We initiate the study of tree structures in the context of scenario-based robust optimization. Specifically, we study Binary Search Trees (BSTs) and Huffman coding, two fundamental techniques for efficiently managing and encoding data based on a known set of frequencies of keys. Given a number of distinct scenarios, each defined by a frequency distribution over the keys, our objective is to compute a single tree of best-possible performance, relative to any scenario. We consider, as performance metrics, the competitive ratio, which compares multiplicatively the cost of the solution to the tree of least cost among all scenarios, as well as the regret, which induces a similar, but additive comparison. For BSTs, we show that the problem is NP-hard across both metrics. We also obtain an optimal competitive ratio that is logarithmic in the number of scenarios. For Huffman Trees, we likewise prove NP-hardness, and we present an algorithm with logarithmic regret, which we prove to be near-optimal by showing a corresponding lower bound. Last, we give a polynomial-time algorithm for computing Pareto-optimal BSTs with respect to their regret, assuming scenarios defined by uniform distributions over the keys. This setting captures, in particular, the first study of fairness in the context of data structures. We provide an experimental evaluation of all algorithms. To this end, we also provide mixed integer linear program formulation for computing optimal trees.

One-max search is a classic problem in online decision-making, in which a trader acts on a sequence of revealed prices and accepts one of them irrevocably to maximise its profit. The problem has been studied both in probabilistic and in worst-case settings, notably through competitive analysis, and more recently in learning-augmented settings in which the trader has access to a prediction on the sequence. However, existing approaches either lack smoothness, or do not achieve optimal worst-case guarantees: they do not attain the best possible trade-off between the consistency and the robustness of the algorithm. We close this gap by presenting the first algorithm that simultaneously achieves both of these important objectives. Furthermore, we show how to leverage the obtained smoothness to provide an analysis of one-max search in stochastic learning-augmented settings which capture randomness in both the observed prices and the prediction.

Online bidding is a classic optimization problem, with several applications in online decision-making, the design of interruptible systems, and the analysis of approximation algorithms. In this work, we study online bidding under learning-augmented settings that incorporate stochasticity, in either the prediction oracle or the algorithm itself. In the first part, we study bidding under distributional predictions, and find Pareto-optimal algorithms that offer the best-possible tradeoff between the consistency and the robustness of the algorithm. In the second part, we study the power and limitations of randomized bidding algorithms, by presenting upper and lower bounds on the consistency/robustness tradeoffs. Previous works focused predominantly on oracles that do not leverage stochastic information on the quality of the prediction, and deterministic algorithms.

The study of online algorithms with machine-learned predictions has gained considerable prominence in recent years. One of the common objectives in the design and analysis of such algorithms is to attain (Pareto) optimal tradeoffs between the {\em consistency} of the algorithm, i.e., its performance assuming perfect predictions, and its {\em robustness}, i.e., the performance of the algorithm under adversarial predictions. In this work, we demonstrate that this optimization criterion can be extremely brittle, in that the performance of Pareto-optimal algorithms may degrade dramatically even in the presence of imperceptive prediction error. To remedy this drawback, we propose a new framework in which the smoothness in the performance of the algorithm is enforced by means of a {\em user-specified profile}. This allows us to regulate the performance of the algorithm as a function of the prediction error, while simultaneously maintaining the analytical notion of consistency/robustness tradeoffs, adapted to the profile setting. We apply this new approach to a well-studied online problem, namely the {\em one-way trading} problem. For this problem, we further address another limitation of the state-of-the-art Pareto-optimal algorithms, namely the fact that they are tailored to worst-case, and extremely pessimistic inputs. We propose a new Pareto-optimal algorithm that leverages any deviation from the worst-case input to its benefit, and introduce a new metric that allows us to compare any two Pareto-optimal algorithms via a {\em dominance} relation.

