46,784 research outputs found

    Permutation Strikes Back: The Power of Recourse in Online Metric Matching

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    In the classical Online Metric Matching problem, we are given a metric space with kk 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 2k12k-1, 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 O(logk)O(\log k) competitive algorithm~\cite{raghvendra2018optimal}. Obtaining (or refuting) O(logk)O(\log k)-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} O(logk)O(\log k)-competitive algorithm with O(logk)O(\log k)-amortized recourse, an exponential improvement over the 2k12k-1 lower bound when no recourse is allowed. We next consider the line metric, and present a deterministic algorithm which is 33-competitive and has O(logk)O(\log k)-recourse, again a substantial improvement over the best known O(logk)O(\log k)-competitive algorithm when no recourse is allowed

    On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter

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    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 O(n1ε)O(n^{1-\varepsilon}) exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the O(m1/2ε)O(m^{1/2-\varepsilon}) bound for sparse graphs of Henzinger et al. [STOC'15] and O(n1/3ε)O(n^{1/3-\varepsilon}) 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 O(mn)O(m\sqrt{n}) exists for the former by Bosek et al. [FOCS'14]. -- No algorithm with amortized update time O(m1ε)O(m^{1-\varepsilon}) 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 O(n1ε)O(n^{1-\varepsilon}) exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time O(n1/2ε)O(n^{1/2 - \varepsilon}) exists for incremental or decremental (4/3ε)(4/3-\varepsilon')-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

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    Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for qq-ary alphabet and string length nn is shown to be of size at most qnq(q1)(n1)\frac{q^n-q}{(q-1)(n-1)}. 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

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    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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