Online Knapsack Problem Research Articles

Online Unit Profit Knapsack with Predictions

A variant of the online knapsack problem is considered in the setting of predictions. In Unit Profit Knapsack, the items have unit profit, i.e., the goal is to pack as many items as possible. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. The prediction available to the online algorithm is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio r=aa^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=\\frac{a}{\\hat{a}}$$\\end{document} where a is the actual value for this average size and a^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{a}$$\\end{document} is the prediction. We give an algorithm which is e-1e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{e-1}{e}$$\\end{document}-competitive, if r=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r=1$$\\end{document}, and this is best possible among online algorithms knowing a and nothing else. More generally, the algorithm has a competitive ratio of e-1er\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{e-1}{e}r$$\\end{document}, if r≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r \\le 1$$\\end{document}, and e-rer\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{e-r}{e}r$$\\end{document}, if 1≤r<e\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le r < e$$\\end{document}. Any algorithm with a better competitive ratio for some r<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<1$$\\end{document} will have a worse competitive ratio for some r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. To obtain a positive competitive ratio for all r, we adjust the algorithm, resulting in a competitive ratio of 12r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2r}$$\\end{document} for r≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 1$$\\end{document} and r2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{r}{2}$$\\end{document} for r≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\le 1$$\\end{document}. We show that improving the result for any r<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r< 1$$\\end{document} leads to a worse result for some r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}.

Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack

In “Tight Guarantees for Multiunit Prophet Inequalities and Online Stochastic Knapsack,” Jiang, Ma, and Zhang analyze the multiunit prophet inequalities and online knapsack problems. They formulate the problems as the online contention resolution scheme problems. Then, they develop linear programming formulations to represent the online contention resolution scheme problems. They characterize the optimal solution of the linear programming and derive the corresponding optimal policies. For the multiunit prophet inequalities, they are able to obtain the tight ratios, which improved over the previous ones. For the online stochastic knapsack problem, their ratio is also optimal and the best up to date. They finally generalize their results to the multiresource setting and expand the applicability of their approach.

The Online Knapsack Problem with Departures

The online knapsack problem is a classic online resource allocation problem in networking and operations research. Its basic version studies how to pack online arriving items of different sizes and values into a capacity-limited knapsack. In this paper, we study a general version that includes item departures, while also considering multiple knapsacks and multi-dimensional item sizes. We design a threshold-based online algorithm and prove that the algorithm can achieve order-optimal competitive ratios. Beyond worst-case optimized algorithms, we also propose a data-driven online algorithm that can achieve near-optimal average performance under typical instances while guaranteeing the worst-case performance.

The Online Knapsack Problem with Departures

The online knapsack problem is a classic online resource allocation problem in networking and operations research. Its basic version studies how to pack online arriving items of different sizes and values into a capacity-limited knapsack. In this paper, we study a general version that includes item departures, while also considering multiple knapsacks and multi-dimensional item sizes. We design a threshold-based online algorithm and prove that the algorithm can achieve order-optimal competitive ratios. Beyond worst-case performance guarantees, we also aim to achieve near-optimal average performance under typical instances. Towards this goal, we propose a data-driven online algorithm that learns within a policy-class that guarantees a worst-case performance bound. In trace-driven experiments, we show that our data-driven algorithm outperforms other benchmark algorithms in an application of online knapsack to job scheduling for cloud computing.

Competitive Algorithms for the Online Multiple Knapsack Problem with Application to Electric Vehicle Charging

We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce a new algorithm that achieves a competitive ratio within an additive factor of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. Finally, in the full version of this paper, we illustrate the proposed algorithm via trace-based experiments of EV charging.

A Near-Optimal Bidding Strategy for Real-Time Display Advertising Auctions

This article introduces a near-optimal bidding algorithm for use in real-time display advertising auctions. These auctions constitute a dominant distribution channel for internet display advertising and a potential funding model for addressable media. The proposed efficient, implementable learning algorithm is proven to rapidly converge to the optimal strategy while achieving zero regret and constituting a competitive equilibrium. This is the first algorithmic solution to the online knapsack problem to offer such theoretical guarantees without assuming a priori knowledge of object values or costs. Furthermore, it meets advertiser requirements by accommodating any valuation metric while satisfying budget constraints. Across a series of 100 simulated and 10 real-world campaigns, the algorithm delivers 98% of the value achievable with perfect foresight and outperforms the best available alternative by 11%. Finally, we show how the algorithm can be augmented to simultaneously estimate impression values and learn the bidding policy. Across a series of simulations, we show that the total regret delivered under this dual objective is less than that from any competing algorithm required only to learn the bidding policy.

