In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete. Furthermore we present a subclass of graphs with diameter 2 that admits a polynomial algorithm for 3-coloring. For graphs with diameter 3 we establish the complexity of 3-coloring by proving that for every $\varepsilon \in [0,1)$, 3-coloring is NP-complete on triangle-free graphs of diameter 3 and radius 2 with $n$ vertices and minimum degree $δ=Θ(n^{\varepsilon})$. Moreover, assuming ETH, we provide three different amplifications of our hardness results to obtain for every $\varepsilon \in [0,1)$ subexponential lower bounds for the complexity of 3-coloring on triangle-free graphs with diameter 3 and minimum degree $δ=Θ(n^{\varepsilon})$. Finally, we provide a 3-coloring algorithm with running time $2^{O(\min\{δΔ,\frac{n}δ\logδ\})}$ for graphs with diameter 3, where $δ$ (resp. $Δ$) is the minimum (resp. maximum) degree of the input graph. To the best of our knowledge, this algorithm is the first subexponential algorithm for graphs with $δ=ω(1)$ and for graphs with $δ=O(1)$ and $Δ=o(n)$. Due to the above lower bounds of the complexity of 3-coloring, the running time of this algorithm is asymptotically almost tight when the minimum degree if the input graph is $δ=Θ(n^{\varepsilon})$, where $\varepsilon \in [1/2,1)$.
Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices. Computing the Diameter of a graph with $m$ edges requires $m^{2-o(1)}$ time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a $2$-approximation for Diameter in $\tilde{O}(m)$ time. Additionally, a line of work concludes with a $3/2$-approximation algorithm for Diameter in weighted directed graphs that runs in $\tilde{O}(m^{3/2})…
A Tight Meta-theorem for LOCAL Certification of MSO$_2$ Properties within Bounded Treewidth Graphs
Korteweg-de Vries Institute of Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；School of Computing, KAIST（韩国科学技术院计算机学院）；CNRS（法国国家科学研究中心）；Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院信息学研究所）
Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important $ST$-variant considers two subsets $S$ and $T$ of the vertex set and lets the $ST$-diameter be the maximum distance between a node in $S$ and a node in $T$, and the $ST$-radius be the minimum distance for a node of $S$ to reach all nodes of $T$. The bichromatic variant is the special case in which $S$ and $T$ partition the vertex set. In this paper we present a comp…
In STOC'95 [ADMSS'95] Arya et al. showed that any set of $n$ points in $\mathbb R^d$ admits a $(1+ε)$-spanner with hop-diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges). They also gave a general upper bound tradeoff of hop-diameter at most $k$ and $O(n α_k(n))$ edges, for any $k \ge 2$. The function $α_k$ is the inverse of a certain Ackermann-style function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $α_0(n) = \lceil n/2 \rceil$, $α_1(n) = \left\lceil \sqrt{n} \right\rceil$, $α_2(n) = \lceil \log{n} \rceil$, $α_3…
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 $O(n^{2-ε})$ time algorithm can achieve an approximation factor better than $3/2$ i…
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a natural question to ask how much can we sparsify a graph and still guarantee that its diameter remains preserved within an approximation $t$. This property is captured by the notion of extremal-distance spanners. Given a graph $G=(V,E)$, a subgraph $H=(V,E_H)$ is defined to be a $t$-diameter spanner if the diameter of $H$ is at most $t$ times the diameter of $G$.…
We prove several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any $\varepsilon>0$, assuming the Strong Exponential Time Hypothesis (SETH), there are no near-linear time $2-\varepsilon$-approximation algorithms for the Diameter of a sparse directed graph, even in unweighted graphs. This result shows that a simple near-linear time 2-approximation algorithm for Diameter is optimal under SETH, answering a question from a survey of Rubinstein and Vassilevska-Williams (SIGACT '19) for the case of directed graphs. In the same survey,…
Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing $\cal{T}$ is the largest diameter of any tree in $\cal{T}$. A desirable property of a tree packing, that is both sufficient and necessary for leveraging the high connectivity of a graph in distributed communication, is that its diameter i…
Finding important nodes in a graph and measuring their importance is a fundamental problem in the analysis of social networks, transportation networks, biological systems, etc. Among popular such metrics are graph centrality, betweenness centrality (BC), and reach centrality (RC). These measures are also very related to classic notions like diameter and radius. Roditty and Vassilevska Williams~[STOC'13] showed that no algorithm can compute a (3/2-δ)-approximation of the diameter in sparse and unweighted graphs faster that n^{2-o(1)} time unless the widely believed strong exponential time hypot…
Certificates in P and Subquadratic-Time Computation of Radius, Diameter, and all Eccentricities in Graphs
Kent State University（肯特州立大学）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学学院）；National Institute for Research and Development in Informatics, Romania（罗马尼亚信息与计算发展国家研究所）；Inria, DI ENS, Paris, France（法国国家信息与自动化研究所，巴黎高等师范学院）
Consider the classical problem of information dissemination: one (or more) nodes in a network have some information that they want to distribute to the remainder of the network. In this paper, we study the cost of information dissemination in networks where edges have latencies, i.e., sending a message from one node to another takes some amount of time. We first generalize the idea of conductance to weighted graphs by defining $ϕ_*$ to be the "critical conductance" and $\ell_*$ to be the "critical latency". One goal of this paper is to argue that $ϕ_*$ characterizes the connectivity of a weigh…
We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between $u$ and $v$ is the minimum of the shortest path distances from $u$ to $v$ and from $v$ to $u$. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help. By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we…
We present a deterministic $(1+o(1))$-approximation $(n^{1/2+o(1)}+D^{1+o(1)})$-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here $n$ is the number of nodes in the network and $D$ is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized $(1+o(1))$-approximation $\tilde O(n^{1/2}D^{1/4}+D)$-time algorithm of Nanongkai [STOC 2014] by a factor of as large as $n^{1/8}$, and (ii) the $O(ε^{-1}\log ε^{-1})$-approximation factor of …
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. In this paper we prove conditional lower bounds for sensitivity problems. For example, we show that under the Boolean Matrix Multiplication (BM…
