The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. In the online setting, items are revealed one by one and the decision, if the current item is packed or discarded forever, must be done immediately and irrevocably upon arrival. We study the online variant in the random order model where the input sequence is a uniform random permutation of the item set.
We develop a randomized (1/6.65)-competitive algorithm for this problem, outperforming the current best algorithm of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)]. Our algorithm is based on two new insights: We introduce a novel algorithmic approach that employs two given algorithms, optimized for restricted item classes, sequentially on the input sequence. In addition, we study and exploit the relationship of the knapsack problem to the 2-secretary problem.
The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. We show that in the same online setting, applying the proposed sequential approach yields a (1/6.99)-competitive randomized algorithm for GAP. Again, our proposed algorithm outperforms the current best result of competitive ratio 1/8.06 [Kesselheim et al. SIAM J. Comp. 47(5)].
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 perfor…
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 a…
In the online simple knapsack problem items are presented in an iterative fashion and an algorithm has to decide for each item whether to reject or permanently include it into the knapsack without any knowledge about the rest of the instance. The goal is to pack the knapsack as full as possible. In this work, we introduce the option of reserving items for the cost of a fixed fraction $α$ of their size. An algorithm may pay this fraction in order to postpone its decision on whether to include or reject these items until after the last item of the instance was presented. While the classical onli…
The fractional knapsack problem is one of the classical problems in combinatorial optimization, which is well understood in the offline setting. However, the corresponding online setting has been handled only briefly in the theoretical computer science literature so far, although it appears in several applications. Even the previously best known guarantee for the competitive ratio was worse than the best known for the integral problem in the popular random order model. We show that there is an algorithm for the online fractional knapsack problem that admits a competitive ratio of 4.39. Our res…
We study two canonical online optimization problems under capacity/budget constraints: the fractional one-way trading problem (OTP) and the integral online knapsack problem (OKP) under an infinitesimal assumption. Under the competitive analysis framework, it is well-known that both problems have the same optimal competitive ratio. However, these two problems are investigated by distinct approaches under separate contexts in the literature. There is a gap in understanding the connection between these two problems and the nature of their online algorithm design. This paper provides a unified fra…
Online knapsack problem is considered, where items arrive in a sequential fashion that have two attributes; value and weight. Each arriving item has to be accepted or rejected on its arrival irrevocably. The objective is to maximize the sum of the value of the accepted items such that the sum of their weights is below a budget/capacity. Conventionally a hard budget/capacity constraint is considered, for which variety of results are available. In modern applications, e.g., in wireless networks, data centres, cloud computing, etc., enforcing the capacity constraint in expectation is sufficient. …
A variant of the online knapsack problem is considered in the settings of trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit profit, and it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, previous work on online algorithms with untrusted predictions generally studied problems where an online algorithm with a constant competitive ratio is known. The prediction, possibly obtained from a machine learning source, that our algo…
In this paper, we introduce online knapsack problems with a resource buffer. In the problems, we are given a knapsack with capacity $1$, a buffer with capacity $R\ge 1$, and items that arrive one by one. Each arriving item has to be taken into the buffer or discarded on its arrival irrevocably. When every item has arrived, we transfer a subset of items in the current buffer into the knapsack. Our goal is to maximize the total value of the items in the knapsack. We consider four variants depending on whether items in the buffer are removable (i.e., we can remove items in the buffer) or non-remo…
Two related online problems: knapsack and truthful bipartite matching are considered. For these two problems, the common theme is how to `match' an arriving left vertex in an online fashion with any of the available right vertices, if at all, so as to maximize the sum of the value of the matched edges, subject to satisfying a sum-weight constraint on the matched left vertices. Assuming that the left vertices arrive in an uniformly random order (secretary model), two almost similar algorithms are proposed for the two problems, that are $2e$ competitive and $24$ competitive, respectively. The pr…