Competitive Algorithms for the Online Multiple Knapsack Problem with Application to Electric Vehicle Charging

We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce a new algorithm that achieves a competitive ratio within an additive factor of one of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. Finally, we illustrate the proposed algorithm via trace-based experiments of EV charging.

A Competitive Analysis of Online Knapsack Problems with Unit Density

We study an online knapsack problem where the items arrive sequentially and must be either immediately packed into the knapsack or irrevocably discarded. Each item has a different size and the objective is to maximize the total size of items packed. We focus on the class of randomized algorithms which initially draw a threshold from some distribution, and then pack every fitting item whose size is at least that threshold. Threshold policies satisfy many desiderata including simplicity, fairness, and incentive-alignment. We derive two optimal threshold distributions, the first of which implies a competitive ratio of 0.432 relative to the optimal offline packing, and the second of which implies a competitive ratio of 0.428 relative to the optimal fractional packing. We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack and must be placed in at most one. This is equivalent to the AdWords problem where item truncation is not allowed. We derive a randomized threshold algorithm for this problem which is 0.214-competitive. We also show that any randomized algorithm for this problem cannot be more than 0.461-competitive, providing the first upper bound which is strictly less than 0.5. This online knapsack problem finds applications in many areas, like supply chain ordering, online advertising, and healthcare scheduling, refugee integration, and crowdsourcing. We show how our optimal threshold distributions can be naturally implemented in the warehouses for a Latin American chain department store. We run simulations on their large-scale order data, which demonstrate the robustness of our proposed algorithms.

Online knapsack problem under concave functions

In this paper, we address an online knapsack problem under concave function f(x), i.e., an item with size x has its profit f(x). We first obtain a simple lower bound max⁡{q,f′(0)f(1)}, where holden ratio q≈1.618, then show that this bound is not tight, and give an improved lower bound. Finally, we find the online algorithm for linear function can be employed to the concave case, and prove its competitive ratio is f′(0)f(1/q), then we give a refined online algorithm with a competitive ratio f′(0)f(1)+1 when f′(0)/f(1) is very large. And we also give optimal algorithms for some specific piecewise linear functions.

Cache Policies for Linear Utility Maximization

Cache policies to minimize the content retrieval cost have been studied through competitive analysis when the miss costs are additive and the sequence of content requests is arbitrary. More recently, a cache utility maximization problem has been introduced, where contents have stationary popularities and utilities are strictly concave in the hit rates. This paper bridges the two formulations, considering linear costs and content popularities. We show that minimizing the retrieval cost corresponds to solving an online knapsack problem, and we propose new dynamic policies inspired by simulated annealing, including DynqLRU, a variant of qLRU. We prove that DynqLRU asymptotically asymptotic converges to the optimum under the characteristic time approximation. In a real scenario, popularities vary over time and their estimation is very difficult. DynqLRU does not require popularity estimation, and our realistic, trace-driven evaluation shows that it significantly outperforms state-of-the-art policies, with up to 45% cost reduction.

Online knapsack of unknown capacity: How to optimize energy consumption in smartphones

We propose a new variant of the standard online knapsack problem where the only information missing to the provided instances is the capacity B of the knapsack. We refer to this problem as the online Knapsack of Unknown Capacity (KUC) problem. Any algorithm solving the KUC problem must provide a strategy for filling online the knapsack until its capacity is revealed. When the knapsack capacity is revealed, no other item can be inserted and also the last inserted item is discarded if it does not completely fit in the knapsack.Apart for the interest in a new version of the fundamental knapsack problem, the motivations that lead to define this new variant come from energy consumption constraints in smartphones communications. We provide lower and upper bounds to the problem for various cases. In general, we design an optimal algorithm admitting a 12-competitive ratio. When all items admit uniform ratio of profit over size, our algorithm provides a 4986=.569… competitive ratio that leaves some gap with the provided bound of 1φ=.618…, the inverse of the golden number. We then conduct experimental analysis for the competitive ratio guaranteed algorithms compared to the optimum and to various heuristics.

The online knapsack problem with incremental capacity

We consider an online knapsack problem with incremental capacity. In each time period, a set of items, each with a specific weight and value, is revealed and, without knowledge of future items, it has to be decided which of these items to accept. Additionally, the knapsack capacity is not fully available from the start but increases by a constant amount in each time period. The goal is to maximize the overall value of the accepted items. This setting extends the basic online knapsack problem by introducing a dynamic instead of a static knapsack capacity and is applicable to classic problems such as resource allocation or one-way trading. In contrast to the basic online knapsack problem, for which no competitive algorithms exist, the setting of incremental capacity facilitates the development of competitive algorithms for a bounded time horizon. We provide a competitive analysis of deterministic and randomized online algorithms for the online knapsack problem with incremental capacity and present lower bounds on the competitive ratio achievable by online algorithms for the problem. Most of these lower bounds match the competitive ratios achieved by our online algorithms exactly or differ only by a constant factor.

