Using a geometric lens to find k disjoint shortest paths

07/24/2020
by   Matthias Bentert, et al.
0

Given an undirected n-vertex graph and k pairs of terminal vertices (s_1,t_1), …, (s_k,t_k), the k-Disjoint Shortest Paths (k-DSP)-problem asks whether there are k pairwise vertex-disjoint paths P_1,…, P_k such that P_i is a shortest s_i-t_i-path for each i ∈ [k]. Recently, Lochet [arXiv 2019] provided an algorithm that solves k-DSP in n^O(k^4^k) time, answering a 20-year old question about the computational complexity of k-DSP for constant k. On the one hand, we present an improved O(k n^12k · k! + k + 1)-time algorithm based on a novel geometric view on this problem. For the special case k=2, we show that the running time can be further reduced to O(n^2m) by small modifications of the algorithm and a further refined analysis. On the other hand, we show that k-DSP is W[1]-hard with respect to k, showing that the dependency of the degree of the polynomial running time on the parameter k is presumably unavoidable.

READ FULL TEXT

page 1

page 2

page 3

page 4

12/22/2019

The shortest disjoint paths problem

For any fixed k, we show the existence of a polynomial-time algorithm de...
09/11/2018

The Undirected Two Disjoint Shortest Paths Problem

The k disjoint shortest paths problem (k-DSPP) on a graph with k source-...
09/30/2021

Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

For vertices u and v of an n-vertex graph G, a uv-trail of G is an induc...
06/23/2021

Finding single-source shortest p-disjoint paths: fast computation and sparse preservers

Let G be a directed graph with n vertices, m edges, and non-negative edg...
02/05/2018

Shortest k-Disjoint Paths via Determinants

The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-...
06/01/2021

Parallel Polynomial Permanent Mod Powers of 2 and Shortest Disjoint Cycles

We present a parallel algorithm for permanent mod 2^k of a matrix of uni...
02/15/2011

An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains

We present the first polynomial time approximation algorithm for computi...