In the online multiple knapsack problem, an algorithm faces a stream of items, and each item has to be either rejected or stored irrevocably in one of $n$ bins (knapsacks) of equal size. The gain of an~algorithm is equal to the sum of sizes of accepted items and the goal is to maximize the total gain. So far, for this natural problem, the best solution was the $0.5$-competitive algorithm First Fit (the result holds for any $n \geq 2$). We present the first algorithm that beats this ratio, achieving the competitive ratio of $1/(1+\ln(2))-O(1/n) \approx 0.5906 - O(1/n)$. Our algorithm is determi…
The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor $1+\varepsilon$ runs in $\tilde O(n + (1/\varepsilon)^{12/5})$ time, where $\tilde O$ hides polylogarithmic factors. In this paper we present an improved algorithm in $\tilde O(n+(1/\varepsilon)^{9/4})$ time, with only a $(1/\varepsilon)^{1/4}$ gap from the quadratic conditional lower bound based on $(\min,+)$-convolution. Our improvement comes from a multi-level extension of Chan's number-theoretic cons…
The design of online algorithms has tended to focus on algorithms with worst-case guarantees, e.g., bounds on the competitive ratio. However, it is well-known that such algorithms are often overly pessimistic, performing sub-optimally on non-worst-case inputs. In this paper, we develop an approach for data-driven design of online algorithms that maintain near-optimal worst-case guarantees while also performing learning in order to perform well for typical inputs. Our approach is to identify policy classes that admit global worst-case guarantees, and then perform learning using historical data …
Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is g…
In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed non-overlappingly into the knapsack. The currently best known polynomial-time approximation factor for 2DK is $17/9+\varepsilon<1.89$ and there is a $(3/2+\varepsilon)$-approximation algorithm if we are allowed to rotate items by 90 degrees~{[}Gálvez et al., FOCS 2017{]}. In this paper, we give $(4/3+\varepsilon)$-approximation algorithms in polynomial time for bot…
We consider the {\em matroid median} problem \cite{KrishnaswamyKNSS11}, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation in \cite{KrishnaswamyKNSS11}. Our techniques illustrate that much of the work involved in the…
We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed $ε> 0$, there exists a polynomial-time algorithm with an approximation ratio $1-c/e-ε$, where $c \in [0,1]$ is the (total) curvature of the input function. This approximation ratio is tight up to $ε$ for any $c \in [0,1]$. To the best of our knowledge, this is the first result for a knapsack constraint that incorporates the curvature to obtain an approximation ratio better than $1-1/e$, which is tight for general submodular functions. As an application of our result, we…
We consider a variant of the knapsack problem, where items are available with different possible weights. Using a separate budget for these item improvements, the question is: Which items should be improved to which degree such that the resulting classic knapsack problem yields maximum profit? We present a detailed analysis for several cases of improvable knapsack problems, presenting constant factor approximation algorithms and two PTAS.
Optimal regret bounds for Multi-Armed Bandit problems are now well documented. They can be classified into two categories based on the growth rate with respect to the time horizon $T$: (i) small, distribution-dependent, bounds of order of magnitude $\ln(T)$ and (ii) robust, distribution-free, bounds of order of magnitude $\sqrt{T}$. The Bandits with Knapsacks model, an extension to the framework allowing to model resource consumption, lacks this clear-cut distinction. While several algorithms have been shown to achieve asymptotically optimal distribution-free bounds on regret, there has been l…
We study the online problem of reading articles that are listed in an aggregated form in a dynamic stream, e.g., in news feeds, as abbreviated social media posts, or in the daily update of new articles on arXiv. In such a context, the brief information on an article in the listing only hints at its content. We consider readers who want to maximize their information gain within a limited time budget, hence either discarding an article right away based on the hint or accessing it for reading. The reader can decide at any point whether to continue with the current article or skip the remaining pa…
University of Bonn（波恩大学）；Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
We introduce online learning with vector costs (\OLVCp) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^{d}$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study \OLVCp in both stochastic and adversarial arrival settings, and give…
