In the k-Disjoint Shortest Paths problem, a set of terminal pairs of vertices $\{(s_i,t_i)\mid 1\le i\le k\}$ is given and we are asked to find paths $P_1,\ldots,P_k$ such that each path $P_i$ is a shortest path from $s_i$ to $t_i$ and every vertex of the graph routes at most one of them. We introduce a generalization of the problem, namely, $k$-Disjoint Shortest Paths with Congestion-$c$ where every vertex is allowed to route up to $c$ paths.
We provide a simple algorithm to solve the problem in time $f(k) n^{O(k-c)}$ on DAGs. Using the techniques for DAGs, we show the problem is solvable in time $f(k) n^{O(k)}$ on general undirected graphs. Our algorithm for DAGs is based on the earlier algorithm for $k$-Disjoint Paths with Congestion-$c$[IPL2019], but we significantly simplify their argument.
Then we prove that it is not possible to improve the algorithm significantly by showing that for every constant $c$ the problem is W[1]-hard w.r.t.\ parameter $k-c$. We also consider the problem on acyclic planar graphs, but this time we restrict ourselves to the edge-disjoint shortest paths problem. We show that even on acyclic planar graphs there is no $f(k)n^{o(k)}$ algorithm for the problem unless ETH fails.
Changing a given configuration in a graph into another one is known as a re- configuration problem. Such problems have recently received much interest in the context of algorithmic graph theory. We initiate the theoretical study of the following reconfiguration problem: How to reroute $k$ unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner? This problem finds immediate applications, e.g., in traffic engineering in computer networks. We show that the problem is generally NP-hard already for $k = 2$ f…
Shortest path queries over graphs are usually considered as isolated tasks, where the goal is to return the shortest path for each individual query. In practice, however, such queries are typically part of a system (e.g., a road network) and their execution dynamically affects other queries and network parameters, such as the loads on edges, which in turn affects the shortest paths. We study the problem of collectively processing shortest path queries, where the objective is to optimize a collective objective, such as minimizing the overall cost. We define a temporal load-aware network that dy…
Let $G = (V,E,w)$ be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source $s$ and the goal is to maintain a data structure that can answer path-queries $s \rightarrowtail v$ for any $v \in V$. In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to $v$, and in the SCC problem the goal is to maintain strongly connected components of $G$ and to answer path queries within each component. All of these problems have been very acti…
The min-distance between two nodes $u, v$ is defined as the minimum of the distance from $v$ to $u$ or from $u$ to $v$, and is a natural distance metric in DAGs. As with the standard distance problems, the Strong Exponential Time Hypothesis [Impagliazzo-Paturi-Zane 2001, Calabro-Impagliazzo-Paturi 2009] leaves little hope for computing min-distance problems faster than computing All Pairs Shortest Paths, which can be solved in $\tilde{O}(mn)$ time. So it is natural to resort to approximation algorithms in $\tilde{O}(mn^{1-ε})$ time for some positive $ε$. Abboud, Vassilevska W., and Wang [SODA …
A straightforward dynamic programming method for the single-source shortest paths problem (SSSP) in an edge-weighted directed acyclic graph (DAG) processes the vertices in a topologically sorted order. First, we similarly iterate this method alternatively in a breadth-first search sorted order and the reverse order on an input directed graph with both positive and negative real edge weights, $n$ vertices and $m$ edges. For a positive integer $t,$ after $O(t)$ iterations in $O(tm)$ time, we obtain for each vertex $v$ a path distance from the source to $v$ not exceeding that yielded by the short…
We study reachability and shortest paths problems in dynamic directed graphs. Whereas algebraic dynamic data structures supporting edge updates and reachability/distance queries have been known for quite a long time, they do not, in general, allow reporting the underlying paths within the same time bounds, especially against an adaptive adversary. In this paper we develop the first known fully dynamic reachability data structures working against an adaptive adversary and supporting edge updates and path queries for two natural variants: (1) point-to-point path reporting, and (2) single-source …
In the Directed Disjoint Paths problem, we are given a digraph $D$ and a set of requests $\{(s_1, t_1), \ldots, (s_k, t_k)\}$, and the task is to find a collection of pairwise vertex-disjoint paths $\{P_1, \ldots, P_k\}$ such that each $P_i$ is a path from $s_i$ to $t_i$ in $D$. This problem is NP-complete for fixed $k=2$ and W[1]-hard with parameter $k$ in DAGs. A few positive results are known under restrictions on the input digraph, such as being planar or having bounded directed tree-width, or under relaxations of the problem, such as allowing for vertex congestion. Positive results are sc…
