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Register a new userTowards Tight Bounds for Spectral Sparsification of Hypergraphs
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Jakab Tardos
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may offer new applications. However, the current bounds on the size of hypergraph sparsifiers are not as tight as the corresponding bounds for graphs. Our first result is a polynomial-time algorithm that, given a hypergraph on $n$ vertices with maximum hyperedge size $r$, outputs an $epsilon$-spectral sparsifier with $O^*(nr)$ hyperedges, where $O^*$ suppresses $(epsilon^{-1} log n)^{O(1)}$ factors. This size bound improves the two previous bounds: $O^*(n^3)$ [Soma and Yoshida, SODA19] and $O^*(nr^3)$ [Bansal, Svensson and Trevisan, FOCS19]. Our main technical tool is a new method for proving concentration of the nonlinear analogue of the quadratic form of the Laplacians for hypergraph expanders. We complement this with lower bounds on the bit complexity of any compression scheme that $(1+epsilon)$-approximates all the cuts in a given hypergraph, and hence also on the bit complexity of every $epsilon$-cut/spectral sparsifier. These lower bounds are based on Ruzsa-Szemeredi graphs, and a particular instantiation yields an $Omega(nr)$ lower bound on the bit complexity even for fixed constant $epsilon$. This is tight up to polylogarithmic factors in $n$, due to recent hypergraph cut sparsifiers of [Chen, Khanna and Nagda, FOCS20]. Finally, for directed hypergraphs, we present an algorithm that computes an $epsilon$-spectral sparsifier with $O^*(n^2r^3)$ hyperarcs, where $r$ is the maximum size of a hyperarc. For small $r$, this improves over $O^*(n^3)$ known from [Soma and Yoshida, SODA19], and is getting close to the trivial lower bound of $Omega(n^2)$ hyperarcs.
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Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this parameter can be approximated. Chechik et al. showed that the diameter can be approximated within a multiplicative factor of $3/2$ in $tilde{O}(m^{3/2})$ time. Furthermore, Roditty and Vassilevska W. showed that unless the Strong Exponential Time Hypothesis (SETH) fails, no $O(n^{2-epsilon})$ time algorithm can achieve an approximation factor better than $3/2$ in sparse graphs. Thus the above algorithm is essentially optimal for sparse graphs for approximation factors less than $3/2$. It was, however, completely plausible that a $3/2$-approximation is possible in linear time. In this work we conditionally rule out such a possibility by showing that unless SETH fails no $O(m^{3/2-epsilon})$ time algorithm can achieve an approximation factor better than $5/3$. Another fundamental set of graph parameters are the Eccentricities. The Eccentricity of a vertex $v$ is the distance between $v$ and the farthest vertex from $v$. Chechik et al. showed that the Eccentricities of all vertices can be approximated within a factor of $5/3$ in $tilde{O}(m^{3/2})$ time and Abboud et al. showed that no $O(n^{2-epsilon})$ algorithm can achieve better than $5/3$ approximation in sparse graphs. We show that the runtime of the $5/3$ approximation algorithm is also optimal under SETH. We also show that no near-linear time algorithm can achieve a better than $2$ approximation for the Eccentricities and that this is essentially tight: we give an algorithm that approximates Eccentricities within a $2+delta$ factor in $tilde{O}(m/delta)$ time for any $0<delta<1$. This beats all Eccentricity algorithms in Cairo et al.
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We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $ell$ servers, where servers have capacity $k$ and we assume that the graph has $ell cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $varepsilon k$ for constant $varepsilon>0$. Whenever $operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $operatorname{ONL}$ should minimize the competitive ratio: the total cost $operatorname{ONL}$ incurs compared to an optimal offline algorithm $operatorname{OPT}$. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(log ell + log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $Omega(log ell + log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $Theta(ell lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. Our algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.
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