Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform's algorithm may miss the users most disposed to do so. This mismatch decreases the platform's revenue and the advertiser's chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer.Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models.
Motivated by maximizing spot instances in cloud shared systems, in this work, we consider the problem of taking advantage of unused resources in highly dynamic cloud environments while preserving users’ performance. We introduce an online model for sharing resources that captures basic properties of cloud systems, such as unpredictable users’ demand patterns, very limited feedback from the system, and service level agreement (SLA) between the users and the cloud provider. We provide a simple and efficient algorithm for the single-resource case. For any demand patterns, our algorithm guarantees near-optimal resource utilization as well as high users’ performance compared with their SLA baseline. In addition to this, we validate empirically the performance of our algorithm using synthetic data and data obtained from Microsoft’s systems.
Motivated by bursty bandwidth allocation and by the allocation of virtual machines to servers in the cloud, we consider the online problem of packing items with random sizes into unit-capacity bins. Items arrive sequentially, but on arrival, an item’s actual size is unknown; only its probabilistic information is available to the decision maker. Without knowing this size, the decision maker must irrevocably pack the item into an available bin or place it in a new bin. Once packed in a bin, the decision maker observes the item’s actual size, and overflowing the bin is a possibility. An overflow incurs a large penalty cost, and the corresponding bin is unusable for the rest of the process. In practical terms, this overflow models delayed services, failure of servers, and/or loss of end-user goodwill. The objective is to minimize the total expected cost given by the sum of the number of opened bins and the overflow penalty cost. We present an online algorithm with expected cost at most a constant factor times the cost incurred by the optimal packing policy when item sizes are drawn from an independent and identically distributed (i.i.d.) sequence of unknown length. We give a similar result when item size distributions are exponential with arbitrary rates. We also study the offline model, where distributions are known in advance but must be packed sequentially. We construct a soft-capacity polynomial-time approximation scheme for this problem and show that the complexity of computing the optimal offline cost is [Formula: see text]-hard. Finally, we provide an empirical study of our online algorithm’s performance.
Cloud computing has motivated renewed interest in resource allocation problems with new consumption models. A common goal is to share a resource, such as CPU or I/O bandwidth, among distinct users with different demand patterns as well as different quality of service requirements. To ensure these service requirements, cloud offerings often come with a service level agreement (SLA) between the provider and the users. An SLA specifies the amount of a resource a user is entitled to utilize. In many cloud settings, providers would like to operate resources at high utilization while simultaneously respecting individual SLAs. There is typically a tradeoff between these two objectives; for example, utilization can be increased by shifting away resources from idle users to "scavenger" workload, but with the risk of the former then becoming active again. We study this fundamental tradeoff by formulating a resource allocation model that captures basic properties of cloud computing systems, including SLAs, highly limited feedback about the state of the system, and variable and unpredictable input sequences. Our main result is a simple and practical algorithm that achieves near-optimal performance on the above two objectives. First, we guarantee nearly optimal utilization of the resource even if compared to the omniscient offline dynamic optimum. Second, we simultaneously satisfy all individual SLAs up to a small error. The main algorithmic tool is a multiplicative weight update algorithm, and a duality argument to obtain its guarantees. Experiments on both synthetic and real production traces demonstrate the merits of our algorithm in practical settings.
The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of n nodes. In this paper we consider the situation where the joint input to the nodes is an nnode labeled graph G, i.e., the local input received by each node is the indicator function of its neighborhood in G. Nodes execute an algorithm, communicating with each other in synchronous rounds and their goal is to compute some function that depends on G. In every round, each of the n nodes may send up to n − 1 different b-bit messages through each of its n − 1 communication links. We denote by R the number of rounds of the algorithm. The product Rb, that is, the total number of bits received by a node through one link, is the cost of the algorithm.The most difficult problem we could attempt to solve is the reconstruction problem, where nodes are asked to recover all the edges of the input graph G. Formally, given a class of graphs G, the problem is defined as follows: if G / ∈ G, then every node must reject; on the other hand, if G ∈ G, then every node must end up, after the R rounds, knowing all the edges of G. It is not difficult to see that the cost Rb of any algorithm that solves this problem (even with public coins) is at least Ω(log |G n |/n), where G n is the subclass of all n-node labeled graphs in G. In this paper we prove that previous bound is tight and that it is possible to achieve it with only R = 2 rounds. More precisely, we exhibit (i) a one-round algorithm that achieves this bound for hereditary graph classes; and (ii) a two-round algorithm that achieves this bound for arbitrary graph classes. Later, we show that the bound Ω(log |G n |/n) cannot be achieved in one-round for arbitrary graph classes, and we give tight algorithms for that case.From (i) we recover all known results concerning the reconstruction of graph classes in one round and bandwidth O(log n): forests, planar graphs, cographs, etc. But we also get new one-round algorithms for other hereditary graph classes such as unit disc graphs, interval graphs, etc. From (ii), we can conclude that any problem restricted to a class of graphs of size 2 O(n log n) can be solved in the congested clique model in two rounds, with bandwidth O(log n). Moreover, our general two-round algorithm is valid for any set of labeled graphs, not only for graph classes (which are sets of labeled graphs closed under isomorphims).
Diversity plays a crucial role in multiple contexts such as team formation, representation of minority groups and generally when allocating resources fairly. Given a community composed by individuals of different types, we study the problem of partitioning this community such that the global diversity is preserved as much as possible in each subgroup. We consider the diversity metric introduced by Simpson in his influential work that, roughly speaking, corresponds to the inverse probability that two individuals are from the same type when taken uniformly at random, with replacement, from the community of interest. We provide a novel perspective by reinterpreting this quantity in geometric terms. We characterize the instances in which the optimal partition exactly preserves the global diversity in each subgroup. When this is not possible, we provide an efficient polynomial-time algorithm that outputs an optimal partition for the problem with two types. Finally, we discuss further challenges and open questions for the problem that considers more than two types.
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