We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with congestion $O(δD \log n)$ and dilation $O(δD)$, where $δ$ is the maximum edge-density of any minor of $G$. Our proof is simple, elementary, and constructive - featuring a $\tildeΘ(δD)$-round distributed construction algorithm. Our results are tight up to $\tilde{O}(1)$ factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant $δ$, only a $\tilde{O}(D^2)$ bound was known based on a very technical proof that relies on the Ro…
We study the problem of quickly computing point-to-point shortest paths in massive road networks with traffic predictions. Incorporating traffic predictions into routing allows, for example, to avoid commuter traffic congestions. Existing techniques follow a two-phase approach: In a preprocessing step, an index is built. The index depends on the road network and the traffic patterns but not on the path start and end. The latter are the input of the query phase, in which shortest paths are computed. All existing techniques have large index size, slow query running times or may compute suboptima…
APSP with small integer weights in undirected graphs [Seidel'95, Galil and Margalit'97] has an $\tilde{O}(n^ω)$ time algorithm, where $ω<2.373$ is the matrix multiplication exponent. APSP in directed graphs with small weights however, has a much slower running time that would be $Ω(n^{2.5})$ even if $ω=2$ [Zwick'02]. To understand this $n^{2.5}$ bottleneck, we build a web of reductions around directed unweighted APSP. We show that it is fine-grained equivalent to computing a rectangular Min-Plus product for matrices with integer entries; the dimensions and entry size of the matrices depend on …
In this paper, we show that given a weighted, directed planar graph $G$, and any $ε>0$, there exists a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm that computes the shortest path between two fixed vertices in $G$. We also consider the {\RB} problem, which states that given a graph $G$ whose edges are colored either red or blue and two fixed vertices $s$ and $t$ in $G$, is there a path from $s$ to $t$ in $G$ that alternates between red and blue edges. The {\RB} problem in planar DAGs is {\NL}-complete. We exhibit a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm (for a…
We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset $X$ of vertices of a $δ$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk …
We introduce a model for differentially private analysis of weighted graphs in which the graph topology $(V,E)$ is assumed to be public and the private information consists only of the edge weights $w:E\to\mathbb{R}^+$. This can express hiding congestion patterns in a known system of roads. Differential privacy requires that the output of an algorithm provides little advantage, measured by privacy parameters $ε$ and $δ$, for distinguishing between neighboring inputs, which are thought of as inputs that differ on the contribution of one individual. In our model, two weight functions $w,w'$ are …
In the decremental Single-Source Shortest Path problem (SSSP), we are given a weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source vertex $r \in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the graph. The goal is to obtain a data structure that maintains shortest paths from $r$ to all vertices in $V$ and can answer distance queries in $O(1)$ time, as well as return the corresponding path $P$ in $O(|P|)$ time. This problem was first considered by Even and Shiloach [JACM'81], who provided an algorithm with total update time $O(mn)$ for unweighted undirected gra…
Algorithms and Hardness for Geodetic Set on Tree-like Digraphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学、法国国家科学研究中心、克莱蒙奥弗涅理工大学、圣艾蒂安国立高等矿业学校、信息与系统优化实验室）；School of Mathematics, Statistics, and Computer Science, University of Tehran（德黑兰大学数学、统计与计算机科学学院）；Indian Institute of Science Education and Research Pune（浦那科学教育与研究所）
This paper studies the fundamental problem of how to reroute $k$ unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner and fast. This scheduling problem has applications in traffic engineering in communication networks and has recently received much attention in software-defined networks, in which updates are distributed over an asynchronous network by a software controller. However, existing algorithms for this problem either have a super-polynomial runtime or only compute feasible schedules, which d…
Traditional navigation services find the fastest route for a single driver. Though always using the fastest route seems desirable for every individual, selfish behavior can have undesirable effects such as higher energy consumption and avoidable congestion, even leading to higher overall and individual travel times. In contrast, strategic routing aims at optimizing the traffic for all agents regarding a global optimization goal. We introduce a framework to formalize real-world strategic routing scenarios as algorithmic problems and study one of them, which we call Single Alternative Path (SAP)…
Structural parameterizations of Geodetic Set on directed (acyclic) graphs
Université Clermont-Auvergne（克莱蒙奥弗涅大学）；CNRS（法国国家科学研究中心）；Mines de Saint-Étienne（圣艾蒂安国立高等矿业学院）；Clermont-Auvergne-INP（克莱蒙奥弗涅理工学院）；LIMOS（LIMOS实验室）；Indian Institute of Science Education and Research Pune（印度科学教育研究学院浦那分校）
Distributed network optimization algorithms, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require $\tildeΩ(\sqrt n)$ rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this l…
