In this paper, we augment online algorithms for the knapsack problem using the total weight information. The conventional optimal online algorithm achieves the ln(U/L)+1 competitive ratio where L and U are the upper and lower bounds of the value-to-weight ratio. However, it does not consider that decision makers can know the total weight information or obtain it through machine-learned predictions. To fill this gap, we first propose the Known Weight Algorithm (KWA) which uses the exact total weight information to achieve a competitive ratio of W((U-L)/(eL))+1, where W denotes the Lambert-W function. We prove that it is optimal and tight. After that, we extend KWA to the Predicted Weight Algorithm (PWA), a learning-augmented online algorithm that uses predicted total weight. We show the consistency and robustness of PWA, and prove that its competitive ratio degrades gracefully as the prediction error grows. Finally, we introduce the Limited Volume Algorithm (LWA), which achieves a better competitive ratio than ln(U/L)+1 when the total weight is less than twice the capacity.

We introduce and study the Combinatorial Ski Rental (CSR) problem, which involves multiple items that can be rented or purchased, either individually or in combination. At each time step, a decision-maker must make an irrevocable buy-or-rent decision for items that have not yet been purchased, without knowing the end of the time horizon. We propose a randomized online algorithm, Sorted Optimal Amortized Cost (SOAC), that achieves the optimal competitive ratio. Moreover, SOAC can be extended to address various well-known ski rental variants, including the multi-slope, multi-shop, multi-commodity ski rental and CSR with upgrading problems. Building on the proposed SOAC algorithm, we further develop a learning-augmented algorithm that leverages machine-learned predictions to improve the performance of CSR. This algorithm is capable of recovering or improving upon existing results of learning-augmented algorithms in both the classic ski rental and multi-shop ski rental problems. Experimental results validate our theoretical analysis and demonstrate the advantages of our algorithms over baseline methods for ski rental problems.

As the demand for faster and more reliable content delivery escalates, Content Delivery Networks (CDNs) face significant challenges in managing content placement across their increasingly complex, multi-tiered structures to balance performance, complexity, and scalability, while addressing the transient nature of data and the unpredictability of internet traffic. Addressing these challenges, this study introduces a novel multi-tier CDN caching strategy that navigates spatial and temporal trade-offs in cache placement, considering the cache placement cost diminishes with the content lifetime, and the uncertainty of future data demands. We design a distributed online algorithm that evaluates each incoming request and places new caches when the total content delivery cost exceeds a threshold. Our competitive analysis shows a tight and optimal$\mathtt {Tiers}+1$competitive ratio. Additionally, our algorithm has low complexity by passing$O(\mathtt {Tiers})$number of reference messages for each request, which enhances its practical applicability. Empirical validation through numerical simulations and trace-driven experiments confirms the superiority of our approach to existing benchmarks in real-world CDN settings.

We investigate an extended version of the classical ski-rental problem with multiple commodities. A customer uses a set of commodities altogether, and he/she needs to choose payment options to cover the usage of each commodity without the knowledge of the future. The payment options of each commodity include (1) renting: to pay for an on-demand usage and (2) buying: to pay for the lifetime usage. It is a novel extension of the classical ski-rental problem which deals with only one commodity. To address this problem, we propose a new online algorithm called the Multi-Object Break-Even (MOBE) algorithm and conduct competitive analysis. We show that the tight lower and upper bounds of MOBE algorithm's competitive ratio are e/e-1 and 2 respectively against adaptive adversary under arbitrary renting and buying prices. We further prove that MOBE algorithm is an optimal online algorithm if commodities have the same rent-to-buy ratio. Numerical results verify our theoretical conclusion and demonstrate the advantages of MOBE in a real-world scenario.

In this paper, we study a two-level ski-rental problem. There are multiple commodities, each one can be “rented” (paying for on-demand usage) or “purchased” (paying for life-time usage). There is also a combo purchase available so that all commodities can be purchased as a combo. Since the usages of the commodities in future are not known in advance, to minimize the overall cost, we design an online algorithm to decide if we rent a commodity, purchase a commodity, or make a combo purchase. We first propose a deterministic online algorithm. It can achieve 3 competitive ratio, which is optimal and tight. Next, we further propose a randomized online algorithm, leading to a e^σ/(e^σ-1) competitive ratio, where σ is the ratio between the price of a single commodity and the price of combo purchase. Finally, we apply simulation to verify the theoretical competitive ratios and evaluate the actual performance against benchmarks.