Knapsack Secretary with Bursty Adversary

06/20/2020
by   Thomas Kesselheim, et al.
0

The random-order or secretary model is one of the most popular beyond-worst case model for online algorithms. While it avoids the pessimism of the traditional adversarial model, in practice we cannot expect the input to be presented in perfectly random order. This has motivated research on “best of both worlds” (algorithms with good performance on both purely stochastic and purely adversarial inputs), or even better, on inputs that are a mix of both stochastic and adversarial parts. Unfortunately the latter seems much harder to achieve and very few results of this type are known. Towards advancing our understanding of designing such robust algorithms, we propose a random-order model with bursts of adversarial time steps. The assumption of burstiness of unexpected patterns is reasonable in many contexts, since changes (e.g. spike in a demand for a good) are often triggered by a common external event. We then consider the Knapsack Secretary problem in this model: there is a knapsack of size k (e.g., available quantity of a good), and in each of the n time steps an item comes with its value and size in [0,1] and the algorithm needs to make an irrevocable decision whether to accept or reject the item. We design an algorithm that gives an approximation of 1 - Õ(Γ/k) when the adversarial time steps can be covered by Γ≥√(k) intervals of size Õ(n/k). In particular, setting Γ = √(k) gives a (1 - O(ln^2 k/√(k)))-approximation that is resistant to up to a ln^2 k/√(k)-fraction of the items being adversarial, which is almost optimal even in the absence of adversarial items. Also, setting Γ = Ω̃(k) gives a constant approximation that is resistant to up to a constant fraction of items being adversarial.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
10/13/2022

Online Algorithms for the Santa Claus Problem

The Santa Claus problem is a fundamental problem in fair division: the g...
research
07/11/2019

Competitive Analysis with a Sample and the Secretary Problem

We extend the standard online worst-case model to accommodate past exper...
research
09/28/2022

Repeated Prophet Inequality with Near-optimal Bounds

In modern sample-driven Prophet Inequality, an adversary chooses a seque...
research
12/04/2017

Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack

The stochastic knapsack problem is the stochastic variant of the classic...
research
02/25/2020

Random-Order Models

This chapter introduces the random-order model in online algorithms. In ...
research
11/17/2019

Robust Algorithms for the Secretary Problem

In classical secretary problems, a sequence of n elements arrive in a un...
research
11/15/2017

Online Allocation with Traffic Spikes: Mixing Adversarial and Stochastic Models

Motivated by Internet advertising applications, online allocation proble...

Please sign up or login with your details

Forgot password? Click here to reset