elements for all the functions to have value above a certain threshold as soon on average as possible, assuming we choose one element per time. The problem is flexible enough to capture various applications in machine learning, including decision trees. This paper considers the min-max version of SR where multiple instances share the ground set. With the view of each instance being associated with an agent, the min-max problem is to order the common elements to minimize the maximum objective of all agents -- thus, finding a fair solution for all agents. We give approximation algorithms for this problem and demonstrate their effectiveness in the application of finding a decision tree for multiple agents., Comment: To appear in AAAI 2023

Given a connected graph on whose edges we can build roads to connect the nodes, a number of agents hold possibly different perspectives on which edges should be selected by assigning different edge weights. Our task is to build a minimum number of roads so that every agent has a spanning tree in the built subgraph whose weight is the same as a minimum spanning tree in the original graph. We first show that this problem is NP-hard and does not admit better than $((1-o(1))\ln k)$-approximation polynomial-time algorithms unless P=NP, where $k$ is the number of agents. We then give a simple voting algorithm with an optimal approximation ratio. Moreover, our algorithm only needs to access the agents' rankings on the edges. Finally, we extend our results to submodular objective functions and Matroid rank constraints., Comment: To appear in AAAI 2023