7,796 research outputs found
Permutation Strikes Back: The Power of Recourse in Online Metric Matching
In the classical Online Metric Matching problem, we are given a metric space
with servers. A collection of clients arrive in an online fashion, and upon
arrival, a client should irrevocably be matched to an as-yet-unmatched server.
The goal is to find an online matching which minimizes the total cost, i.e.,
the sum of distances between each client and the server it is matched to. We
know deterministic algorithms~\cite{KP93,khuller1994line} that achieve a
competitive ratio of , and this bound is tight for deterministic
algorithms. The problem has also long been considered in specialized metrics
such as the line metric or metrics of bounded doubling dimension, with the
current best result on a line metric being a deterministic
competitive algorithm~\cite{raghvendra2018optimal}. Obtaining (or refuting)
-competitive algorithms in general metrics and constant-competitive
algorithms on the line metric have been long-standing open questions in this
area.
In this paper, we investigate the robustness of these lower bounds by
considering the Online Metric Matching with Recourse problem where we are
allowed to change a small number of previous assignments upon arrival of a new
client. Indeed, we show that a small logarithmic amount of recourse can
significantly improve the quality of matchings we can maintain. For general
metrics, we show a simple \emph{deterministic} -competitive
algorithm with -amortized recourse, an exponential improvement over
the lower bound when no recourse is allowed. We next consider the line
metric, and present a deterministic algorithm which is -competitive and has
-recourse, again a substantial improvement over the best known
-competitive algorithm when no recourse is allowed
Efficient Classification for Metric Data
Recent advances in large-margin classification of data residing in general
metric spaces (rather than Hilbert spaces) enable classification under various
natural metrics, such as string edit and earthmover distance. A general
framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004]
left open the questions of computational efficiency and of providing direct
bounds on generalization error.
We design a new algorithm for classification in general metric spaces, whose
runtime and accuracy depend on the doubling dimension of the data points, and
can thus achieve superior classification performance in many common scenarios.
The algorithmic core of our approach is an approximate (rather than exact)
solution to the classical problems of Lipschitz extension and of Nearest
Neighbor Search. The algorithm's generalization performance is guaranteed via
the fat-shattering dimension of Lipschitz classifiers, and we present
experimental evidence of its superiority to some common kernel methods. As a
by-product, we offer a new perspective on the nearest neighbor classifier,
which yields significantly sharper risk asymptotics than the classic analysis
of Cover and Hart [IEEE Trans. Info. Theory, 1967].Comment: This is the full version of an extended abstract that appeared in
Proceedings of the 23rd COLT, 201
A Match in Time Saves Nine: Deterministic Online Matching With Delays
We consider the problem of online Min-cost Perfect Matching with Delays
(MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number
of requests appear in a metric space at different times and the goal of an
online algorithm is to match them in pairs. In contrast to traditional online
matching problems, in MPMD all requests appear online and an algorithm can
match any pair of requests, but such decision may be delayed (e.g., to find a
better match). The cost is the sum of matching distances and the introduced
delays.
We present the first deterministic online algorithm for this problem. Its
competitive ratio is , where is the
number of requests. This is polynomial in the number of metric space points if
all requests are given at different points. In particular, the bound does not
depend on other parameters of the metric, such as its aspect ratio. Unlike
previous (randomized) solutions for the MPMD problem, our algorithm does not
need to know the metric space in advance
Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
Low-distortional metric embeddings are a crucial component in the modern
algorithmic toolkit. In an online metric embedding, points arrive sequentially
and the goal is to embed them into a simple space irrevocably, while minimizing
the distortion. Our first result is a deterministic online embedding of a
general metric into Euclidean space with distortion (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , generalizing previous
results (Indyk et al.\ (2010), Bartal et al.\ (2020)).
Next, we study the \emph{online minimum-weight perfect matching} problem,
where a sequence of metric points arrive in pairs, and one has to maintain
a perfect matching at all times. We allow recourse (as otherwise the order of
arrival determines the matching). The goal is to return a perfect matching that
approximates the \emph{minimum-weight} perfect matching at all times, while
minimizing the recourse. Our third result is a randomized algorithm with
competitive ratio and recourse against an
oblivious adversary, this result is obtained via our new stochastic online
embedding. Our fourth result is a deterministic algorithm against an adaptive
adversary, using recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24
Metric Embedding via Shortest Path Decompositions
We study the problem of embedding shortest-path metrics of weighted graphs
into spaces. We introduce a new embedding technique based on low-depth
decompositions of a graph via shortest paths. The notion of Shortest Path
Decomposition depth is inductively defined: A (weighed) path graph has shortest
path decomposition (SPD) depth . General graph has an SPD of depth if it
contains a shortest path whose deletion leads to a graph, each of whose
components has SPD depth at most . In this paper we give an
-distortion embedding for graphs of SPD
depth at most . This result is asymptotically tight for any fixed ,
while for it is tight up to second order terms.
As a corollary of this result, we show that graphs having pathwidth embed
into with distortion . For
, this improves over the best previous bound of Lee and Sidiropoulos that
was exponential in ; moreover, for other values of it gives the first
embeddings whose distortion is independent of the graph size . Furthermore,
we use the fact that planar graphs have SPD depth to give a new
proof that any planar graph embeds into with distortion . Our approach also gives new results for graphs with bounded treewidth,
and for graphs excluding a fixed minor
Faster Clustering via Preprocessing
We examine the efficiency of clustering a set of points, when the
encompassing metric space may be preprocessed in advance. In computational
problems of this genre, there is a first stage of preprocessing, whose input is
a collection of points ; the next stage receives as input a query set
, and should report a clustering of according to some
objective, such as 1-median, in which case the answer is a point
minimizing .
We design fast algorithms that approximately solve such problems under
standard clustering objectives like -center and -median, when the metric
has low doubling dimension. By leveraging the preprocessing stage, our
algorithms achieve query time that is near-linear in the query size ,
and is (almost) independent of the total number of points .Comment: 24 page
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