73 research outputs found
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
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
Maintaining Perfect Matchings at Low Cost
The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage 2k new nodes appear and we must adapt the solution to a perfect matching for the new instance.
We say that an algorithm is (alpha,beta)-robust if the solutions constructed in both stages are alpha-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most beta k. Hence, alpha measures the quality of the algorithm and beta its robustness. In this setting we aim to balance both measures by deriving algorithms for constant alpha and beta. We show that there exists an algorithm that is (3,1)-robust for any metric if one knows the number 2k of arriving nodes in advance. For the case that k is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a (10,2)-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy
Online Steiner Tree with Deletions
In the online Steiner tree problem, the input is a set of vertices that
appear one-by-one, and we have to maintain a Steiner tree on the current set of
vertices. The cost of the tree is the total length of edges in the tree, and we
want this cost to be close to the cost of the optimal Steiner tree at all
points in time. If we are allowed to only add edges, a tight bound of
on the competitiveness is known. Recently it was shown that if
we can add one new edge and make one edge swap upon every vertex arrival, we
can maintain a constant-competitive tree online.
But what if the set of vertices sees both additions and deletions? Again, we
would like to obtain a low-cost Steiner tree with as few edge changes as
possible. The original paper of Imase and Waxman had also considered this
model, and it gave a greedy algorithm that maintained a constant-competitive
tree online, and made at most edge changes for the first
requests. In this paper give the following two results.
Our first result is an online algorithm that maintains a Steiner tree only
under deletions: we start off with a set of vertices, and at each time one of
the vertices is removed from this set: our Steiner tree no longer has to span
this vertex. We give an algorithm that changes only a constant number of edges
upon each request, and maintains a constant-competitive tree at all times. Our
algorithm uses the primal-dual framework and a global charging argument to
carefully make these constant number of changes.
We then study the natural greedy algorithm proposed by Imase and Waxman that
maintains a constant-competitive Steiner tree in the fully-dynamic model (where
each request either adds or deletes a vertex). Our second result shows that
this algorithm makes only a constant number of changes per request in an
amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc
Online Maximum Matching with Recourse
We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter k. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most k such actions per edge take place, where k is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. [Avitabile et al., 2013], whereas the special case k=2 was studied by Boyar et al. [Boyar et al., 2017].
In the first part of this paper, we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP given in [Avitabile et al., 2013], by exploiting the structure of the matching problem. In addition, we extend the result of [Boyar et al., 2017] and show that the greedy algorithm has competitive ratio 3/2 for every even k and ratio 2 for every odd k. Moreover, we present and analyze an improvement of the greedy algorithm which we call L-Greedy, and we show that for small values of k it outperforms the algorithm of [Avitabile et al., 2013]. In terms of lower bounds, we show that no deterministic algorithm better than 1+1/(k-1) exists, improving upon the lower bound of 1+1/k shown in [Avitabile et al., 2013].
The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of L-Greedy and AMP carry through in this model; moreover we show a lower bound of (k^2-3k+6)/(k^2-4k+7) for all even k >= 4. For k in {2,3}, the competitive ratio is 3/2
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
Online traveling salesman problems with rejection options
In this article, we consider online versions of the traveling salesman problem on metric spaces for which requests to visit points are not mandatory. Associated with each request is a penalty (if rejected). Requests are revealed over time (at their release dates) to a server who must decide which requests to accept and serve in order to minimize a linear combination of the time to serve all accepted requests and the total penalties of all rejected requests. In the basic online version of the problem, a request can be accepted any time after its release date. In the real-time online version, a request must be accepted or rejected at the time of its release date. For the basic version, we provide a best possible 2-competitive online algorithm for the problem on a general metric space. For the real-time version, we first consider special metric spaces: on the nonnegative real line, we provide a best possible 2.5-competitive polynomial time online algorithm; on the real line, we prove a Ω(√ln n) lower bound of 2.64 on any competitive ratios and give a 3-competitive online algorithm. We then consider the case of a general metric space and prove a inline image lower bound on the competitive ratio of any online algorithms. Finally, among the restricted class of online algorithms with prior knowledge about the total number of requests n, we propose an asymptotically best possible O(√ln n)-competitive algorithm.United States. Office of Naval Research (Grant N00014-09-1-0326)United States. Office of Naval Research (Grant N00014-12-1-0033)United States. Air Force Office of Scientific Research (Grant FA9550-10-1-0437
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