Related papers: Successive shortest paths in complete graphs with …
A solution of the $k$ shortest paths problem may output paths that are identical up to a single edge. On the other hand, a solution of the $k$ independent shortest paths problem consists of paths that share neither an edge nor an…
Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the…
It is well known that finding extremal values and structures can be hard in weighted graphs. However, if the weights are random, this problem can become way easier. In this paper, we examine the minimal weight of a union of $k$…
We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has…
We determine, asymptotically in $n$, the distribution and mean of the weight of a minimum-weight $k$-clique (or any strictly balanced graph $H$) in a complete graph $K_n$ whose edge weights are independent random values drawn from the…
Recall that Janson showed that if the edges of the complete graph $K_n$ are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal…
We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct…
We consider a constrained version of the shortest path problem on the complete graphs whose edges have independent random lengths and costs. We establish the asymptotic value of the minimum length as a function of the cost-budget within a…
Assume that the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ edges weights. We consider the expected minimum total weight $\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show that…
Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size…
We construct a family of planar graphs $\{G_n\}_{n\geq 4}$, where $G_n$ has $n$ vertices including a source vertex $s$ and a sink vertex $t$, and edge weights that change linearly with a parameter $\lambda$ such that, as $\lambda$ varies in…
We present an all-pairs shortest path algorithm whose running time on a complete directed graph on $n$ vertices whose edge weights are chosen independently and uniformly at random from $[0,1]$ is $O(n^2)$, in expectation and with high…
The shortest path problem in graphs is a cornerstone of AI theory and applications. Existing algorithms generally ignore edge weight computation time. We present a generalized framework for weighted directed graphs, where edge weight can be…
In minimum power network design problems we are given an undirected graph $G=(V,E)$ with edge costs $\{c_e:e \in E\}$. The goal is to find an edge set $F\subseteq E$ that satisfies a prescribed property of minimum power $p_c(F)=\sum_{v \in…
In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let…
Given a graph $G=(V,E)$ and a set $T=\{ (s_i, t_i) : 1\leq i\leq k \}\subseteq V\times V$ of $k$ pairs, the $k$-vertex-disjoint-paths (resp. $k$-edge-disjoint-paths) problem asks to determine whether there exist~$k$ pairwise vertex-disjoint…
The mean weight of a cycle in an edge-weighted graph is the sum of the cycle's edge weights divided by the cycle's length. We study the minimum mean-weight cycle on the complete graph on n vertices, with random i.i.d. edge weights drawn…
Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the…
In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $\mathcal{T}=\{(s_1,t_1),\dots,(s_k,t_k)\}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,\dots,P_k$ in $G$ so that each $P_i$…
We consider the problem of computing all pairs shortest paths (APSP) and shortest paths for k sources in a weighted graph in the distributed CONGEST model. For graphs with non-negative integer edge weights (including zero weights) we build…