A $(1 + {\varepsilon})$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs
Andreas Emil Feldmann, Wai Shing Fung, Jochen Könemann, Ian Post
备注：
A preliminary version of this paper appeared in the proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP), 2015
Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location.
To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.
In this paper, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser (1959), we are given a graph $G=(V,E)$ with metric edges costs, a depot $r\in V$, and a vehicle of bounded capacity $Q$. The goal is to find minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most $Q$ nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node $v$ has a demand $d_v$ and the total demand o…
The concept of bounded highway dimension was developed to capture observed properties of the metrics of road networks. We show that a graph with bounded highway dimension, for any vertex, can be embedded into a a graph of bounded treewidth in such a way that the distance between $u$ and $v$ is preserved up to an additive error of $ε$ times the distance from $u$ or $v$ to the selected vertex. We show that this theorem yields a PTAS for Bounded-Capacity Vehicle Routing in graphs of bounded highway dimension. In this problem, the input specifies a depot and a set of clients, each with a location …
We consider the $k$-Center problem and some generalizations. For $k$-Center a set of $k$ center vertices needs to be found in a graph $G$ with edge lengths, such that the distance from any vertex of $G$ to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we …
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time $O(n^{tw(H)+1})$ [Alon, Yuster, Zwick'95], where $n$ is the number of vertices of the host graph $G$. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of $O(n^{tw(H)+1-\varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are…
Many combinatorial problems can be solved in time $O^*(c^{tw})$ on graphs of treewidth $tw$, for a problem-specific constant $c$. In several cases, matching upper and lower bounds on $c$ are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time, and Dominating Set cannot be solved in $O…
The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (SIAM J. Comput. 2005) gave an $O(2^tn^3)$-time algorithm when given a tree decomposition of width $t$ of the input graph, where $n$ is the number of vertices of the input graph. Eiben et al. (ESA 2019) improved …
The Glauber dynamics on the colourings of a graph is a random process which consists in recolouring at each step a random vertex of a graph with a new colour chosen uniformly at random among the colours not already present in its neighbourhood. It is known that when the total number of colours available is at least $Δ+2$, where $Δ$ is the maximum degree of the graph, this process converges to a uniform distribution on the set of all the colourings. Moreover, a well known conjecture is that the time it takes for the convergence to happen, called the mixing time, is polynomial in the size of the…
A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is assigned with a list $L(v)$ of vertices of $H$. We ask whether there exists a homomorphism $h$ from $G$ to $H$, which respects lists $L$, i.e., for every $v \in V(G)$ it holds that $h(v) \in L(v)$. The complexity dichotomy for LHom($H$) was proven by Feder, Hell, and Huang [JGT 2003]. We are interested in the complexity of the problem, paramete…
DAG Covers for Structured Graphs: The Steiner Point Effect
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Dartmouth College（达特茅斯学院）；Bar-Ilan University（巴伊兰大学）；POSTECH（浦项科技大学）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Institute of Science and Technology Austria（奥地利科学技术研究所）
For any fixed measure $H$ that maps graphs to real numbers, the MinH problem is defined as follows: given a graph $G$, an integer $k$, and a target $τ$, is there a set $S$ of $k$ vertices that can be deleted, so that $H(G - S)$ is at most $τ$? In this paper, we consider the MinH problem on trees. We call $H$ "balanced on trees" if, whenever $G$ is a tree, there is an optimal choice of $S$ such that the components of $G-S$ have sizes bounded by a polynomial in $n/k$. We show that MinH on trees is FPT for parameter $n/k$, and furthermore, can be solved in subexponential time, and polynomial spac…
We consider the problem of computing a Steiner tree of minimum cost under a hop constraint which requires the depth of the tree to be at most $k$. Our main result is an exact algorithm for metrics induced by graphs with bounded treewidth that runs in time $n^{O(k)}$. For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if $k$ is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the $k$-hop constraint by at most one hop for more general metrics induced by graphs of boun…
In this paper we study the hardness of the $k$-Center problem on inputs that model transportation networks. For the problem, a graph $G=(V,E)$ with edge lengths and an integer $k$ are given and a center set $C\subseteq V$ needs to be chosen such that $|C|\leq k$. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs.…
In this paper, we refine the (almost) \emph{existentially optimal} distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS `21) into an (almost) \emph{universally optimal} distributed Laplacian solver. Specifically, when the topology is known, we show that any Laplacian system on an $n$-node graph with \emph{shortcut quality} $\text{SQ}(G)$ can be solved within $n^{o(1)} \text{SQ}(G) \log(1/\varepsilon)$ rounds, where $\varepsilon$ is the required accuracy. This almost matches our lower bound which guarantees that any correct algorithm on $G$ requires …
Clustering under Constraints: Efficient Parameterized Approximation Schemes
Indian Institute of Technology Bombay（孟买印度理工学院）；CISPA Helmholtz Center for Information Security（CISPA赫尔姆霍兹信息安全中心）；Indian Institute of Technology Jodhpur（焦特普尔印度理工学院）
