73,406 research outputs found
Learning Augmented Online Facility Location
Following the research agenda initiated by Munoz & Vassilvitskii [1] and
Lykouris & Vassilvitskii [2] on learning-augmented online algorithms for
classical online optimization problems, in this work, we consider the Online
Facility Location problem under this framework. In Online Facility Location
(OFL), demands arrive one-by-one in a metric space and must be (irrevocably)
assigned to an open facility upon arrival, without any knowledge about future
demands.
We present an online algorithm for OFL that exploits potentially imperfect
predictions on the locations of the optimal facilities. We prove that the
competitive ratio decreases smoothly from sublogarithmic in the number of
demands to constant, as the error, i.e., the total distance of the predicted
locations to the optimal facility locations, decreases towards zero. We
complement our analysis with a matching lower bound establishing that the
dependence of the algorithm's competitive ratio on the error is optimal, up to
constant factors. Finally, we evaluate our algorithm on real world data and
compare our learning augmented approach with the current best online algorithm
for the problem
Online facility location with timed-requests and congestion
The classic online facility location problem deals with finding the optimal
set of facilities in an online fashion when demand requests arrive one at a
time and facilities need to be opened to service these requests. In this work,
we study two variants of the online facility location problem; (1) timed
requests and (2) congestion. Both of these variants are motivated by the
applications to real life and the previously known results on online facility
location cannot be directly adapted to analyse them.
Timed requests : In this variant, each demand request is a pair where
the is the standard location of the demand while is the corresponding
weight of the request. The cost of servicing request at facility is
where is the set of facilities available at the time of
request . For this variant, we present an online algorithm attaining a
competitive ratio of in the secretarial model for the
timed requests and show that it is optimal.
Congestion : The congestion variant considers the case when there is an
additional congestion cost that grows with the number of requests served by
each request. For this variant, when the congestion cost is a monomial, we show
that there exists an algorithm attaining a constant competitive ratio. This
constant is a function of the exponent of the monomial and the facility opening
cost but independent of the number of requests.Comment: 25 pages, 6 figure
A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location
In the online non-metric variant of the facility location problem, there is a
given graph consisting of a set of facilities (each with a certain opening
cost), a set of potential clients, and weighted connections between them.
The online part of the input is a sequence of clients from , and in response
to any requested client, an online algorithm may open an additional subset of
facilities and must connect the given client to an open facility.
We give an online, polynomial-time deterministic algorithm for this problem,
with a competitive ratio of . The
result is optimal up to loglog factors. Our algorithm improves over the
-competitive
construction that first reduces the facility location instance to a set cover
one and then later solves such instance using the deterministic algorithm by
Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical
scenario where .
We achieve this by a more direct approach: we design an algorithm for a
fractional relaxation of the non-metric facility location problem with
clustered facilities. To handle the constraints of such non-covering LP, we
combine the dual fitting and multiplicative weight updates approach. By
maintaining certain additional monotonicity properties of the created
fractional solution, we can handle the dependencies between facilities and
connections in a rounding routine.
Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the
first deterministic algorithm for the online node-weighted Steiner tree
problem. The resulting competitive ratio is on
graphs of nodes and terminals.Comment: STACS 202
The Online Median Problem
We introduce a natural variant of the (metric uncapacitated) k-median problem that we call the online median problem. Whereas the k-median problem involves optimizing the simultaneous placement of k facilities, the online median problem imposes the following additional constraints: the facilities are placed one at a time, a facility cannot be moved once it is placed, and the total number of facilities to be placed, k, is not known in advance. The objective of an online median algorithm is to minimize the competitive ratio, that is, the worst-case ratio of the cost of an online placement to that of an optimal offline placement. Our main result is a constant-competitive algorithm for the online median problem running in time that is linear in the input size. In addition, we present a related, though substantially simpler, constant-factor approximation algorithm for the (metric uncapacitated) facility location problem that runs in time linear in the input size. The latter algorithm is similar in spirit to the recent primal-dual-based facility location algorithm of Jain and Vazirani, but our approach is more elementary and yields an improved running time. While our primary focus is on problems which ask us to minimize the weighted average service distance to facilities, we also show that our results can be generalized to hold, to within constant factors, for more general objective functions. For example, we show that all of our approximation results hold, to within constant factors, for the k-means objective function
Facility Reallocation on the Line
We consider a multi-stage facility reallocation problems on the real line, where a facility is being moved between time stages based on the locations reported by agents. The aim of the reallocation algorithm is to minimise the social cost, i.e., the sum over the total distance between the facility and all agents at all stages, plus the cost incurred for moving the facility. We study this problem both in the offline setting and online setting. In the offline case the algorithm has full knowledge of the agent locations in all future stages, and in the online setting the algorithm does not know these future locations and must decide the location of the facility on a stage-per-stage basis. We derive the optimal algorithm in both cases. For the online setting we show that its competitive ratio is . As neither of these algorithms turns out to yield a strategy-proof mechanism, we propose another strategy-proof mechanism which has a competitive ratio of for odd and for even , which we conjecture to be the best possible. We also consider a generalisation with multiple facilities and weighted agents, for which we show that the optimum can be computed in polynomial time for a fixed number of facilities
Online Network Design Algorithms via Hierarchical Decompositions
We develop a new approach for online network design and obtain improved
competitive ratios for several problems. Our approach gives natural
deterministic algorithms and simple analyses. At the heart of our work is a
novel application of embeddings into hierarchically well-separated trees (HSTs)
to the analysis of online network design algorithms --- we charge the cost of
the algorithm to the cost of the optimal solution on any HST embedding of the
terminals. This analysis technique is widely applicable to many problems and
gives a unified framework for online network design.
In a sense, our work brings together two of the main approaches to online
network design. The first uses greedy-like algorithms and analyzes them using
dual-fitting. The second uses tree embeddings and results in randomized -competitive algorithms, where is the total number of vertices in the
graph. Our approach uses deterministic greedy-like algorithms but analyzes them
via HST embeddings of the terminals. Our proofs are simpler as we do not need
to carefully construct dual solutions and we get competitive
ratios, where is the number of terminals.
In this paper, we apply our approach to obtain deterministic -competitive online algorithms for the following problems.
- Steiner network with edge duplication. Previously, only a randomized
-competitive algorithm was known.
- Rent-or-buy. Previously, only deterministic -competitive and
randomized -competitive algorithms by Awerbuch, Azar and Bartal
(2004) were known.
- Connected facility location. Previously, only a randomized -competitive algorithm by San Felice, Williamson and Lee (2014) was known.
- Prize-collecting Steiner forest. We match the competitive ratio first
achieved by Qian and Williamson (2011) and give a simpler analysis.Comment: Accepted to SODA 201
Efficient Approximations for the Online Dispersion Problem
The dispersion problem has been widely studied in computational geometry and facility location, and is closely related to the packing problem. The goal is to locate n points (e.g., facilities or persons) in a k-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore we study, for the first time in the literature, the online dispersion problem in Euclidean space.
There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly non-trivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance.
For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is (2ln2+epsilon)-competitive, where epsilon>0 can be arbitrarily small and the algorithm\u27s running time is polynomial in 1/epsilon. We show this algorithm is actually optimal. For the same problem in a square, we provide a 1.591-competitive algorithm and a 1.183 lower-bound. Furthermore, for arbitrary k-dimensional polytopes with k>=2, we provide a 2/(1-epsilon)-competitive algorithm and a 7/6 lower-bound. All our lower-bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary k-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions
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