Title: Combinatorial Ski Rental Problem: Robust and Learning-Augmented Algorithms
Venue: NeurIPS 2025
Accepted: yes
Keywords: combinatorial ski rental problem

competitive analysis

learning-augmented algorithm

online optimization
Abstract: 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn the combinatorial ski rental problem, the goal is to either rent or buy a set of items, that are necessary to ski, for a complete skiing season with an unknown duration. At each day of the skiing season, we can decide to either rent or buy the necessary items that have not been bought yet, with the goal of minimizing the total cost at the end of the skiing season. The cost function for buying a subset of items is submodular, giving the problem a combinatorial nature. On the other hand, the cost for renting a subset of items is just the sum of the individual renting costs. Algorithms for this problem are analysed against the optimal offline solution, which knows the duration of the skiing season upfront. This problem generalizes multiple variants of ski rental that have been studied in the literature. The closest related problem in the literature is the problem variant with upgrading [32]: In this variant, the cost for buying a set of items $S$ at time $t$ is $f(S \\cup P) - f(P)$ where $P$ is the set of items bought in previous steps, i.e., we only pay the marginal costs. In contrast, in the variant considered in this paper, the cost is just $f(S)$. The difference between these two variants is reflected in the competitive ratio: For the variant with upgrading, [32] shows a (randomized) competitive ratio of e/(e-1). For the variant of this paper, the competitive ratio can get worse depending on the given cost functions.\n\nThe main results of this paper are (1) a randomized algorithm that achieves the optimal competitive ratio for each fixed pair of cost functions and (2) a learning-augmented algorithm that receives a prediction on the length of the skiing season and achieves improved results when this prediction is somewhat accurate, while still giving worst-case guarantees.  The authors first observe that a deterministic strategy can be represented by a \\emph{purchase path} (sequence of item bundles to be bought) and \\emph{purchase times} (points in time at which the purchases of the purchase path should be made). Consequently, a randomized strategy can be interpreted as a distribution over purchase paths and purchase time pairs. To achieve the first main result, the authors first show how to compute the optimal purchase time distribution for a fixed purchase path by tracking how the optimal solution cost evolves with an increasing time horizon and bounding the expected algorithmic cost against this increase . Afterwards, they use the insights from this algorithm to design an algorithm that simultaneously computes the optimal purchase path distribution and purchase time distribution by using a projected gradient descent method. Finally, they show how to integrate predictions into this algorithmic framework and give empirical experiments.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n* The presented randomized algorithm unifies several existing results for multiple ski rental variants. In a similar way, the learning-augmented algorithm recovers, and thereby unifies, results for multiple previously studied variants. This seems like a significant step towards a general framework for solving problems of this type. In particular, because the theoretical results give the best-possible competitive ratio \\emph{for each} fixed pair of cost functions, which makes them applicable to special cases without loss. \n* In my opinion, the algorithmic ideas for computing the purchase probabilities for a fixed purchase path (Section 3.1) are very elegant. They use a characterization of the offline optimal solution to track how the optimal cost increases with an increasing time horizon. In order to find $\\alpha$-approximative purchase probability, they locally bound the (expected) increase in the algorithm's cost by $\\alpha$ times the increase in the optimal cost. Tracking how the optimal solution evolves over time in this way, is a very nice idea. Similarly, the insight that the purchase path probabilities can be computed with the projected gradient descent method is nice. In total, the paper seems to bring multiple new ideas to the area of ski rental type problems. \n\nWeaknesses:\n* The running time of the presented algorithm can be exponential in the input size. In the conclusion, the authors state that this is due to the combinatorial nature of the problem. It would be nice to formally underline this statement with hardness results.\n* The instances (item number and cost functions) considered in the empirical part seem a bit arbitrary. They are still useful as a proof of concept. However, it is unclear what the empirical results mean for practical instances.\n\nQUESTIONS:\n* Are there hardness results justifying the need for an exponential running time?