The Online Min-Sum Set Cover Problem
OPEN ACCESS
Author / Producer
Date
2020
Publication Type
Conference Paper
ETH Bibliography
yes
Citations
Altmetric
OPEN ACCESS
Data
Rights / License
Abstract
We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on n elements based on subsets S₁, S₂, … arriving online. The algorithm serves each set S_t upon arrival, using its current permutation π_t, incurring an access cost equal to the position of the first element of S_t in π_t. Then, the algorithm may update its permutation to π_{t+1}, incurring a moving cost equal to the Kendall tau distance of π_t to π_{t+1}. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the r-uniform version, where each S_t has cardinality r. List update is the special case where r = 1. We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of (r+1)(1-r/(n+1)) on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a O(r)-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm. Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the k-server problem. We show that its competitive ratio is Ω(r²) and 2^{O(√{log n ⋅ log r})}, and conjecture that it is f(r)-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is Ω(r √n) and O(r^{3/2} √n)-competitive.
Permanent link
Publication status
published
External links
Book title
47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Volume
168
Pages / Article No.
51
Publisher
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Event
47th International Colloquium on Automata, Languages, and Programming (ICALP 2020) (virtual)
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Online Algorithms; Competitive Analysis; Min-Sum Set Cover
Organisational unit
Notes
Due to the Corona virus (COVID-19) the conference was conducted virtually.