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 @math is the number of requests, we show that: (1) A paging algorithm needs @math bits of advice to achieve a competitive ratio better than @math , where @math is the cache size. Previously, it was only known that @math bits of advice were necessary to achieve a constant competitive ratio smaller than @math . (2) Every @math -competitive vertex coloring algorithm must use @math bits of advice. Previously, it was only known that @math bits of advice were necessary to be optimal. For certain online problems, including the MTS, @math -server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic @math -competitive @math -server algorithm with advice complexity @math if and only if there exists a randomized @math -competitive @math -server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Ros 'en [ICALP'09].
As already mentioned, our information theoretical direct product theorem ( mainthm ) is an extension of a previous lower bound technique due to @cite_3 . We will call this technique the . The only known application of this technique is a lower bound for generalized matching pennies @cite_3 , but it is clear that the technique could also be used (together with an @math -round input distribution) for other online problems.
@cite_3
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A strong direct product theorem states that, in order to solve k instances of a problem, if we provide less than k times the resource required to compute one instance, then the probability of overall success is exponentially small in k. In this paper, we consider the model of two-way public-coin communication complexity and show a strong direct product theorem for all relations in terms of the smooth rectangle bound, introduced by Jain and Klauck as a generic lower bound method in this model. Our result therefore uniformly implies a strong direct product theorem for all relations for which an (asymptotically) optimal lower bound can be provided using the smooth rectangle bound, for example Inner Product, Greater-Than, Set-Disjointness, Gap-Hamming Distance etc. Our result also implies near optimal direct product results for several important functions and relations used to show exponential separations between classical and quantum communication complexity, for which near optimal lower bounds are provided using the rectangle bound, for example by Raz [1999], Gavinsky [2008] and Klartag and Regev [2011]. In fact we are not aware of any relation for which it is known that the smooth rectangle bound does not provide an optimal lower bound. This lower bound subsumes many of the other lower bound methods, for example the rectangle bound (a.k.a the corruption bound), the smooth discrepancy bound (a.k.a the bound) which in turn subsumes the discrepancy bound, the subdistribution bound and the conditional min-entropy bound. We show our result using information theoretic arguments. A key tool we use is a sampling protocol due to Braverman [2012], in fact a modification of it used by Kerenidis, Laplante, Lerays, Roland and Xiao [2012]. An instance of the generalized partial cover problem consists of a ground set U and a family of subsets @math . Each element e∈U is associated with a profit p(e), whereas each subset @math has a cost c(S). The objective is to find a minimum cost subcollection @math such that the combined profit of the elements covered by @math is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e∈U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection @math that minimizes the cost of @math plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results. We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within @math assuming @math . These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model. The “direct-product code” of a function @math gives its values on all @math -tuples @math . This basic construct underlies “hardness amplification” in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the “list-decoding” regime. In particular, they give a @math -query test which works for polynomially small success probability @math and show that no such test works below success probability @math . Our main result is a @math -query test which works for exponentially small success probability @math . Our techniques (based on recent simplified decoding algorithms for the same code [R. , Proceeding...