We study the classic problem of searching for a hidden target in the line and the m-ray star, in a setting in which the searcher has some prediction on the hider’s position. We first focus on the main metric for comparing search strategies under predictions; namely, we give positive and negative results on the consistency-robustness tradeoff, where the performance of the strategy is evaluated at extreme situations in which the prediction is either error-free, or adversarially generated, respectively. For the line, we show tight bounds concerning this tradeoff, under the untrusted advice model, in which the prediction is in the form of a k-bit string which encodes the responses to k binary queries. For the star, we give tight, and near-tight tradeoffs in the positional and the directional models, in which the prediction is related to the position of the target within the star, and to the ray on which the target hides, respectively. Last, for all three prediction models, we show how to generalize our study to a setting in which the performance of the strategy is evaluated as a function of the searcher’s desired tolerance to prediction errors, both in terms of positive and inapproximability results.

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 generated adversarially). In this work, we establish connections between games with a lying responder, also known as Rényi-Ulam games, and the design and analysis of online algorithms with imperfect advice. Specifically, we demonstrate how to obtain upper and lower bounds on the competitive ratio for well-studied online problems such as time-series search, online bidding, and fractional knapsack. Our techniques provide the first lower bounds for online problems in this model. We also highlight and exploit connections between competitive analysis with imperfect advice and fault-tolerance in multiprocessor systems. Last, we show how to waive the dependence on the tolerance parameter, by means of resource augmentation and robustification.

We introduce the study of search problems, in a setting in which the searcher has some information, or hint concerning the hiding target. In particular, we focus on one of the fundamental problems in search theory, namely the linear search problem. Here, an immobile target is hidden at some unknown position on an unbounded line, and a mobile searcher, initially positioned at some specific point of the line called the root , must traverse the line so as to locate the target. The objective is to minimize the worst-case ratio of the distance traversed by the searcher to the distance of the target from the root, which is known as the competitive ratio of the search. We consider three settings in regards to the nature of the hint: i) the hint suggests the exact position of the target on the line; ii) the hint suggests the direction of the optimal search (i.e., to the left or the right of the root); and iii) the hint is a general k -bit string that encodes some information concerning the target. Our objective is to study the Pareto -efficiency of strategies in this model, with respect to the tradeoff between consistency and robustness . Namely, we seek optimal, or near-optimal tradeoffs between the searcher's performance if the hint is correct (i.e., provided by a trusted source) and if the hint is incorrect (i.e., provided by an adversary). We prove several results in each of these three settings. For positional hints, we show that the optimal consistency of r -robust strategies is ( b r + 1 ) / ( b r − 1 ) , where b r is defined to be equal to ρ r + ρ r 2 − 4 ρ r 2 , and ρ r = ( r − 1 ) / 2 , for all r ≥ 9 . For directional hints, we show that for every b ≥ 1 and δ ∈ ( 0 , 1 ] , there exists a strategy with consistency equal to c = 1 + 2 ( b 2 b 2 − 1 + δ b 3 b 2 − 1 ) and robustness equal to 1 + 2 ( b 2 b 2 − 1 + 1 δ b 3 b 2 − 1 ) ; furthermore, we show again that this upper bound is tight. Last, for general k -bit hints, we show upper bounds for general k -bit hints, as well as lower bounds: specifically, we show that the consistency of any 9-robust strategy must be at least 5, and that the consistency of r -robust strategies is at least 1 + 2 b r / ( b r − 1 ) , in the case of a natural class of asymptotic strategies.

Contract scheduling is a general technique that allows the design of systems with interruptible capabilities, given an algorithm that is not necessarily interruptible. Previous work on this topic has assumed that the interruption is a worst-case deadline that is unknown to the scheduler. In this work, we study new settings in which the scheduler has access to some imperfect prediction in regards to the interruption. In the first setting, which is inspired by recent advances in learning-enhanced algorithms, the prediction describes the time that the interruption occurs. The second setting introduces a new model in which predictions are elicited as responses to a number of binary queries. For both settings, we investigate trade-offs between the robustness (i.e., the worst-case performance of the schedule if the prediction is generated adversarially) and the consistency (i.e., the performance assuming that the prediction is error-free). We also establish results on the performance of the schedules as a function of the prediction error.

