Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAlmost-linear-time Weighted $ell_p$-norm Solvers in Slightly Dense Graphs via Sparsification
141
0
0.0
(
0
)
Added by
Deeksha Adil
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 ($m ge n^{4/3 + o(1)}$).
rate research
Read More
In this note, we study the expander decomposition problem in a more general setting where the input graph has positively weighted edges and nonnegative demands on its vertices. We show how to extend the techniques of Chuzhoy et al. (FOCS 2020) to this wider setting, obtaining a deterministic algorithm for the problem in almost-linear time.
In this paper, we consider the problem of designing cut sparsifiers and sketches for directed graphs. To bypass known lower bounds, we allow the sparsifier/sketch to depend on the balance of the input graph, which smoothly interpolates between undirected and directed graphs. We give nearly matching upper and lower bounds for both for-all (cf. Benczur and Karger, STOC 1996) and for-each (Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show an interesting application of digraph sparsification via cut balance by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).
We present an $tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m = Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(msqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $tilde O(n^omega)$-time algorithms [Ibarra-Moran 1981] (where currently $omegaapprox 2.373$). On sparser graphs, i.e. when $m = n^{9/8 + delta}$ for any constant $delta>0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $tilde O(m^{4/3+o(1)})$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler $tilde O(n sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $tilde O(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).
We give improved algorithms for the $ell_{p}$-regression problem, $min_{x} |x|_{p}$ such that $A x=b,$ for all $p in (1,2) cup (2,infty).$ Our algorithms obtain a high accuracy solution in $tilde{O}_{p}(m^{frac{|p-2|}{2p + |p-2|}}) le tilde{O}_{p}(m^{frac{1}{3}})$ iterations, where each iteration requires solving an $m times m$ linear system, $m$ being the dimension of the ambient space. By maintaining an approximate inverse of the linear systems that we solve in each iteration, we give algorithms for solving $ell_{p}$-regression to $1 / text{poly}(n)$ accuracy that run in time $tilde{O}_p(m^{max{omega, 7/3}}),$ where $omega$ is the matrix multiplication constant. For the current best value of $omega > 2.37$, we can thus solve $ell_{p}$ regression as fast as $ell_{2}$ regression, for all constant $p$ bounded away from $1.$ Our algorithms can be combined with fast graph Laplacian linear equation solvers to give minimum $ell_{p}$-norm flow / voltage solutions to $1 / text{poly}(n)$ accuracy on an undirected graph with $m$ edges in $tilde{O}_{p}(m^{1 + frac{|p-2|}{2p + |p-2|}}) le tilde{O}_{p}(m^{frac{4}{3}})$ time. For sparse graphs and for matrices with similar dimensions, our iteration counts and running times improve on the $p$-norm regression algorithm by [Bubeck-Cohen-Lee-Li STOC`18] and general-purpose convex optimization algorithms. At the core of our algorithms is an iterative refinement scheme for $ell_{p}$-norms, using the smoothed $ell_{p}$-norms introduced in the work of Bubeck et al. Given an initial solution, we construct a problem that seeks to minimize a quadratically-smoothed $ell_{p}$ norm over a subspace, such that a crude solution to this problem allows us to improve the initial solution by a constant factor, leading to algorithms with fast convergence.
Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study computationally efficient diffusion primitives beyond random walk. We design an $widetilde{O}(m)$-time randomized algorithm for the $ell_2$-norm flow diffusion problem, a recently proposed diffusion model based on network flow with demonstrated graph clustering related applications both in theory and in practice. Examples include finding locally-biased low conductance cuts. Using a known connection between the optimal dual solution of the flow diffusion problem and the local cut structure, our algorithm gives an alternative approach for finding such cuts in nearly linear time. From a technical point of view, our algorithm contributes a novel way of dealing with inequality constraints in graph optimization problems. It adapts the high-level algorithmic framework of nearly linear time Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and accelerated proximal gradient methods with inexact proximal mapping computation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
