In this paper, we exploit linear programming duality in the online setting (i.e., where input arrives on the fly) from the unique perspective of designing lower bounds on the competitive ratio. In particular, we provide a general technique for obtaining online deterministic and randomized lower bounds (i.e., hardness results) on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight lower bounds for three diverse online problems. The three problems we show tight lower bounds for are the Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds.
Our techniques are in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primal-dual scheme. We design a general recipe with the opposite aim of obtaining negative results via duality. The general idea behind our approach is to construct a primal linear program based on a collection of input sequences, where the objective function corresponds to optimizing the competitive ratio. We then obtain the corresponding dual linear program and provide a feasible solution, where the objective function yields a lower bound on the competitive ratio. Online lower bounds are often achieved by adapting the input sequence according to an online algorithm's behavior and doing an appropriate ad hoc case analysis. Using our unifying techniques, we simultaneously combine these cases into one linear program and achieve online lower bounds via a more robust analysis. We are confident that our framework can be successfully applied to produce many more lower bounds for a wide array of online problems.
We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to derive dual programs for problems which could be formulated as convex assignment problems. The duals have intuitive structures as the ones in linear programming. The constraints of the duals explicitly indicate the online decisions and naturally lead to competitive algorithms. Second, we use a dual-fitting approach, which also based on the weak duality, to study …
There are several problems in the theory of online computation where tight lower bounds on the competitive ratio are unknown and expected to be difficult to describe in a short form. A good example is the Online Bin Stretching problem, in which the task is to pack the incoming items online into bins while minimizing the load of the largest bin. Additionally, the optimal load of the entire instance is known in advance. The contribution of this paper is twofold. We use the Coq proof assistant to formalize the Online Bin Stretching problem and provide a program certifying lower bounds of this pro…
Packing a given sequence of items into as few bins as possible in an online fashion is a widely studied problem. We improve lower bounds for packing boxes into bins in two or more dimensions, both for general algorithms for squares and rectangles (in two dimensions) and for an important subclass, so-called Harmonic-type algorithms for hypercubes (in two or more dimensions). Lastly, we show that two adaptions of ideas from a one-dimensional packing algorithm to square packing do not help to break the barrier of 2.
We present space lower bounds for online pattern matching under a number of different distance measures. Given a pattern of length m and a text that arrives one character at a time, the online pattern matching problem is to report the distance between the pattern and a sliding window of the text as soon as the new character arrives. We require that the correct answer is given at each position with constant probability. We give Omega(m) bit space lower bounds for L_1, L_2, L_\infty, Hamming, edit and swap distances as well as for any algorithm that computes the cross-correlation/convolution. We…
We consider the online vector bin packing problem where $n$ items specified by $d$-dimensional vectors must be packed in the fewest number of identical $d$-dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm $A$, there exist instances I, such that $A(I)$, the number of bins used by $A$ to pack $I$, is $Ω(d/\log^2 d)$ times $OPT(I)$, the minimal number of bins to pack $I$. However in those instances, $OPT(I)$ was only $O(\log d)$, which left open the possibility of improved algorithms with better asymptotic competitive ratio when $OPT(I) \gg d$. We rule this out by show…
In the online matching on the line problem, the task is to match a set of requests $R$ online to a given set of servers $S$. The distance metric between any two points in $R\,\cup\, S$ is a line metric and the objective for the online algorithm is to minimize the sum of distances between matched server-request pairs. This problem is well-studied and - despite recent improvements - there is still a large gap between the best known lower and upper bounds: The best known deterministic algorithm for the problem is $O(\log^2n)$-competitive, while the best known deterministic lower bound is $9.001$.…
For online matching with the line metric, we present a lower bound of $Ω(\log n)$ on the approximation ratio of any online (possibly randomized) algorithm. This beats the previous best lower bound of $Ω(\sqrt{\log n})$ and matches the known upper bound of $O(\log n)$.