Randomized algorithms for online knapsack problems

In this paper, we study online knapsack problems. The input is a sequence of items e1,e2,…,en, each of which has a size and a value. Given the ith item ei, we either put ei into the knapsack or reject it. In the removable setting, when ei is put into the knapsack, some items in the knapsack are removed with no cost if the sum of the size of ei and the total size in the current knapsack exceeds the capacity of the knapsack. Our goal is to maximize the profit, i.e., the sum of the values of items in the last knapsack.We present a simple randomized 2-competitive algorithm for the unweighted non-removable case and show that it is the best possible, where knapsack problem is called unweighted if the value of each item is equal to its size.For the removable case, we propose a randomized 2-competitive algorithm despite there is no constant competitive deterministic algorithm. We also provide a lower bound 1+1/e≈1.368 for the competitive ratio. For the unweighted removable case, we propose a 10/7-competitive algorithm and provide a lower bound 1.25 for the competitive ratio.

User context-based data delivery in opportunistic smartphone networks

With the growing number of smartphone users, the ability to ubiquitously construct peer-to-peer autonomous networks, which are defined as opportunistic smartphone networks (OSNs), has also grown. Due to the mobility of smartphone users and the volatile links between diverse network interfaces, connectivity is sporadic. Therefore, the traditional view of a network as a connected graph is not appropriate for modeling OSNs. To accomplish data delivery in such challenging environments, store–carry–forward techniques are generally used, and data is replicated in multiple copies to increase delivery probability. Apart from using the history of inter-encounters, previous works put little effort into designing user context-based message replication. In this paper, we design a user context-based message replication that achieves efficient data delivery with deterministic cost. The model used for message replication is an online knapsack problem. Our scheme efficiently estimates data delivery probabilities of individual nodes through learning and predicting user context information. The number of replicated messages is dynamically adapted with a given delivery threshold. Evaluation based on real-traces, indicates that the proposed algorithm outperforms similar mechanisms, in terms of delivery rate and cost.

The online knapsack problem: Advice and randomization

We study the advice complexity and the random bit complexity of the online knapsack problem. Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm.For this problem, we show that a single bit of advice helps a deterministic online algorithm to become 2-competitive, but that Ω(logn) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. Additionally, we show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing the online algorithm to overpack the knapsack by some small amount, a constant number of advice bits suffices to achieve a near-optimal competitive ratio.We also study the weighted version of the problem proving that, with O(logn) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive. Furthermore, we show that an arbitrary number of random bits does not permit a constant competitive ratio.

Online removable knapsack problem under convex function

In this paper, we address an online knapsack problem under a convex size–profit function. We first give a greedy online algorithm with a competitive ratio 2. Then we propose an improved online algorithm with a competitive ratio 5/3. We also prove that when the convex function has a specific property, our improved online algorithm is (1+5)/2-competitive, which is optimal. Finally, we prove that the lower bound of this problem is (1+5)/2.

On the Advice Complexity of the Knapsack Problem

We study the advice complexity and the random bit complexity of the online knapsack problem: Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm. We show that a single bit of advice helps a deterministic algorithm to become 2-competitive, but that Ω(log n) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. We also show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing a little overpacking of the knapsack, a constant number of advice bits suffices to achieve a near-optimal competitive ratio. We also study the weighted version of the problem proving that, with O(log n) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive.

Average-Case Analysis of Off-Line and On-Line Knapsack Problems

We consider the average-case behavior of the Zero–One Knapsack problem, as well as an on-line version studied by Marchetti-Spaccamela and Vercellis. We allow the capacity of the knapsack to grow proportionally to the number of items, so that the optimum solution tends to be Θ(n). Under fairly general conditions on the distribution, we obtain a description of the expected value of the optimum offline solution which is accurate up to terms which areo(1). We then consider a simple greedy method for the on-line problem, which is calledOnLineGreedyand is allowed to use knowledge of the distribution, and we show that the solution obtained by this algorithm differs from the true optimum by an average of Θ(logn); in fact, we can determine the multiplicative constant hidden by the Θ-notation. Thus on average the cost of being forced to give answers on-line is quite small compared to the optimum solution. We also show that no on-line algorithm can improve overOnLineGreedyby more thano(logn). Our results hold, under fairly general conditions on the input distribution, for either the Zero–One Knapsack problem or its linear programming relaxation.