This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating step-off, a dynamic programming algorithm from 1966, presented the lowest mean time to solve the most recent benchmark from the literature. The threshold and collective dominances are properties of the unbounded knapsack problem first discussed in 1998, and exploited by the current state-of-the-art algorithms. The terminating step-off algorithm did not exploit such domi…
Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde…
The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum $\mathrm{OPT}(I)$, i.e. of value at least $(1-\varepsilon) \mathrm{OPT}(I)$ for $\varepsilon > 0$, and have a running time polynomial in the input length and $\frac{1}{\varepsilon}$. For over thirty years, the best FPTAS was due to Lawler with …
We consider the {\em MST-interdiction} problem: given a multigraph $G = (V, E)$, edge weights $\{w_e\geq 0\}_{e \in E}$, interdiction costs $\{c_e\geq 0\}_{e \in E}$, and an interdiction budget $B\geq 0$, the goal is to remove a set $R\subseteq E$ of edges of total interdiction cost at most $B$ so as to maximize the $w$-weight of an MST of $G-R:=(V,E\setminus R)$. Our main result is a $4$-approximation algorithm for this problem. This improves upon the previous-best $14$-approximation~\cite{Zenklusen15}. Notably, our analysis is also significantly simpler and cleaner than the one in~\cite{Zenk…
One of the most fundamental problems in Computer Science is the Knapsack problem. Given a set of n items with different weights and values, it asks to pick the most valuable subset whose total weight is below a capacity threshold T. Despite its wide applicability in various areas in Computer Science, Operations Research, and Finance, the best known running time for the problem is O(Tn). The main result of our work is an improved algorithm running in time O(TD), where D is the number of distinct weights. Previously, faster runtimes for Knapsack were only possible when both weights and values ar…
Department of Statistics and Data Science, University of Pennsylvania（统计与数据科学系，宾夕法尼亚大学）；Department of Mathematics, The Hong Kong University of Science and Technology（数学系，香港科学与技术大学）；Department of Industrial Engineering & Decision Analytics, The Hong Kong University of Science and Technology（工业工程与决策分析系，香港科学与技术大学）
We study the d-dimensional hypercube knapsack problem where we are given a set of d-dimensional hypercubes with associated profits, and a knapsack which is a unit d-dimensional hypercube. The goal is to find an axis-aligned non-overlapping packing of a subset of hypercubes such that the profit of the packed hypercubes is maximized. For this problem, Harren (ICALP'06) gave an algorithm with an approximation ratio of (1+1/2^d+epsilon). For d=2, Jansen and Solis-Oba (IPCO'08) showed that the problem admits a polynomial-time approximation scheme (PTAS); Heydrich and Wiese (SODA'17) further improve…
We study online multidimensional variants of the generalized assignment problem which are used to model prominent real-world applications, such as the assignment of virtual machines with multiple resource requirements to physical infrastructure in cloud computing. These problems can be seen as an extension of the well known secretary problem and thus the standard online worst-case model cannot provide any performance guarantee. The prevailing model in this case is the random-order model, which provides a useful realistic and robust alternative. Using this model, we study the $d$-dimensional ge…
We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective function. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The online algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The "profit" of a set of accepted requests is its total value minus a convex cost function of its total size. This problem falls within the framework of secretary problems. Unlike previous work in that area,…
The random-order or secretary model is one of the most popular beyond-worst case model for online algorithms. While it avoids the pessimism of the traditional adversarial model, in practice we cannot expect the input to be presented in perfectly random order. This has motivated research on ``best of both worlds'' (algorithms with good performance on both purely stochastic and purely adversarial inputs), or even better, on inputs that are a mix of both stochastic and adversarial parts. Unfortunately the latter seems much harder to achieve and very few results of this type are known. Towards adv…
To find the number of assignments of zeros and ones satisfying a specific Knapsack Problem is $\#P$ hard, so only approximations are envisageable. A Markov chain allowing uniform sampling of all possible solutions is given by Luby, Randall and Sinclair. In 2005, Morris and Sinclair, by using a flow argument, have shown that the mixing time of this Markov chain is $\mathcal{O}(n^{9/2+ε})$, for any $ε> 0$. By using a canonical path argument on the distributive lattice structure of the set of solutions, we obtain an improved bound, the mixing time is given as $τ_{_{x}}(ε) \leq n^{3} \ln (16 ε^{-1…
