Related papers: Long induced paths in a configuration model
A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This…
We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set…
We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to…
In this invited contribution, we revisit the stochastic shortest path problem, and show how recent results allow one to improve over the classical solutions: we present algorithms to synthesize strategies with multiple guarantees on the…
We study the inapproximability of the induced disjoint paths problem on an arbitrary $n$-node $m$-edge undirected graph, which is to connect the maximum number of the $k$ source-sink pairs given in the graph via induced disjoint paths. It…
A minimum path cover (MPC) of a directed acyclic graph (DAG) $G = (V,E)$ is a minimum-size set of paths that together cover all the vertices of the DAG. Computing an MPC is a basic polynomial problem, dating back to Dilworth's and…
Ding (1992) proved that for each integer ${m} \geqslant 0$, and every infinite sequence of finite simple graphs $G_1, G_2, \ldots$, if none of these graphs contains a path of length ${m}$ as a subgraph, then there are indices $i < j$ such…
We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the…
We consider the problem of inferring the topology of a network using the measurements available at the end nodes, without cooperation from the internal nodes. To this end, we provide a simple method to obtain path interference which…
Following [21, 23], the present work investigates a new relative entropy-regularized algorithm for solving the optimal transport on a graph problem within the randomized shortest paths formalism. More precisely, a unit flow is injected into…
We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$. In this setting, each vertex corresponds to a geometric object, and two…
A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are…
It is proven that the connected pathwidth of any graph $G$ is at most $2\cdot\pw(G)+1$, where $\pw(G)$ is the pathwidth of $G$. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width…
Given an undirected, weighted graph, with $n$ vertices and $m$ edges, and two special vertices $s$ and $t$, the problem is to find the shortest path between them. We give two bounded-error quantum algorithms with improved runtime in the…
We consider the parametric shortest paths problem in a linearly interpolated graph. Given two positively-weighted directed graphs $G_0=(V,E,\omega_0)$ and $G_1=(V,E,\omega_1),$ the linearly interpolated graph is the family of graphs…
We propose an approach to solve multi-agent path planning (MPP) problems for complex environments. Our method first designs a special pebble graph with a set of feasibility constraints, under which MPP problems have feasibility guarantee.…
Complex systems of interacting components often can be modeled by a simple graph $\mathcal{G}$ that consists of a set of $n$ nodes and a set of $m$ edges. Such a graph can be represented by an adjacency matrix $A\in\R^{n\times n}$, whose…
There has been much interest in the distribution of the circumference, the length of the longest cycle, of a random graph $G(n,p)$ in the sparse regime, when $p = \Theta\left(\frac{1}{n}\right)$. Recently, the first author and Frieze…
We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs…
This paper considers the problem of inferring the structure of a network from indirect observations. Each observation (a "trace") is the unordered set of nodes which are activated along a path through the network. Since a trace does not…