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Register a new userComputing exact minimum cuts without knowing the graph
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Added by
Tselil Schramm
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query $S subset V$, the oracle returns the size of the cut between $S$ and $V setminus S$. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with $tilde{O}(n^{5/3})$ queries, and computing an exact global minimum cut of $G$ with only $tilde{O}(n)$ queries (while learning the graph requires $tilde{Theta}(n^2)$ queries).
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Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier $V_H$ contains the set of terminals $K$ and for every bipartition $U, K-U $ of the terminals $K$, the size of the minimum cut separating $U$ from $K-U$ in $G$ is exactly equal to the size of the minimum cut separating $U$ from $K-U$ in $H$. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde (1995) who also exhibited a mimicking network of size $2^{2^{k}}$ for every graph with $k$ terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is $k+1$ for graphs with $k$ terminals (Chaudhuri et al. 2000). In this work, we improve both the upper and lower bounds reducing the doubly-exponential gap between them to a single-exponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: 1) Given a graph $G$, we exhibit a construction of mimicking network with at most $(|K|-1)$th Dedekind number ($approx 2^{{(k-1)} choose {lfloor {{(k-1)}/2} rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {it restricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2) There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{frac{k-1}{2}}$. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width.
We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $mgeq n^{1+epsilon}$ for any constant $epsilon>0$, our algorithm requires $O(m log n)$ work and $O(log^3 n)$ depth and succeeds with high probability. Its work matches the best $O(m log n)$ runtime for sequential algorithms [MN STOC 2020, GMW SOSA 2021]. This improves the previous best work by Geissmann and Gianinazzi [SPAA 2018] by $O(log^3 n)$ factor, while matching the depth of their algorithm. To do this, we design a work-efficient approximation algorithm and parallelize the recent sequential algorithms [MN STOC 2020; GMW SOSA 2021] that exploit a connection between 2-respecting minimum cuts and 2-dimensional orthogonal range searching.
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+epsilon$ error must use $Omega(nlog n/epsilon^2)$ bits of space in the worst case, improving the $Omega(n/epsilon^2)$ bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two $d-$regular graphs which approximate each others cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.
We give an algorithm to find a mincut in an $n$-vertex, $m$-edge weighted directed graph using $tilde O(sqrt{n})$ calls to any maxflow subroutine. Using state of the art maxflow algorithms, this yields a directed mincut algorithm that runs in $tilde O(msqrt{n} + n^2)$ time. This improves on the 30 year old bound of $tilde O(mn)$ obtained by Hao and Orlin for this problem.
A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,Vsetminus S]$ of $G$ such that $G[S]$ and $G[Vsetminus S]$ are both connected. Given $s,tin V(G)$, an $st$-bond of $G$ is a bond whose removal disconnects $s$ and $t$. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest $st$-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that {sc Largest Bond} remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless $P = NP$. We also show that {sc Largest Bond} and {sc Largest $st$-Bond} on graphs of clique-width $w$ cannot be solved in time $f(w)times n^{o(w)}$ unless the Exponential Time Hypothesis fails, but they can be solved in time $f(w)times n^{O(w)}$. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP $subseteq$ coNP/poly.
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