In this paper, we investigate the parametric knapsack problem, in which the item profits are affine functions depending on a real-valued parameter. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problem may need to output an exponential number of knapsack solutions. We present a fully polynomial-time approximation scheme (FPTAS) for the problem that, for any desired precision $\varepsilon \in (0,1)$, computes $(1-\varepsilon)$-approximate solutions for all values of the parameter. This is the first FPTAS for the parametric kn…
Semi-online algorithms that are allowed to perform a bounded amount of repacking achieve guaranteed good worst-case behaviour in a more realistic setting. Most of the previous works focused on minimization problems that aim to minimize some costs. In this work, we study maximization problems that aim to maximize their profit. We mostly focus on a class of problems that we call choosing problems, where a maximum profit subset of a set objects has to be maintained. Many known problems, such as Knapsack, MaximumIndependentSet and variations of these, are part of this class. We present a framework…
This paper considers approximation algorithms for generalized $k$-median problems. This class of problems can be informally described as $k$-median with a constant number of extra constraints, and includes $k$-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized $k$-median that outputs a $6.387$-approximate solution, with a constant number of fractional variables. The algorithm builds on the iterative rounding framework introduced by Krishnaswamy, Li, and Sandeep for $k$-median with outliers. The main technical innovation is allo…
We prove new lower bounds for suitable competitive ratio measures of two relaxed online packing problems: online removable multiple knapsack, and a recently introduced online minimum peak appointment scheduling problem. The high level objective in both problems is to pack arriving items of sizes at most 1 into bins of capacity 1 as efficiently as possible, but the exact formalizations differ. In the appointment scheduling problem, every item has to be assigned to a position, which can be seen as a time interval during a workday of length 1. That is, items are not assigned to bins, but only onc…
Vector norms play a fundamental role in computer science and optimization, so there is an ongoing effort to generalize existing algorithms to settings beyond $\ell_\infty$ and $\ell_p$ norms. We show that many online and bandit applications for general norms admit good algorithms as long as the norm can be approximated by a function that is ``gradient-stable'', a notion that we introduce. Roughly it says that the gradient of the function should not drastically decrease (multiplicatively) in any component as we increase the input vector. We prove that several families of norms, including all mo…
We study the nascent setting of online computation with imperfect advice, in which the online algorithm is enhanced by some prediction encoded in the form of a possibly erroneous binary string. The algorithm is oblivious to the advice error, but defines a desired tolerance, namely an upper bound on the number of erroneous advice bits it can tolerate. This is a model that generalizes the untrusted advice model [Angelopoulos et al. ITCS 2020], in which the performance of the algorithm is only evaluated at the extreme values of error (namely, if the advice has either no errors, or if it is genera…
Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we study the following partition problem variant: given a sequence of $n$ items $1,\ldots,n$, where each item $i$ is associated with weight $w_i$ and another parameter $s_i$, partition the sequence into several consecutive subsequences, so that the total weight of each subsequence is no more than a threshold $w_0$, and the sum of the largest $s_i$ in each subsequence is minimized. This problem admits a straightforward solution based on dynamic pro…
Quantum Algorithms for Bandits with Knapsacks with Improved Regret and Time Complexities
Center on Frontiers of Computing Studies, Peking University School of Computer Science, Peking University（前沿计算研究所以及北京大学计算机科学学院，北京大学）；Guanghua School of Management, Peking University（光华管理学院，北京大学）；Department of Management Science and Engineering, Stanford University（管理科学与工程系，斯坦福大学）
In the moldable job scheduling problem one has to assign a set of $n$ jobs to $m$ machines, in order to minimize the time it takes to process all jobs. Each job is moldable, so it can be assigned not only to one but any number of the equal machines. We assume that the work of each job is monotone and that jobs can be placed non-contiguously. In this work we present a $(\frac 3 2 + ε)$-approximation algorithm with a worst-case runtime of ${O(n \log^2(\frac 1 ε+ \frac {\log (εm)} ε) + \frac{n}ε \log(\frac 1 ε) {\log (εm)})}$ when $m\le 16n$. This is an improvement over the best known algorithm o…
