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Register a new userPERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching
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Sai Sandeep
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In the classical Online Metric Matching problem, we are given a metric space with $k$ 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 $2k-1$, 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 $O(log k)$ competitive algorithm~cite{raghvendra2018optimal}. Obtaining (or refuting) $O(log k)$-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} $O(log k)$-competitive algorithm with $O(log k)$-amortized recourse, an exponential improvement over the $2k-1$ lower bound when no recourse is allowed. We next consider the line metric, and present a deterministic algorithm which is $3$-competitive and has $O(log k)$-recourse, again a substantial improvement over the best known $O(log k)$-competitive algorithm when no recourse is allowed.
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We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight $O(log n)$-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of $O(log n)$ has long been conjectured and remains a tantalizing open question. In this paper, we show improved results in the i.i.d arrival model. We show how the i.i.d model can be used to give substantially better algorithms: our main result is an $O((log log log n)^2)$-competitive algorithm in this model. Along the way we give a $9$-competitive algorithm for the line and tree metrics. Both results imply a strict separation between the i.i.d model and the adversarial and random order models, both for general metrics and these much-studied metrics.
Let $G$ be any $n$-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by $1/sqrt{Delta}$ (for example, a random graph $G$ of average degree~$Theta(Delta)$ typically has this property). We show that the $expBig(c frac{log n}{log Delta}Big)$-round Sherali--Adams linear programming hierarchy certifies that the maximum cut in such a~$G$ is at most $50.1%$ (in fact, at most $tfrac12 + 2^{-Omega(c)}$). For example, in random graphs with $n^{1.01}$ edges, $O(1)$ rounds suffice; in random graphs with $n cdot text{polylog}(n)$ edges, $n^{O(1/log log n)} = n^{o(1)}$ rounds suffice. Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for maxcut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali--Adams can strongly refute random Boolean $k$-CSP instances with $n^{lceil k/2 rceil + delta}$ constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.
In this paper we study the facility location problem in the online with recourse and dynamic algorithm models. In the online with recourse model, clients arrive one by one and our algorithm needs to maintain good solutions at all time steps with only a few changes to the previously made decisions (called recourse). We show that the classic local search technique can lead to a $(1+sqrt{2}+epsilon)$-competitive online algorithm for facility location with only $Oleft(frac{log n}{epsilon}logfrac1epsilonright)$ amortized facility and client recourse. We then turn to the dynamic algorithm model for the problem, where the main goal is to design fast algorithms that maintain good solutions at all time steps. We show that the result for online facility location, combined with the randomized local search technique of Charikar and Guha [10], leads to an $O(1+sqrt{2}+epsilon)$ approximation dynamic algorithm with amortized update time of $tilde O(n)$ in the incremental setting. Notice that the running time is almost optimal, since in general metric space it takes $Omega(n)$ time to specify a new clients position. The approximation factor of our algorithm also matches the best offline analysis of the classic local search algorithm. Finally, we study the fully dynamic model for facility location, where clients can both arrive and depart. Our main result is an $O(1)$-approximation algorithm in this model with $O(|F|)$ preprocessing time and $O(log^3 D)$ amortized update time for the HST metric spaces. Using the seminal results of Bartal [4] and Fakcharoenphol, Rao and Talwar [17], which show that any arbitrary $N$-point metric space can be embedded into a distribution over HSTs such that the expected distortion is at most $O(log N)$, we obtain a $O(log |F|)$ approximation with preprocessing time of $O(|F|^2log |F|)$ and $O(log^3 D)$ amortized update time.
In this paper, we study $k$-Way Min-cost Perfect Matching with Delays - the $k$-MPMD problem. This problem considers a metric space with $n$ nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of exactly $k$, such that the space and time cost of all matched requests are minimized. The notion of the space cost requires a definition of an underlying metric space that gives distances of subsets of $k$ elements. For $k>2$, the task of finding a suitable metric space is at the core of our problem: We show that for some known generalizations to $k=3$ points, such as the $2$-metric and the $D$-metric, there exists no competitive randomized algorithm for the $3$-MPMD problem. The $G$-metrics are defined for 3 points and allows for a competitive algorithm for the $3$-MPMD problem. For $k>3$ points, there exist two generalizations of the $G$-metrics known as $n$- and $K$-metrics. We show that neither the $n$-metrics nor the $K$-metrics can be used for the $k$-MPMD problem. On the positive side, we introduce the $H$-metrics, the first metrics to allow for a solution of the $k$-MPMD problem for all $k$. In order to devise an online algorithm for the $k$-MPMD problem on the $H$-metrics, we embed the $H$-metric into trees with an $O(log n)$ distortion. Based on this embedding result, we extend the algorithm proposed by Azar et al. (2017) and achieve a competitive ratio of $O(log n)$ for the $k$-MPMD problem.
This paper studies seeded graph matching for power-law graphs. Assume that two edge-correlated graphs are independently edge-sampled from a common parent graph with a power-law degree distribution. A set of correctly matched vertex-pairs is chosen at random and revealed as initial seeds. Our goal is to use the seeds to recover the remaining latent vertex correspondence between the two graphs. Departing from the existing approaches that focus on the use of high-degree seeds in $1$-hop neighborhoods, we develop an efficient algorithm that exploits the low-degree seeds in suitably-defined $D$-hop neighborhoods. Specifically, we first match a set of vertex-pairs with appropriate degrees (which we refer to as the first slice) based on the number of low-degree seeds in their $D$-hop neighborhoods. This significantly reduces the number of initial seeds needed to trigger a cascading process to match the rest of the graphs. Under the Chung-Lu random graph model with $n$ vertices, max degree $Theta(sqrt{n})$, and the power-law exponent $2<beta<3$, we show that as soon as $D> frac{4-beta}{3-beta}$, by optimally choosing the first slice, with high probability our algorithm can correctly match a constant fraction of the true pairs without any error, provided with only $Omega((log n)^{4-beta})$ initial seeds. Our result achieves an exponential reduction in the seed size requirement, as the best previously known result requires $n^{1/2+epsilon}$ seeds (for any small constant $epsilon>0$). Performance evaluation with synthetic and real data further corroborates the improved performance of our algorithm.
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