46,784 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
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
Non-asymptotic Upper Bounds for Deletion Correcting Codes
Explicit non-asymptotic upper bounds on the sizes of multiple-deletion
correcting codes are presented. In particular, the largest single-deletion
correcting code for -ary alphabet and string length is shown to be of
size at most . An improved bound on the asymptotic
rate function is obtained as a corollary. Upper bounds are also derived on
sizes of codes for a constrained source that does not necessarily comprise of
all strings of a particular length, and this idea is demonstrated by
application to sets of run-length limited strings.
The problem of finding the largest deletion correcting code is modeled as a
matching problem on a hypergraph. This problem is formulated as an integer
linear program. The upper bound is obtained by the construction of a feasible
point for the dual of the linear programming relaxation of this integer linear
program.
The non-asymptotic bounds derived imply the known asymptotic bounds of
Levenshtein and Tenengolts and improve on known non-asymptotic bounds.
Numerical results support the conjecture that in the binary case, the
Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure
Relaxing the Irrevocability Requirement for Online Graph Algorithms
Online graph problems are considered in models where the irrevocability
requirement is relaxed. Motivated by practical examples where, for example,
there is a cost associated with building a facility and no extra cost
associated with doing it later, we consider the Late Accept model, where a
request can be accepted at a later point, but any acceptance is irrevocable.
Similarly, we also consider a Late Reject model, where an accepted request can
later be rejected, but any rejection is irrevocable (this is sometimes called
preemption). Finally, we consider the Late Accept/Reject model, where late
accepts and rejects are both allowed, but any late reject is irrevocable. For
Independent Set, the Late Accept/Reject model is necessary to obtain a constant
competitive ratio, but for Vertex Cover the Late Accept model is sufficient and
for Minimum Spanning Forest the Late Reject model is sufficient. The Matching
problem has a competitive ratio of 2, but in the Late Accept/Reject model, its
competitive ratio is 3/2
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