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1. Differentially Private Online Submodular Maximization

   Perez-Salazar, Sebastian ; Cummings, Rachel ( January 2021 , Proceedings of Machine Learning Research)

   null (Ed.)

   In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of T submodular functions over a common finite ground set U arrives online, and at each time-step the decision maker must choose at most k elements of U before observing the function. The decision maker obtains a profit equal to the function evaluated on the chosen set and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an (𝜀,𝛿)-DP algorithm with expected (1-1/e)-regret bound of 𝑂(𝑘2log|𝑈|𝑇log𝑘/𝛿√𝜀). This algorithm contains k ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an (𝜀,𝛿+𝑂(𝑒−𝑇1/3))-DP algorithm with expected (1-1/e)-regret bound of 𝑂(log𝑘/𝛿√𝜀(𝑘(|𝑈|log|𝑈|)1/3)2𝑇2/3). One challenge for privacy in this setting is that the payoff and feedback of expert i depends on the actions taken by her i-1 predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest. 

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2. Optimal strategies for patrolling fences

   https://doi.org/10.4230/LIPIcs.ICALP.2019.144  

   Haeupler, Bernhard ; Kuhn, Fabian ; Martinsson, Anders ; Petrova, Kalina ; Pfister, Pascal ( April 2019 , International Colloquium on Automata, Languages and Programming)

   A classical multi-agent fence patrolling problem asks: What is the maximum length L of a line fence that k agents with maximum speeds v_1,..., v_k can patrol if each point on the line needs to be visited at least once every unit of time. It is easy to see that L = alpha sum_{i=1}^k v_i for some efficiency alpha in [1/2,1). After a series of works [Czyzowicz et al., 2011; Dumitrescu et al., 2014; Kawamura and Kobayashi, 2015; Kawamura and Soejima, 2015] giving better and better efficiencies, it was conjectured by Kawamura and Soejima [Kawamura and Soejima, 2015] that the best possible efficiency approaches 2/3. No upper bounds on the efficiency below 1 were known. We prove the first such upper bounds and tightly bound the optimal efficiency in terms of the minimum speed ratio s = {v_{max}}/{v_{min}} and the number of agents k. Our bounds of alpha <= 1/{1 + 1/s} and alpha <= 1 - 1/(sqrt{k)+1} imply that in order to achieve efficiency 1 - epsilon, at least k >= Omega(epsilon^{-2}) agents with a speed ratio of s >= Omega(epsilon^{-1}) are necessary. Guided by our upper bounds, we construct a scheme whose efficiency approaches 1, disproving the conjecture stated above. Our scheme asymptotically matches our upper bounds in terms of the maximal speed difference and the number of agents used. A variation of the fence patrolling problem considers a circular fence instead and asks for its circumference to be maximized. We consider the unidirectional case of this variation, where all agents are only allowed to move in one direction, say clockwise. At first, a strategy yielding L = max_{r in [k]} r * v_r where v_1 >= v_2 >= ... >= v_k was conjectured to be optimal by Czyzowicz et al. [Czyzowicz et al., 2011] This was proven not to be the case by giving constructions for only specific numbers of agents with marginal improvements of L. We give a general construction that yields L = 1/{33 log_e log_2(k)} sum_{i=1}^k v_i for any set of agents, which in particular for the case 1, 1/2, ..., 1/k diverges as k - > infty, thus resolving a conjecture by Kawamura and Soejima [Kawamura and Soejima, 2015] affirmatively. 

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3. On Max-Min Fairness of Completion Times for Multi-Task Job Scheduling

   Shafiee, Mehrnoosh ; Ghaderi, Javad ( July 2020 , 2020 IFIP Networking Conference (Networking))

   We study the max-min fairness of multi-task jobs in distributed computing platforms. We consider a setting where each job consists of a set of parallel tasks that need to be processed on different servers, and the job is completed once all its tasks finish processing. Each job is associated with a utility which is a decreasing function of its completion time, and captures how sensitive it is to latency. The objective is to schedule tasks in a way that achieves max-min fairness for jobs' utilities, i.e., an optimal schedule in which any attempt to improve the utility of a job necessarily results in hurting the utility of some other job with smaller or equal utility. We first show a strong result regarding NP-hardness of finding the max-min fair vector of job utilities. The implication of this result is that achieving max-min fairness in many other distributed scheduling problems (e.g., coflow scheduling) is NP-hard. We then proceed to define two notions of approximation solutions: one based on finding a certain number of elements of the max-min fair vector, and the other based on a single-objective optimization whose solution gives the max-min fair vector. We develop scheduling algorithms that provide guarantees under these approximation notions, using dynamic programming and random perturbation of tasks' processing times. We verify the performance of our algorithms through extensive simulations, using a real traffic trace from a large Google cluster. 

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4. On Max-Min Fairness of Completion Times for Multi-Task Job Scheduling

   Shafiee, Mehrnoosh ; Ghaderi, Javad ( July 2020 , IFIP Networking Conference)

   We study the max-min fairness of multi-task jobs in distributed computing platforms. We consider a setting where each job consists of a set of parallel tasks that need to be processed on different servers, and the job is completed once all its tasks finish processing. Each job is associated with a utility which is a decreasing function of its completion time, and captures how sensitive it is to latency. The objective is to schedule tasks in a way that achieves max-min fairness for jobs’ utilities, i.e., an optimal schedule in which any attempt to improve the utility of a job necessarily results in hurting the utility of some other job with smaller or equal utility.We first show a strong result regarding NP-hardness of finding the max-min fair vector of job utilities. The implication of this result is that achieving max-min fairness in many other distributed scheduling problems (e.g., coflow scheduling) is NP-hard. We then proceed to define two notions of approximation solutions: one based on finding a certain number of elements of the max-min fair vector, and the other based on a single-objective optimization whose solution gives the max-min fair vector. We develop scheduling algorithms that provide guarantees under these approximation notions, using dynamic programming and random perturbation of tasks’ processing times. We verify the performance of our algorithms through extensive simulations, using a real traffic trace from a large Google cluster. 

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5. Search Algorithms for Multi-Agent Teamwise Cooperative Path Finding

   https://doi.org/10.1109/ICRA48891.2023.10160864  

   Ren, Zhongqiang ; Zhang, Chaoran ; Rathinam, Sivakumar ; Choset, Howie ( May 2023 , IEEE)

   Multi-Agent Path Finding (MA-PF) computes a set of collision-free paths for multiple agents from their respective starting locations to destinations. This paper considers a generalization of MA-PF called Multi-Agent Teamwise Cooperative Path Finding (MA-TC-PF), where agents are grouped as multiple teams and each team has its own objective to be minimized. For example, an objective can be the sum or max of individual arrival times of the agents. In general, there is more than one team, and MA-TC-PF is thus a multi-objective planning problem with the goal of finding the entire Paretooptimal front that represents all possible trade-offs among the objectives of the teams. To solve MA-TC-PF, we propose two algorithms TC-CBS and TC-M*, which leverage the existing CBS and M* for conventional MA-PF. We discuss the conditions under which the proposed algorithms are complete and are guaranteed to find the Pareto-optimal front. We present numerical results for several types of MA-TC-PF problems. 

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