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Register a new userAlmost-Linear-Time Algorithms for Markov Chains and New Spectral Primitives for Directed Graphs
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Jonathan A. Kelner
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper we introduce a notion of spectral approximation for directed graphs. While there are many potential ways one might define approximation for directed graphs, most of them are too strong to allow sparse approximations in general. In contrast, we prove that for our notion of approximation, such sparsifiers do exist, and we show how to compute them in almost linear time. Using this notion of approximation, we provide a general framework for solving asymmetric linear systems that is broadly inspired by the work of [Peng-Spielman, STOC`14]. Applying this framework in conjunction with our sparsification algorithm, we obtain an almost linear time algorithm for solving directed Laplacian systems associated with Eulerian Graphs. Using this solver in the recent framework of [Cohen-Kelner-Peebles-Peng-Sidford-Vladu, FOCS`16], we obtain almost linear time algorithms for solving a directed Laplacian linear system, computing the stationary distribution of a Markov chain, computing expected commute times in a directed graph, and more. For each of these problems, our algorithms improves the previous best running times of $O((nm^{3/4} + n^{2/3} m) log^{O(1)} (n kappa epsilon^{-1}))$ to $O((m + n2^{O(sqrt{log{n}loglog{n}})}) log^{O(1)} (n kappa epsilon^{-1}))$ where $n$ is the number of vertices in the graph, $m$ is the number of edges, $kappa$ is a natural condition number associated with the problem, and $epsilon$ is the desired accuracy. We hope these results open the door for further studies into directed spectral graph theory, and will serve as a stepping stone for designing a new generation of fast algorithms for directed graphs.
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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.
Let $G$ be a graph and $S, T subseteq V(G)$ be (possibly overlapping) sets of terminals, $|S|=|T|=k$. We are interested in computing a vertex sparsifier for terminal cuts in $G$, i.e., a graph $H$ on a smallest possible number of vertices, where $S cup T subseteq V(H)$ and such that for every $A subseteq S$ and $B subseteq T$ the size of a minimum $(A,B)$-vertex cut is the same in $G$ as in $H$. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlstrom (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier $H$ with $O(k^3)$ vertices can be computed in randomized polynomial time, even for arbitrary digraphs $G$. However, since then, no improvements on the size $O(k^3)$ have been shown. In this paper, we draw inspiration from the renowned Bollobass Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlstroms methods. This new perspective allows us to construct a sparsifier $H$ of $Theta(k^2)$ vertices for the case that $G$ is a DAG. We also show how to compute $H$ in time near-linear in the size of $G$, improving on the previous $O(n^{omega+1})$. Furthermore, $H$ recovers the closest min-cut in $G$ for every partition $(A,B)$, which was not previously known. Finally, we show that a sparsifier of size $Omega(k^2)$ is required, both for DAGs and for undirected edge cuts.
We study the algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. We identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^{O(sqrt{k}log k)}cdot n^{O(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.
In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, OSullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, OSullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014].
We give almost-linear-time algorithms for constructing sparsifiers with $n poly(log n)$ edges that approximately preserve weighted $(ell^{2}_2 + ell^{p}_p)$ flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit $ell_p$ weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find $(1+2^{-text{poly}(log n)})$ approximations for weighted $ell_p$-norm minimizing flows or voltages in $p(m^{1+o(1)} + n^{4/3 + o(1)})$ time for $p=omega(1),$ which is almost-linear for graphs that are slightly dense ($m ge n^{4/3 + o(1)}$).
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