Bin packing is a classic optimization problem with a wide range of applications from load balancing to supply chain management. In this work, we study the online variant of the problem, in which a sequence of items of various sizes must be placed into a minimum number of bins of uniform capacity. The online algorithm is enhanced with a (potentially erroneous) prediction concerning the frequency of item sizes in the sequence. We design and analyze online algorithms with efficient tradeoffs between the consistency (i.e., the competitive ratio assuming no prediction error) and the robustness (i.e., the competitive ratio under adversarial error), and whose performance degrades near-optimally as a function of the prediction error. This is the first theoretical and experimental study of online bin packing in the realistic setting of learnable predictions. Previous work addressed only extreme cases with respect to the prediction error, and relied on overly powerful and error-free oracles.

In the online (time-series) search problem, a player is presented with a sequence of prices which are revealed in an online manner. In the standard definition of the problem, for each revealed price, the player must decide irrevocably whether to accept or reject it, without knowledge of future prices (other than an upper and a lower bound on their extreme values), and the objective is to minimize the competitive ratio, namely the worst case ratio between the maximum price in the sequence and the one selected by the player. The problem formulates several applications of decision-making in the face of uncertainty on the revealed samples. Previous work on this problem has largely assumed extreme scenarios in which either the player has almost no information about the input, or the player is provided with some powerful, and error-free advice. In this work, we study learning-augmented algorithms, in which there is a potentially erroneous prediction concerning the input. Specifically, we consider two different settings: the setting in which the prediction is related to the maximum price in the sequence, as well as well as the setting in which the prediction is obtained as a response to a number of binary queries. For both settings, we provide tight, or near-tight upper and lower bounds on the worst-case performance of search algorithms as a function of the prediction error. We also provide experimental results on data obtained from stock exchange markets that confirm the theoretical analysis, and explain how our techniques can be applicable to other learning-augmented applications.

Contract scheduling is a general technique that allows to design a system with interruptible capabilities, given an algorithm that is not necessarily interruptible. Previous work on this topic has largely assumed that the interruption is a worst-case deadline that is unknown to the scheduler. In this work, we study the setting in which there is a potentially erroneous prediction concerning the interruption. Specifically, we consider the setting in which the prediction describes the time that the interruption occurs, as well as the setting in which the prediction is obtained as a response to a single or multiple binary queries. For both settings, we investigate tradeoffs between the robustness (i.e., the worst-case performance assuming adversarial prediction) and the consistency (i.e, the performance assuming that the prediction is error-free), both from the side of positive and negative results.

The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model.

We consider the setting in which executions of contract algorithms are scheduled in a processor so as to produce an interruptible system. Such algorithms offer a trade off between the quality of output and the available computation time, provided that the latter is known in advance. Previous work on this setting has provided strict performance guarantees for several variants of this setting, assuming that an interruption can occur arbitrarily ahead in the future. In practice, however, one expects that the schedule will reach a point beyond which further progress will only be marginal, hence it can be deemed complete. In this work we show how to optimize the time at which the system reaches a desired performance objective, while maintaining interruptible guarantees throughout the entire execution. The resulting schedule is provably optimal, and it guarantees that upon completion each individual contract algorithm has attained a predefined end guarantee.

In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the input is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice. Motivated by the above, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.

We study the problem of searching for a hidden target in an environment that is modeled by an edge-weighted graph. A sequence of edges is chosen starting from a given root vertex such that each edge is adjacent to a previously chosen edge. This search paradigm, known as expanding search was recently introduced by Alpern and Lidbetter (2013) for modeling problems such as searching for coal or minesweeping in which the cost of re-exploration is negligible. It can also be used to model a team of searchers successively splitting up in the search for a hidden adversary or explosive device, for example. We define the search ratio of an expanding search as the maximum over all vertices of the ratio of the time taken to reach the vertex and the shortest-path cost to it from the root. This can be interpreted as a measure of the multiplicative regret incurred in searching, and similar objectives have previously been studied in the context of conventional (pathwise) search. In this paper we address algorithmic and computational issues of minimizing the search ratio over all expanding searches, for a variety of search environments, including general graphs, trees and star-like graphs. Our main results focus on the problem of finding the randomized expanding search with minimum expected search ratio, which is equivalent to solving a zero-sum game between a Searcher and a Hider. We solve these problems for certain classes of graphs, and obtain constant-factor approximations for others.