In the online bin packing problem, items of sizes in (0,1] arrive online to be packed into bins of size 1. The goal is to minimize the number of used bins. In this paper, we present an online bin packing algorithm with asymptotic competitive ratio of 1.5813. This is the first improvement in fifteen years and reduces the gap to the lower bound by 15%. Within the well-known SuperHarmonic framework, no competitive ratio below 1.58333 can be achieved. We make two crucial changes to that framework. First, some of our algorithm's decisions depend on exact sizes of items, instead of only their types.…
Modern data centers face a key challenge of effectively serving user requests that arrive online. Such requests are inherently multi-dimensional and characterized by demand vectors over multiple resources such as processor cycles, storage space, and network bandwidth. Typically, different resources require different objectives to be optimized, and $L_r$ norms of loads are among the most popular objectives considered. To address these problems, we consider the online vector scheduling problem in this paper. Introduced by Chekuri and Khanna (SIAM J of Comp. 2006), vector scheduling is a generali…
In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players, Bob, his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures. We first present a tight lower bound for the online set intersection problem in the online c…
We consider a variant of the online buffer management problem in network switches, called the $k$-frame throughput maximization problem ($k$-FTM). This problem models the situation where a large frame is fragmented into $k$ packets and transmitted through the Internet, and the receiver can reconstruct the frame only if he/she accepts all the $k$ packets. Kesselman et al.\ introduced this problem and showed that its competitive ratio is unbounded even when $k=2$. They also introduced an "order-respecting" variant of $k$-FTM, called $k$-OFTM, where inputs are restricted in some natural way. They…
Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of $11/8 = 1.375$ for three bins. Additionally, we present a lower bound of $45/33 = 1.\overline{36}$ for Online Bin Stretching on three bins and a lower bound of $19/14$ for four and five bins that were discovered using a computer search.
Cardinality constrained bin packing or bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k \geq 2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem in the sense that we prove a lower bound of 2 on the overall asymptotic competitive ratio. This closes this long standing open problem, since an algorithm of an absolute competitive ratio 2 is known. Addit…
We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.
In this paper, we consider the Online Traveling Salesperson Problem (OLTSP) where the locations of the requests are known in advance, but not their arrival times. We study both the open variant, in which the algorithm is not required to return to the origin when all the requests are served, as well as the closed variant, in which the algorithm has to return to the origin after serving all the requests. Our aim is to measure the impact of the extra knowledge of the locations on the competitiveness of the problem. We present an online 3/2-competitive algorithm for the general case and a matching…
In the model of online caching with machine learned advice, introduced by Lykouris and Vassilvitskii, the goal is to solve the caching problem with an online algorithm that has access to next-arrival predictions: when each input element arrives, the algorithm is given a prediction of the next time when the element will reappear. The traditional model for online caching suffers from an $Ω(\log k)$ competitive ratio lower bound (on a cache of size $k$). In contrast, the augmented model admits algorithms which beat this lower bound when the predictions have low error, and asymptotically match the…
In 2005 Li et al. gave a phi-competitive deterministic online algorithm for scheduling of packets with agreeable deadlines with a very interesting analysis. This is known to be optimal due to a lower bound by Hajek. We claim that the algorithm by Li et al. can be slightly simplified, while retaining its competitive ratio. Then we introduce randomness to the modified algorithm and argue that the competitive ratio against oblivious adversary is at most 4/3. Note that this still leaves a gap between the best known lower bound of 5/4 by Chin et al. for randomised algorithms against oblivious adver…
In this paper, we study two variants of the online metric matching problem. The first problem is the online metric matching problem where all the servers are placed at one of two positions in the metric space. We show that a simple greedy algorithm achieves the competitive ratio of 3 and give a matching lower bound. The second problem is the online facility assignment problem on a line, where servers have capacities, servers and requests are placed on 1-dimensional line, and the distances between any two consecutive servers are the same. We show lower bounds $1+ \sqrt{6}$ $(> 3.44948)$, $\frac…
We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.
We consider the file maintenance problem (also called the online labeling problem) in which n integer items from the set {1,...,r} are to be stored in an array of size m >= n. The items are presented sequentially in an arbitrary order, and must be stored in the array in sorted order (but not necessarily in consecutive locations in the array). Each new item must be stored in the array before the next item is received. If r<=m then we can simply store item j in location j but if r>m then we may have to shift the location of stored items to make space for a newly arrived item. The algorithm is ch…
Although many authors have considered how many ternary comparisons it takes to sort a multiset $S$ of size $n$, the best known upper and lower bounds still differ by a term linear in $n$. In this paper we restrict our attention to online stable sorting and prove upper and lower bounds that are within (o (n)) not only of each other but also of the best known upper bound for offline sorting. Specifically, we first prove that if the number of distinct elements (σ= o (n / \log n)), then ((H + 1) n + o (n)) comparisons are sufficient, where $H$ is the entropy of the distribution of the elements in …
We improve the lower bound on the asymptotic competitive ratio of any online algorithm for bin packing to above 1.54278. We demonstrate for the first time the advantage of branching and the applicability of full adaptivity in the design of lower bounds for the classic online bin packing problem. We apply a new method for weight based analysis, which is usually applied only in proofs of upper bounds. The values of previous lower bounds were approximately 1.5401 and 1.5403.
