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Register a new userSurvivable Network Design for Group Connectivity in Low-Treewidth Graphs
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Added by
Daniel Vaz
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In the Group Steiner Tree problem (GST), we are given a (vertex or edge)-weighted graph $G=(V,E)$ on $n$ vertices, a root vertex $r$ and a collection of groups ${S_i}_{iin[h]}: S_isubseteq V(G)$. The goal is to find a min-cost subgraph $H$ that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group $S_i$ has a demand $k_iin[k],kinmathbb N$, and we wish to find a min-cost $Hsubseteq G$ such that, for each group $S_i$, there is a vertex in $S_i$ connected to the root via $k_i$ (vertex or edge) disjoint paths. While GST admits $O(log^2 nlog h)$ approximation, its high connectivity variants are Label-Cover hard, and for the vertex-weighted version, the hardness holds even when $k=2$. Previously, positive results were known only for the edge-weighted version when $k=2$ [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where the disjoint paths may end at different vertices in a group [Chalermsook et al., SODA 2015]. Our main result is an $O(log nlog h)$ approximation for Restricted Group SNDP that runs in time $n^{f(k, w)}$, where $w$ is the treewidth of $G$. This nearly matches the lower bound when $k$ and $w$ are constant. The key to achieving this result is a non-trivial extension of the framework in [Chalermsook et al., SODA 2017], which embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.
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In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V,E)$ with costs on edges, along with a connectivity requirement $r(u,v)$ for each pair $u,v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u,v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 log n)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $n$ denotes the number of vertices. We also give a simple proof of the recently discovered $O(k^2 log n)$-approximation result for the single-source version of vertex-connectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the $log n$ term in the approximation guarantee can be replaced with a $log tau$ term where $tau$ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-fragile graph classes, a property that is true for many natural graph classes with sublinear separators. We also provide quasipolynomial-time approximation schemes for these problems in all classes with sublinear separators.
We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{epsilon, k}(mathrm{tw}(G))$ where $mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.
Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A fundamental idea that helps both to understand the structural properties of these metrics and lead to strong algorithmic results is to construct a small-complexity graph that approximately preserves distances between pairs of points of the metric. We show the two following structural results for minor-free metrics: 1. Construction of a light subset spanner. Given a subset of vertices called terminals, and $epsilon$, in polynomial time we construct a subgraph that preserves all pairwise distances between terminals up to a multiplicative $1+epsilon$ factor, of total weight at most $O_{epsilon}(1)$ times the weight of the minimal Steiner tree spanning the terminals. 2. Construction of a stochastic metric embedding into low treewidth graphs with expected additive distortion $epsilon D$. Namely, given a minor free graph $G=(V,E,w)$ of diameter $D$, and parameter $epsilon$, we construct a distribution $mathcal{D}$ over dominating metric embeddings into treewidth-$O_{epsilon}(log n)$ graphs such that the additive distortion is at most $epsilon D$. One of our important technical contributions is a novel framework that allows us to reduce emph{both problems} to problems on simpler graphs of bounded diameter. Our results have the following algorithmic consequences: (1) the first efficient approximation scheme for subset TSP in minor-free metrics; (2) the first approximation scheme for vehicle routing with bounded capacity in minor-free metrics; (3) the first efficient approximation scheme for vehicle routing with bounded capacity on bounded genus metrics.
It has long been known that Feedback Vertex Set can be solved in time $2^{mathcal{O}(wlog w)}n^{mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{mathcal{O}(w)}n^{mathcal{O}(1)}$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class $mathcal{P}$ of graphs, the Bounded $mathcal{P}$-Block Vertex Deletion problem asks, given a graph~$G$ on $n$ vertices and positive integers~$k$ and~$d$, whether $G$ contains a set~$S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $mathcal{P}$. Assuming that $mathcal{P}$ is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of $d$: if $mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{mathcal{O}(wd^2)} n^{mathcal{O}(1)}$, and if $mathcal{P}$ contains a graph with an induced cycle of length $ellge 4$, then the problem is not solvable in time $2^{o(wlog w)} n^{mathcal{O}(1)}$ even for fixed $d=ell$, unless the ETH fails. We also study a similar problem, called Bounded $mathcal{P}$-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if $d$ is part of the input and $mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.
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