A contract algorithm is an algorithm which is given, as part of the input, a specified amount of allowable computation time. The algorithm must then complete its execution within the allotted time. An interruptible algorithm, in contrast, can be interrupted at an arbitrary point in time, at which point it must report its currently best solution. It is known that contract algorithms can simulate interruptible algorithms using iterative deepening techniques. This simulation is done at a penalty in the performance of the solution, as measured by the so-called acceleration ratio. In this paper we give matching (i.e., optimal) upper and lower bounds for the acceleration ratio under such a simulation. We assume the most general setting in which n problem instances must be solved by means of scheduling executions of contract algorithms in $m$ identical parallel processors. This resolves an open conjecture of Bernstein, Filkenstein, and Zilberstein who gave an optimal schedule under the restricted setting of round robin and length-increasing schedules, but whose optimality in the general unrestricted case remained open. Lastly, we show how to evaluate the average acceleration ratio of the class of exponential strategies in the setting of n problem instances and m parallel processors. This is a broad class of schedules that tend to be either optimal or near-optimal, for several variants of the basic problem.

It has long been known that for the paging problem in its standard form, competitive analysis cannot adequately distinguish algorithms based on their performance: there exists a vast class of algorithms that achieve the same competitive ratio, ranging from extremely naive and inefficient strategies (such as Flush-When-Full), to strategies of excellent performance in practice (such as Least-Recently-Used and some of its variants). A similar situation arises in the list update problem: in particular, under the cost formulation studied by Martínez and Roura [2000] and Munro [2000] every list update algorithm has, asymptotically, the same competitive ratio. Several refinements of competitive analysis, as well as alternative performance measures have been introduced in the literature, with varying degrees of success in narrowing this disconnect between theoretical analysis and empirical evaluation. In this article, we study these two fundamental online problems under the framework of bijective analysis [Angelopoulos et al. 2007, 2008]. This is an intuitive technique that is based on pairwise comparison of the costs incurred by two algorithms on sets of request sequences of the same size. Coupled with a well-established model of locality of reference due to Albers et al. [2005], we show that Least-Recently-Used and Move-to-Front are the unique optimal algorithms for paging and list update, respectively. Prior to this work, only measures based on average-cost analysis have separated LRU and MTF from all other algorithms. Given that bijective analysis is a fairly stringent measure (and also subsumes average-cost analysis), we prove that in a strong sense LRU and MTF stand out as the best (deterministic) algorithms.

It has long been known that for the paging problem in its standard form, competitive analysis cannot adequately distinguish algorithms based on their performance: there exists a vast class of algorithms which achieve the same competitive ratio, ranging from extremely naive and inefficient strategies (such as Flush-When-Full), to strategies of excellent performance in practice (such as Least-Recently-Used and some of its variants).A similar situation arises in the list update problem: in particular, under the cost formulation studied by Martínez and Roura [TCS 2000] and Munro [ESA 2000] every list update algorithm has, asymptotically, the same competitive ratio.Several refinements of competitive analysis, as well as alternative performance measures have been introduced in the literature, with varying degrees of success in narrowing this disconnect between theoretical analysis and empirical evaluation.In this paper we study these two fundamental online problems under the framework of bijective analysis [Angelopoulos, Dorrigiv and López-Ortiz, SODA 2007 and LATIN 2008].This is an intuitive technique which is based on pairwise comparison of the costs incurred by two algorithms on sets of request sequences of the same size.Coupled with a well-established model of locality of reference due to Albers, Favrholdt and Giel [JCSS 2005], we show that Least-Recently-Used and Move-to-Front are the unique optimal algorithms for paging and list update, respectively.Prior to this work, only measures based on average-cost analysis have separated LRU and MTF from all other algorithms.Given that bijective analysis is a fairly stringent measure (and also subsumes average-cost analysis), we prove that in a strong sense LRU and MTF stand out as the best algorithms.