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Combinatorial Ski Rental (CSR) problem, a generalization of classic ski rental where multiple items can be rented/purchased individually or in bundles, with submodular purchase costs and additive rental rates. The key contributions include an optimal randomized online algorithm achieving the best possible competitive ratio for CSR (which is called SOAC). The authors also study the learning-augmented setting, leveraging predictions to balance consistency (accuracy with good predictions) and robustness (worst-case guarantees). Finally, the authors also did numerical validation of SOAC/LA-SOAC superiority over baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. I like the proposed CSR model; it elegantly unifies several ski rental variants and addresses the open non-upgradable case (where prior work assumed upgradable purchases). The submodular cost model captures real-world bundle discounts. I also appreciate that the authors draw Figure 1 to describe the related works, and it is helpful.\n\n2. I usually work on offline approximation algorithms. But, as far as I can see, the algorithm SOAC is novel, using amortized cost tracking against an offline optimal benchmark and convex optimization over purchase paths via an augmented path construction. Thus, I think that the technical contribution is sufficient.\n\n3. The paper is well-written and organized. I enjoy reading the paper.\n\nWeakness:\n\n 1. SOAC’s runtime scales exponentially with the number of super items, which is stated in section 6. While the authors note that real-world discounts are often sparse, scalability for large item sets remains a concern.\n\n2. The additive rental rates and submodular costs are well-justified but may not hold in all applications (e.g., interdependent rental costs).\n\nSummary:\n\nThis is a good paper, and I believe that it should be accepted.\n\nQUESTIONS:\nCould you comment on what will happen if the rental rate is also a submodular function? Is the problem becoming much harder? When the rental rate is additive, the offline optimal solution is equivalent to minimizing a non-monotone submodular function. When the rental rate is submodular, as far as I  can see, the offline problem no longer seems in P, but I am not sure how hard both the online and offline versions are.\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper brings and studies CSR problem, which is an online combinatorial optimization problem.\nTo solve this problem, the authors propose Sorted Optimal Amortized Cost (SOAC).\nThis SOAC algorithm has good performance and is capable of solving multiple variants of the CSR problem.\nThe authors also propose a learning-based augmentation for further improvement.\nThey conduct experiments for validation.\n\nSTRENGTHS AND WEAKNESSES:\nS1. The figures are easy to read.\n\nS2. Theoretical analysis are provided.\n\nS3. The paper is conducted on a \n\nW1. The abstract could be improved. In the abstract, the authors introduce the CSR problem, however they do not clarify the optimization goal (ie minimizing rental or purchase cost) of the problem.\n\nW2. The idea is not original enough. Improving average performance instead of worst case performance has long been proposed and studied. [Ref1]\n\nW3. The problem abstraction seems adopt a lot of assumptions, which largely confines the potential of learning-based methods.\n\nW4. The theoretical derivation is largely drawn from spatio temporal research community, which is not original enough. [Ref2] [Ref3]\n\n\n[Ref1] Yongxin Tong, Jieying She, Bolin Ding, Lei Chen, Tianyu Wo, Ke Xu. \"Online Minimum Matching in Real-Time Spatial Data: Experiments and Analysis\" , Proceedings of the VLDB Endowment, 9(12): 1053-1064, 2016.\n\n[Ref2] Dickerson J P, Sankararaman K A, Srinivasan A, et al. Allocation problems in ride-sharing platforms: Online matching with offline reusable resources[J]. ACM Transactions on Economics and Computation (TEAC), 2021, 9(3): 1-17.\n\n[Ref3] Nanda V, Xu P, Sankararaman K A, et al. Balancing the tradeoff between profit and fairness in rideshare platforms during high-demand hours[C]//Proceedings of the AAAI conference on artificial intelligence. 2020, 34(02): 2210-2217.\n\nQUESTIONS:\nQ1. The problem seems adopts a lot of settings, could you please introduce the most simple and basic version of the problem?\n\nQ2. Could the author provide some examples or cases to intuitively explain why their algorithm is better?\n\nLIMITATIONS:\n\"yes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a generalization of the ski rental problem called Combinatorial Ski Rental (CSR). Like the traditional ski rental problem, the proposed problem is an online optimization problem in which ski equipments must be either rented (for a day) or purchased (for good) at the beginning of each day, until the end of the last day which is not known to the algorithm in advance. The difference of CSR from the traditional problem is that the required ski equipments are a set of multiple items. Each day, we may decide to purchase a subset of the items, and those items that haven't been purchased must be rented. After observing that CSR generalizes several ski-rental type of problems that were previously studied, the paper goes on to give a competitive algorithm and a learning-augmented algorithm for CSR.