We present the first non-trivial online algorithms for the non-uniform, multicommodity buy-at-bulk (MC-BB) network design problem in undirected and directed graphs. Our competitive ratios qualitatively match the best known approximation factors for the corresponding offline problems. The main engine for our results is an online reduction theorem of MC-BB problems to their single-sink (SS-BB) counterparts. We use the concept of junction-tree solutions (Chekuri et al., FOCS 2006) that play an important role in solving the offline versions of the problem via a greedy subroutine -- an inherently offline procedure. Our main technical contribution is in designing an online algorithm using only the existence of good junction-trees to reduce an MC-BB instance to multiple SS-BB sub-instances. Along the way, we also give the first non-trivial online node-weighted/directed single-sink buy-at-bulk algorithms. In addition to the new results, our generic reduction also yields new proofs of recent results for the online node-weighted Steiner forest and online group Steiner forest problems.
The online (uniform) buy-at-bulk network design problem asks us to design a network, where the edge-costs exhibit economy-of-scale. Previous approaches to this problem used tree- embeddings, giving us randomized algorithms. Moreover, the optimal results with a logarithmic competitive ratio requires the metric on which the network is being built to be known up-front; the competitive ratios then depend on the size of this metric (which could be much larger than the number of terminals that arrive). We consider the buy-at-bulk problem in the least restrictive model where the metric is not known i…
In the oblivious buy-at-bulk network design problem in a graph, the task is to compute a fixed set of paths for every pair of source-destinations in the graph, such that any set of demands can be routed along these paths. The demands could be aggregated at intermediate edges where the fusion-cost is specified by a canonical (non-negative concave) function $f$. We give a novel algorithm for planar graphs which is oblivious with respect to the demands, and is also oblivious with respect to the fusion function $f$. The algorithm is deterministic and computes the fixed set of paths in polynomial t…
We consider the problem of constructing a single spanning tree for the single-source buy-at-bulk network design problem for doubling-dimension graphs. We compute a spanning tree to route a set of demands (or data) along a graph to or from a designated root node. The demands could be aggregated at (or symmetrically distributed to) intermediate nodes where the fusion-cost is specified by a non-negative concave function $f$. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number of data sources (or demands) and cost function at int…
In the Multislope Ski Rental problem, the user needs a certain resource for some unknown period of time. To use the resource, the user must subscribe to one of several options, each of which consists of a one-time setup cost (``buying price''), and cost proportional to the duration of the usage (``rental rate''). The larger the price, the smaller the rent. The actual usage time is determined by an adversary, and the goal of an algorithm is to minimize the cost by choosing the best option at any point in time. Multislope Ski Rental is a natural generalization of the classical Ski Rental problem…
Let $G$ be a graph with edge set $(e_1,e_2,...e_N)$. We independently associate to each edge $e_i$ of $G$ a cost ${x}_i$ that is drawn from a Uniform [0, 1] distribution. Suppose $\mathcal{F}$ is a set of targeted structures that consists of subgraphs of $G$. We would like to buy a subset of $\mathcal{F}$ at small cost, however we do not know a priori the values of the random variables ${x}_1,...,{x}_N$. Instead, we inspect the random variables $x_i$ one at a time. As soon as we inspect the random variable associated with the cost of an edge we have to decide whether we want to buy that edge o…
We consider the single-source (or single-sink) buy-at-bulk problem with an unknown concave cost function. We want to route a set of demands along a graph to or from a designated root node, and the cost of routing x units of flow along an edge is proportional to some concave, non-decreasing function f such that f(0) = 0. We present a polynomial time algorithm that finds a distribution over trees such that the expected cost of a tree for any f is within an O(1)-factor of the optimum cost for that f. The previous best simultaneous approximation for this problem, even ignoring computation time, wa…
We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized fo…
In the context of investment analysis, we formulate an abstract online computing problem called a planning game and develop general tools for solving such a game. We then use the tools to investigate a practical buy-and-hold trading problem faced by long-term investors in stocks. We obtain the unique optimal static online algorithm for the problem and determine its exact competitive ratio. We also compare this algorithm with the popular dollar averaging strategy using actual market data.
Binghamton University（宾汉姆顿大学）；State University of New York（纽约州立大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
We study the problem of augmenting online algorithms with machine learned (ML) advice. In particular, we consider the \emph{multi-shop ski rental} (MSSR) problem, which is a generalization of the classical ski rental problem. In MSSR, each shop has different prices for buying and renting a pair of skis, and a skier has to make decisions on when and where to buy. We obtain both deterministic and randomized online algorithms with provably improved performance when either a single or multiple ML predictions are used to make decisions. These online algorithms have no knowledge about the quality or…
Microsoft Research（微软研究院）；The University of Hong Kong（香港大学）；Stanford University（斯坦福大学）
We consider revenue maximization in online auction/pricing problems. A seller sells an identical item in each period to a new buyer, or a new set of buyers. For the online posted pricing problem, we show regret bounds that scale with the best fixed price, rather than the range of the values. We also show regret bounds that are almost scale free, and match the offline sample complexity, when comparing to a benchmark that requires a lower bound on the market share. These results are obtained by generalizing the classical learning from experts and multi-armed bandit problems to their multi-scale …
Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival process and the matching process. The Online Stochastic Matching with Timeouts problem introduced by Bansal, et al., (Algorithmica, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a …
We consider fractional online covering problems with $\ell_q$-norm objectives. The problem of interest is of the form $\min\{ f(x) \,:\, Ax\ge 1, x\ge 0\}$ where $f(x)=\sum_{e} c_e \|x(S_e)\|_{q_e} $ is the weighted sum of $\ell_q$-norms and $A$ is a non-negative matrix. The rows of $A$ (i.e. covering constraints) arrive online over time. We provide an online $O(\log d+\log ρ)$-competitive algorithm where $ρ= \frac{\max a_{ij}}{\min a_{ij}}$ and $d$ is the maximum of the row sparsity of $A$ and $\max |S_e|$. This is based on the online primal-dual framework where we use the dual of the above c…
A popular line of recent research incorporates ML advice in the design of online algorithms to improve their performance in typical instances. These papers treat the ML algorithm as a black-box, and redesign online algorithms to take advantage of ML predictions. In this paper, we ask the complementary question: can we redesign ML algorithms to provide better predictions for online algorithms? We explore this question in the context of the classic rent-or-buy problem, and show that incorporating optimization benchmarks in ML loss functions leads to significantly better performance, while mainta…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education), Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学数学科学学院，教育部MEA重点实验室，上海市PMMP重点实验室）
We study online combinatorial auctions with production costs proposed by Blum et al. using the online primal dual framework. In this model, buyers arrive online, and the seller can produce multiple copies of each item subject to a non-decreasing marginal cost per copy. The goal is to allocate items to maximize social welfare less total production cost. For arbitrary (strictly convex and differentiable) production cost functions, we characterize the optimal competitive ratio achievable by online mechanisms/algorithms. We show that online posted pricing mechanisms, which are incentive compatible…
University of Pennsylvania（宾夕法尼亚大学）；California Institute of Technology（加州理工学院）
We consider the problem of learning from revealed preferences in an online setting. In our framework, each period a consumer buys an optimal bundle of goods from a merchant according to her (linear) utility function and current prices, subject to a budget constraint. The merchant observes only the purchased goods, and seeks to adapt prices to optimize his profits. We give an efficient algorithm for the merchant's problem that consists of a learning phase in which the consumer's utility function is (perhaps partially) inferred, followed by a price optimization step. We also consider an alternat…