We consider the online bin packing problem under the advice complexity model where the 'online constraint' is relaxed and an algorithm receives partial information about the future requests. We provide tight upper and lower bounds for the amount of advice an algorithm needs to achieve an optimal packing. We also introduce an algorithm that, when provided with log n + o(log n) bits of advice, achieves a competitive ratio of 3/2 for the general problem. This algorithm is simple and is expected to find real-world applications. We introduce another algorithm that receives 2n + o(n) bits of advice …
We give cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream. In this model, a fixed string of $n$ symbols is given and one $δ$-bit symbol arrives at a time in a stream. After each symbol arrives, the distance between the fixed string and a suffix of most recent symbols of the stream is reported. The cell-probe model is perhaps the strongest model of computation for showing data structure lower bounds, subsuming in particular the popular word-RAM model. * We first give an $Ω((δ\log n)/(w+\log\log n))$ lower bound for t…
In this paper, we initiate the study of the weighted paging problem with predictions. This continues the recent line of work in online algorithms with predictions, particularly that of Lykouris and Vassilvitski (ICML 2018) and Rohatgi (SODA 2020) on unweighted paging with predictions. We show that unlike unweighted paging, neither a fixed lookahead nor knowledge of the next request for every page is sufficient information for an algorithm to overcome existing lower bounds in weighted paging. However, a combination of the two, which we call the strong per request prediction (SPRP) model, suffic…
A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.
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…
In the online labeling problem with parameters n and m we are presented with a sequence of n keys from a totally ordered universe U and must assign each arriving key a label from the label set {1,2,...,m} so that the order of labels (strictly) respects the ordering on U. As new keys arrive it may be necessary to change the labels of some items; such changes may be done at any time at unit cost for each change. The goal is to minimize the total cost. An alternative formulation of this problem is the file maintenance problem, in which the items, instead of being labeled, are maintained in sorted…
In this work, we consider online vector bin packing. It is known that no algorithm can have a competitive ratio of $o(d/\log^2 d)$ in the absolute sense, though upper bounds for this problem were always shown in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds on the asymptotic competitive ratio. The existing lower bounds prior to this work were much smaller than $3$ even for very large dimensions. We significantly improve t…
The online unit clustering problem was proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009), which is defined as follows: "Points" are given online in the $d$-dimensional Euclidean space one by one. An algorithm creates a "cluster," which is a $d$-dimensional rectangle. The initial length of each edge of a cluster is 0. An algorithm can extend an edge until it reaches unit length independently of other dimensions. The task of an algorithm is to cover a new given point either by creating a new cluster and assigning it to the point, or by extending edges of a…
Columbia University（哥伦比亚大学）；UC Berkeley（加州大学伯克利分校）；Google（谷歌）
We provide the first sub-linear space and sub-linear regret algorithm for online learning with expert advice (against an oblivious adversary), addressing an open question raised recently by Srinivas, Woodruff, Xu and Zhou (STOC 2022). We also demonstrate a separation between oblivious and (strong) adaptive adversaries by proving a linear memory lower bound of any sub-linear regret algorithm against an adaptive adversary. Our algorithm is based on a novel pool selection procedure that bypasses the traditional wisdom of leader selection for online learning, and a generic reduction that transform…
We show tight bounds for online Hamming distance computation in the cell-probe model with word size w. The task is to output the Hamming distance between a fixed string of length n and the last n symbols of a stream. We give a lower bound of Omega((d/w)*log n) time on average per output, where d is the number of bits needed to represent an input symbol. We argue that this bound is tight within the model. The lower bound holds under randomisation and amortisation.
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…
We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $Ω(n)$ bits of advice to achieve a competitive ratio better than $H_k…
We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. …
We study online mechanisms for preemptive scheduling with deadlines, with the goal of maximizing the total value of completed jobs. This problem is fundamental to deadline-aware cloud scheduling, but there are strong lower bounds even for the algorithmic problem without incentive constraints. However, these lower bounds can be circumvented under the natural assumption of deadline slackness, i.e., that there is a guaranteed lower bound $s > 1$ on the ratio between a job's size and the time window in which it can be executed. In this paper, we construct a truthful scheduling mechanism with a con…
Technion - Israel Institute of Technology（以色列理工学院）；IBM T. J. Watson Research Center（IBM托马斯·J·沃森研究中心）；Hebrew University（希伯来大学）
In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (beta,tau)-decomposability, and derive an efficient…
There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional bounds with unconditional ones. In this paper we make progress in this direction by studying the cell probe complexity of two conjectured to be hard problems of particular importance: matrix-vector multiplication and a version of dynamic set disjoin…