\n\nThe \"output\" or the \"strategy\" of an algorithm can be fully described by a partition of the entire set of items and a (randomized) schedule that specifies on which day each subset will be purchased. As such, the paper starts by giving an algorithm (or a subroutine) called OAC that, given a partition, determines a schedule of the purchases within a finite time horizon. The paper then presents an algorithm called SOAC that uses OAC as a subroutine: the algorithm finds an optimal probability distribution over the partitions, samples a partition from it, and then invokes OAC on that partition. The paper shows that this algorithm gives an \"optimal\" competitive algorithm.\n\nThen, to give a learning-augmented algorithm, the paper presents an algorithm that adjusts the finite time horizon mentioned above according to the given prediction. This adjustment allows us to write the robustness and consistency (bound) of the algorithm indirectly in terms of its performance (or \"competitive ratio\") for a particular choice of the last day.\n\nFinally, the paper evaluates the performance of the proposed algorithms through experiments.\n\nSTRENGTHS AND WEAKNESSES:\n### Strength\n\n1. The paper gives a unified framework that solves a number of problems, generalized by CSR.\n\n### Weakness\n\n1. In order for the unified framework to have true advantage, the unified algorithm must have a good performance, both in terms of the performance guarantee and running time. For many online optimization problems, it is possible for the algorithm to compute the \"optimal\" strategy when given enough amount of running time. But it is not often considered in the literature because either the running time is prohibitively large or an explicit competitive analysis is difficult. However in this paper,\n\n1-a. The competitive algorithm does not come with an analysis that explicitly gives a competitive ratio.\n\n1-b. It appears that the algorithm may require a significant amount of running time, as each iteration of the gradient descent method would involve solving a large system of equations. But the paper does not give theoretical/experimental analysis of the running time.\n\n1-b'. In Section 6, it is briefly mentioned that the running time is exponential, but I don't see why that is the case. A naive implementation's running time may depend on the number of input bits, possibly doubly exponentially.\n\n1-c. For the learning-augmented algorithm, the unification is somewhat incomplete in that we sometimes hand-pick $T_{OPT}^{(1)}$ and $T_{OPT}^{(2)}$.\n\n2. The experiment seems to be limited.\n\n2-a. It uses a fixed set of three items, which is another reason why the experiment would not properly assess the running time issues explained above.\n\n2-b. It is also not clear why this particular choice of the item costs were used in the experiments.\n\n2-c. The experiment only introduces Gaussian noise to the prediction. I'm not sure if this would be a good evaluation of the algorithm's robustness.\n\nQUESTIONS:\n- Line 133, \"Additionally, ... to enforce the order of purchase in $\\sigma$.\": Why do we do this? Suppose for example we have two items 1 and 2, and we decided to purchase 1 and then later 2. The following two strategies both respect the order of purchase:\n\na) buy item 1 on day 10 and item 2 on day 20;\n\nb) buy item 1 on day 30 and item 2 on day 40.\n\nSo, choosing a) with probability 1/2 and b) with 1/2 is a perfectly valid strategy, but the condition would rule out this possibility. Why is the constraint \"$p_n(S_{i+1} | \\sigma)=0$ when $F_n(S_i | \\sigma)<1$\" a safe one to impose?\n\n- The paper claims that the algorithm can handle submodular rental rates, too. Then why don't we present that version instead (especially given that the difference is supposed to be minimal)?\n\n- Line 177, \"We define an optimal amortized cost strategy...\": the current phrasing sounds as if OAC is just defined to be that specific strategy, rather than the \"optimal\" one among all possible strategies (independent from Definition 2). The appendix makes it clear that the latter is the intention, but this should be clarified in the main body.\n\n- Line 730, doesn't the inequality need to contain equality to handle ties?\n\n- Does the experiments actually run the algorithm, or analytically calculate the competitive ratios? This was not clear from the experiment description.\n\nLet me write a few minor comments here too:\n\n- On page 2, it is mentioned that CSR generalizes a number of problems. A brief explanation would be nice.\n\n- Don't we need some additional assumption on $f$, like $f(\\emptyset)=0$?\n\n- In Definition 2, it seems that $T_A$ also is a variable (that we need to determine by solving the system of equations), but that was not clear at a glance.\n\nLIMITATIONS:\nPlease see weaknesses above, especially Item 1. While Section 6 does mention running time as a limitation, I believe a more thorough discussion would be desirable. Item 2 in the weaknesses section discusses limitations of the experiments.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}