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1. Tight Bounds for Parallel Paging and Green Paging

   https://doi.org/10.1137/1.9781611976465.180  

   Agrawal, Kunal ; Bender, Michael A. ; Das, Rathish ; Kuszmaul, William ; Peserico, Enoch ; Scquizzato, Michele ( January 2021 , Proc.\ 32th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   In the parallel paging problem, there are $\pP$ processors that share a cache of size $k$. The goal is to partition the cache among the \procs over time in order to minimize their average completion time. For this long-standing open problem, we give tight upper and lower bounds of $\Theta(\log \p)$ on the competitive ratio with $O(1)$ resource augmentation. A key idea in both our algorithms and lower bounds is to relate the problem of parallel paging to the seemingly unrelated problem of green paging. In green paging, there is an energy-optimized processor that can temporarily turn off one or more of its cache banks (thereby reducing power consumption), so that the cache size varies between a maximum size $k$ and a minimum size $k/\p$. The goal is to minimize the total energy consumed by the computation, which is proportional to the integral of the cache size over time. We show that any efficient solution to green paging can be converted into an efficient solution to parallel paging, and that any lower bound for green paging can be converted into a lower bound for parallel paging, in both cases in a black-box fashion. We then show that, with $O(1)$ resourcemore »augmentation, the optimal competitive ratio for deterministic online green paging is $\Theta(\log \p)$, which, in turn, implies the same bounds for deterministic online parallel paging.« less
2. A general framework for handling commitment in online throughput maximization

   https://doi.org/10.1007/s10107-020-01469-2  

   Chen, Lin ; Eberle, Franziska ; Megow, Nicole ; Schewior, Kevin ; Stein, Cliff ( September 2020 , Mathematical Programming)

   Abstract We study a fundamental online job admission problem where jobs with deadlines arrive online over time at their release dates, and the task is to determine a preemptive single-server schedule which maximizes the number of jobs that complete on time. To circumvent known impossibility results, we make a standard slackness assumption by which the feasible time window for scheduling a job is at least $$1+\varepsilon $$ 1 + ε times its processing time, for some $$\varepsilon >0$$ ε > 0 . We quantify the impact that different provider commitment requirements have on the performance of online algorithms. Our main contribution is one universal algorithmic framework for online job admission both with and without commitments. Without commitment, our algorithm with a competitive ratio of  $$\mathcal {O}(1/\varepsilon )$$ O ( 1 / ε ) is the best possible (deterministic) for this problem. For commitment models, we give the first non-trivial performance bounds. If the commitment decisions must be made before a job’s slack becomes less than a $$\delta $$ δ -fraction of its size, we prove a competitive ratio of $$\mathcal {O}(\varepsilon /((\varepsilon -\delta )\delta ^2))$$ O ( ε / ( ( ε - δ ) δ 2 ) ) ,more »for $$0<\delta <\varepsilon $$ 0 < δ < ε . When a provider must commit upon starting a job, our bound is  $$\mathcal {O}(1/\varepsilon ^2)$$ O ( 1 / ε 2 ) . Finally, we observe that for scheduling with commitment the restriction to the “unweighted” throughput model is essential; if jobs have individual weights, we rule out competitive deterministic algorithms.« less
3. Competitive Algorithms for Online Multidimensional Knapsack Problems

   https://doi.org/10.1145/3547353.3522627  

   Yang, Lin ; Zeynali, Ali ; Hajiesmaili, Mohammad H. ; Sitaraman, Ramesh K. ; Towsley, Don ( June 2022 , ACM SIGMETRICS Performance Evaluation Review)

   In this work, we study the online multidimensional knapsack problem (called OMdKP) in which there is a knapsack whose capacity is represented in m dimensions, each dimension could have a different capacity. Then, n items with different scalar profit values and m-dimensional weights arrive in an online manner and the goal is to admit or decline items upon their arrival such that the total profit obtained by admitted items is maximized and the capacity of knapsack across all dimensions is respected. This is a natural generalization of the classic single-dimension knapsack problem with several relevant applications such as in virtual machine allocation, job scheduling, and all-or-nothing flow maximization over a graph. We develop an online algorithm for OMdKP that uses an exponential reservation function to make online admission decisions. Our competitive analysis shows that the proposed online algorithm achieves the competitive ratio of O(log (Θ α)), where α is the ratio between the aggregate knapsack capacity and the minimum capacity over a single dimension and θ is the ratio between the maximum and minimum item unit values. We also show that the competitive ratio of our algorithm with exponential reservation function matches the lower bound up to a constant factor.
4. Online Hypergraph Matching with Delays

   Pavone, Marco ; Saberi, Amin. ; Schiffer, Maximilian ; Tsao, Matthew ( December 2020 , Conference on Web and Internet Economics)

   We study an online hypergraph matching problem with delays, motivated by ridesharing applications. In this model, users enter a marketplace sequentially, and are willing to wait up to $d$ timesteps to be matched, after which they will leave the system in favor of an outside option. A platform can match groups of up to $k$ users together, indicating that they will share a ride. Each group of users yields a match value depending on how compatible they are with one another. As an example, in ridesharing, $k$ is the capacity of the service vehicles, and $d$ is the amount of time a user is willing to wait for a driver to be matched to them. We present results for both the utility maximization and cost minimization variants of the problem. In the utility maximization setting, the optimal competitive ratio is $\frac{1}{d}$ whenever $k \geq 3$, and is achievable in polynomial-time for any fixed $k$. In the cost minimization variation, when $k = 2$, the optimal competitive ratio for deterministic algorithms is $\frac{3}{2}$ and is achieved by a polynomial-time thresholding algorithm. When $k>2$, we show that a polynomial-time randomized batching algorithm is $(2 - \frac{1}{d}) \log k$-competitive, and it is NP-hardmore »to achieve a competitive ratio better than $\log k - O (\log \log k)$.« less
5. Near-optimal Algorithms for Explainable k-Medians and k-Means

   Makarychev, Konstantin ; Shan, Liren ( July 2021 , International Conference on Machine Learning)

   We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with k-medians with ℓ1 norm and O(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for k-medians with ℓ2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ2 norm.