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Register a new userApproximating Spanners and Directed Steiner Forest: Upper and Lower Bounds
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Michael Dinitz
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Godwin SODA 16, Godwin-Williams SODA 16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an $O(n^{3/5 + epsilon})$-approximation for distance preservers and pairwise spanners (for arbitrary constant $epsilon > 0$). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an $O(log n)$-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an $O(1)$-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an $O(n^{3/5 + epsilon})$-approximation for the Directed Steiner Forest problem (for arbitrary constant $epsilon > 0$) when all edges have uniform costs, improving the previous best $O(n^{2/3 + epsilon})$-approximation due to Berman et al.~[ICALP 11] (which holds for general edge costs).
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We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $tilde{O}(n^{2/3+epsilon})$-competitive algorithm, and an efficient deterministic $tilde{O}(k^{1/2+epsilon})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $tilde{O}(n^{3/5 + epsilon})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $tilde{O}(n^{2/3 + epsilon})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $tilde{O}(k^{1/2 + epsilon})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $tilde{O}(n^{26/45 + epsilon})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.
The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted $m$-edge and $n$-node graphs require $Omega(min{n^{omega}, mn})$ time (for $2leqomega<2.373$). In this paper, we drastically improve these runtimes as follows: * Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in $widetilde{O}(m)$ time computes an $widetilde{O}(1)$-multiplicative approximation of the girth as well as an $widetilde{O}(1)$-multiplicative roundtrip spanner with $widetilde{O}(n)$ edges with high probability (w.h.p). * Nearly Tight Additive Approximations: For unweighted graphs and any $alpha in (0,1)$ we give an algorithm that in $widetilde{O}(mn^{1 - alpha})$ time computes an $O(n^alpha)$-additive approximation of the girth w.h.p, and partially derandomize it. We show that the runtime of our algorithm cannot be significantly improved without a breakthrough in combinatorial Boolean matrix multiplication. Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than $Omega(min{n^omega, mn})$ time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.
We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any $O(n)$-size shortcut set cannot bring the diameter below $Omega(n^{1/6})$, and that any $O(m)$-size shortcut set cannot bring it below $Omega(n^{1/11})$. These improve Hesses [Hesse03] lower bound of $Omega(n^{1/17})$. By combining these constructions with Abboud and Bodwins [AbboudB17] edge-splitting technique, we get additive stretch lower bounds of $+Omega(n^{1/11})$ for $O(n)$-size spanners and $+Omega(n^{1/18})$ for $O(n)$-size emulators. These improve Abboud and Bodwins $+Omega(n^{1/22})$ lower bounds.
Classic dynamic data structure problems maintain a data structure subject to a sequence S of updates and they answer queries using the latest version of the data structure, i.e., the data structure after processing the whole sequence. To handle operations that change the sequence S of updates, Demaine et al. (TALG 2007) introduced retroactive data structures. A retroactive operation modifies the update sequence S in a given position t, called time, and either creates or cancels an update in S at time t. A partially retroactive data structure restricts queries to be executed exclusively in the latest version of the data structure. A fully retroactive data structure supports queries at any time t: a query at time t is answered using only the updates of S up to time t. If the sequence S only consists of insertions, the resulting data structure is an incremental retroactive data structure. While efficient retroactive data structures have been proposed for classic data structures, e.g., stack, priority queue and binary search tree, the retroactive version of graph problems are rarely studied. In this paper we study retroactive graph problems including connectivity, minimum spanning forest (MSF), maximum degree, etc. We provide fully retroactive data structures for maintaining the maximum degree, connectivity and MSF in $tilde{O}(n)$ time per operation. We also give an algorithm for the incremental fully retroactive connectivity with $tilde{O}(1)$ time per operation. We compliment our algorithms with almost tight hardness results. We show that under the OMv conjecture (proposed by Henzinger et al. (STOC 2015)), there does not exist fully retroactive data structures maintaining connectivity or MSF, or incremental fully retroactive data structure maintaining the maximum degree with $O(n^{1-epsilon})$ time per operation, for any constant $epsilon > 0$.
We study the quantum query complexity of two problems. First, we consider the problem of determining if a sequence of parentheses is a properly balanced one (a Dyck word), with a depth of at most $k$. We call this the $Dyck_{k,n}$ problem. We prove a lower bound of $Omega(c^k sqrt{n})$, showing that the complexity of this problem increases exponentially in $k$. Here $n$ is the length of the word. When $k$ is a constant, this is interesting as a representative example of star-free languages for which a surprising $tilde{O}(sqrt{n})$ query quantum algorithm was recently constructed by Aaronson et al. Their proof does not give rise to a general algorithm. When $k$ is not a constant, $Dyck_{k,n}$ is not context-free. We give an algorithm with $Oleft(sqrt{n}(log{n})^{0.5k}right)$ quantum queries for $Dyck_{k,n}$ for all $k$. This is better than the trival upper bound $n$ for $k=oleft(frac{log(n)}{loglog n}right)$. Second, we consider connectivity problems on grid graphs in 2 dimensions, if some of the edges of the grid may be missing. By embedding the balanced parentheses problem into the grid, we show a lower bound of $Omega(n^{1.5-epsilon})$ for the directed 2D grid and $Omega(n^{2-epsilon})$ for the undirected 2D grid. The directed problem is interesting as a black-box model for a class of classical dynamic programming strategies including the one that is usually used for the well-known edit distance problem. We also show a generalization of this result to more than 2 dimensions.
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