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Register a new userPath-contractions, edge deletions and connectivity preservation
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Gregory Gutin
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(klog k)} n^{O(1)}$-algorithm that solves {sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting.
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We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity with high probability: -- Two sequential algorithms with complexities $O(m log n)$ and $O(m+n log^3 n)$. These improve on a long line of developments including a celebrated $O(m log^3 n)$ algorithm of Karger [STOC96] and the state of the art $O(m log^2 n (loglog n)^2)$ algorithm of Henzinger et al. [SODA17]. Moreover, our $O(m+n log^3 n)$ algorithm is optimal whenever $m = Omega(n log^3 n)$. Within our new time bounds, whp, we can also construct the cactus representation of all minimal cuts. -- An $~O(n^{0.8} D^{0.2} + n^{0.9})$ round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC19], which achieved a round complexity of $~O(n^{1-1/353}D^{1/353} + n^{1-1/706})$, hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. -- The first $O(1)$ round algorithm for the massively parallel computation setting with linear memory per machine.
We investigate the parameterized complexity in $a$ and $b$ of determining whether a graph~$G$ has a subset of $a$ vertices and $b$ edges whose removal disconnects $G$, or disconnects two prescribed vertices $s, t in V(G)$.
Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be found in time $m(clog n)^{O(c)}$. We also give a separate algorithm that constructs such graphs with $k cdot O(c)^{2c}$ edges in time $mc^{O(c)}log^{O(1)}n$. These results lead to the first data structures for answering fully dynamic offline $c$-edge-connectivity queries for $c ge 4$ in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.
A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such that the largest strong connectivity component in $D-X$ has at most $ell$ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $ell=1.$ We study parametered complexity of DCOC for general and semicomplete digraphs with the following parameters: $k, ell,ell+k$ and $n-ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$ unless the Exponential Time Hypothesis (ETH) fails. gutin{The upper bound $O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-ell)})$ for the parameter $n-ell.$ We complement the latter by showing that there is no algorithm of time complexity $O^*(2^{o({n-ell})})$ unless ETH fails.} Finally, we improve viol{(in dependency on $ell$)} the upper bound of G{{o}}ke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $ell+k$ on general digraphs from $O^*(2^{O(kelllog (kell))})$ to $O^*(2^{O(klog (kell))}).$ Note that Drange, Dregi and van t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $O^*(2^{o(klog ell)})$ unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $O^*(2^{o(klog k)}).$
The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph $G$ and a spanning tree $T$ for it, and the goal is to augment $T$ with a minimum set of edges $Aug$ from $G$, such that $T cup Aug$ is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and J{a}J{a}, SICOMP 1981. Recently, a 3/2-approximation was given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018], and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017; Fiorini et al., SODA 2018]. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed $2$-approximation for weighted TAP which completes in $O(h)$ rounds, where $h$ is the height of $T$. When $h$ is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in $O(D+sqrt{n}log^*{n})$ rounds, where $n$ is the number of vertices and $D$ is the diameter of $G$. Immediate consequences of our results are an $O(D)$-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an $O(h_{MST}+sqrt{n}log^{*}{n})$-round 3-approximation algorithm for the weighted case, where $h_{MST}$ is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP.
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