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تسجيل مستخدم جديدBibliography of distributed approximation beyond bounded degree
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تأليف
Laurent Feuilloley
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This document is an informal bibliography of the papers dealing with distributed approximation algorithms. A classic setting for such algorithms is bounded degree graphs, but there is a whole set of techniques that have been developed for other classes. These later classes are the focus of the current work. These classes have a geometric nature (planar, bounded genus and unit-disk graphs) and/or have bounded parameters (arboricity, expansion, growth, independence) or forbidden structures (forbidden minors).
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We address the fundamental network design problem of constructing approximate minimum spanners. Our contributions are for the distributed setting, providing both algorithmic and hardness results. Our main hardness result shows that an $alpha$-appro
ximation for the minimum directed $k$-spanner problem for $k geq 5$ requires $Omega(n /sqrt{alpha}log{n})$ rounds using deterministic algorithms or $Omega(sqrt{n }/sqrt{alpha}log{n})$ rounds using randomized ones, in the CONGEST model of distributed computing. Combined with the constant-round $O(n^{epsilon})$-approximation algorithm in the LOCAL model of [Barenboim, Elkin and Gavoille, 2016], as well as a polylog-round $(1+epsilon)$-approximation algorithm in the LOCAL model that we show here, our lower bounds for the CONGEST model imply a strict separation between the LOCAL and CONGEST models. Notably, to the best of our knowledge, this is the first separation between these models for a local approximation problem. Similarly, a separation between the directed and undirected cases is implied. We also prove a nearly-linear lower bound for the minimum weighted $k$-spanner problem for $k geq 4$, and we show lower bounds for the weighted 2-spanner problem. On the algorithmic side, apart from the aforementioned $(1+epsilon)$-approximation algorithm for minimum $k$-spanners, our main contribution is a new distributed construction of minimum 2-spanners that uses only polynomial local computations. Our algorithm has a guaranteed approximation ratio of $O(log(m/n))$ for a graph with $n$ vertices and $m$ edges, which matches the best known ratio for polynomial time sequential algorithms [Kortsarz and Peleg, 1994], and is tight if we restrict ourselves to polynomial local computations. Our approach allows us to extend our algorithm to work also for the directed, weighted, and client-server variants of the problem.
Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al.~2006), this is essentially the only class of linear pro
grams for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call distance-bounded network design convex programs. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in $O( (D/epsilon) log n)$ rounds in the $mathcal{LOCAL}$ model and finds a $(1+epsilon)$-approximation to the optimal LP solution for any $0 < epsilon leq 1$, where $D$ is the largest distance constraint. While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic $3$- and $4$-Spanner, Directed $k$-Spanner, Lowest Degree $k$-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any heavy computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.
We investigate graph problems in the following setting: we are given a graph $G$ and we are required to solve a problem on $G^2$. While we focus mostly on exploring this theme in the distributed CONGEST model, we show new results and surprising conne
ctions to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on $G^2$ would be quite difficult to solve efficiently on $G$, due to congestion. However, we show that the picture is both more complicated and more interesting. Specifically, we encounter two phenomena acting in opposing directions: (i) slowdown due to congestion and (ii) speedup due to structural properties of $G^2$. We demonstrate these two phenomena via two fundamental graph problems, namely, Minimum Vertex Cover (MVC) and Minimum Dominating Set (MDS). Among our many contributions, the highlights are the following. - In the CONGEST model, we show an $O(n/epsilon)$-round $(1+epsilon)$-approximation algorithm for MVC on $G^2$, while no $o(n^2)$-round algorithm is known for any better-than-2 approximation for MVC on $G$. - We show a centralized polynomial time $5/3$-approximation algorithm for MVC on $G^2$, whereas a better-than-2 approximation is UGC-hard for $G$. - In contrast, for MDS, in the CONGEST model, we show an $tilde{Omega}(n^2)$ lower bound for a constant approximation factor for MDS on $G^2$, whereas an $Omega(n^2)$ lower bound for MDS on $G$ is known only for exact computation. In addition to these highlighted results, we prove a number of other results in the distributed CONGEST model including an $tilde{Omega}(n^2)$ lower bound for computing an exact solution to MVC on $G^2$, a conditional hardness result for obtaining a $(1+epsilon)$-approximation to MVC on $G^2$, and an $O(log Delta)$-approximation to the MDS problem on $G^2$ in $mbox{poly}log n$ rounds.
The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present
two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $hat d = O(dlog{n})$. It requires $O((D + sqrt{n}) log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexity for the related minimum spanning tree problem. Our second result refines this approximation factor by constructing a tree with maximum degree $hat d = O(d + log{n})$, though at the cost of additional polylogarithmic factors in the round complexity. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990s, our results are first efficient distributed solutions for this problem.
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the minimum sp
anning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for $k geq 2$ less is known in the distributed setting. In this paper, we present fast randomized distributed approximation algorithms for $k$-ECSS in the CONGEST model. Our first contribution is an $widetilde{O}(D + sqrt{n})$-round $O(log{n})$-approximation for 2-ECSS, for a graph with $n$ vertices and diameter $D$. The time complexity of our algorithm is almost tight and almost matches the time complexity of the MST problem. For larger constant values of $k$ we give an $widetilde{O}(n)$-round $O(log{n})$-approximation. Additionally, in the special case of unweighted 3-ECSS we show how to improve the time complexity to $O(D log^3{n})$ rounds. All our results significantly improve the time complexity of previous algorithms.
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