4 research outputs found
An Improved Algorithm For Online Min-Sum Set Cover
We study a fundamental model of online preference aggregation, where an
algorithm maintains an ordered list of elements. An input is a stream of
preferred sets . Upon seeing and without
knowledge of any future sets, an algorithm has to rerank elements (change the
list ordering), so that at least one element of is found near the list
front. The incurred cost is a sum of the list update costs (the number of swaps
of neighboring list elements) and access costs (position of the first element
of on the list). This scenario occurs naturally in applications such as
ordering items in an online shop using aggregated preferences of shop
customers. The theoretical underpinning of this problem is known as Min-Sum Set
Cover.
Unlike previous work (Fotakis et al., ICALP 2020, NIPS 2020) that mostly
studied the performance of an online algorithm ALG against the static optimal
solution (a single optimal list ordering), in this paper, we study an arguably
harder variant where the benchmark is the provably stronger optimal dynamic
solution OPT (that may also modify the list ordering). In terms of an online
shop, this means that the aggregated preferences of its user base evolve with
time. We construct a computationally efficient randomized algorithm whose
competitive ratio (ALG-to-OPT cost ratio) is and prove the existence
of a deterministic -competitive algorithm. Here, is the maximum
cardinality of sets . This is the first algorithm whose ratio does not
depend on : the previously best algorithm for this problem was -competitive and is a lower bound on the
performance of any deterministic online algorithm.Comment: Presented at AAAI 202
On the Approximability of Multistage Min-Sum Set Cover
We investigate the polynomial-time approximability of the multistage version
of Min-Sum Set Cover (), a natural and intriguing generalization
of the classical List Update problem. In , we maintain a
sequence of permutations on elements, based
on a sequence of requests . We aim to minimize the total
cost of updating to , quantified by the Kendall tau
distance , plus the total cost of
covering each request with the current permutation , quantified by
the position of the first element of in .
Using a reduction from Set Cover, we show that does not admit
an -approximation, unless , and that any
(resp. ) approximation to implies a
sublogarithmic (resp. ) approximation to Set Cover (resp. where each
element appears at most times). Our main technical contribution is to show
that can be approximated in polynomial-time within a factor of
in general instances, by randomized rounding, and within a factor
of , if all requests have cardinality at most , by deterministic
rounding
The Online Min-Sum Set Cover Problem
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.ISSN:1868-896