Distributed Approximation Algorithms for the Combinatorial Motion Planning Problem

We present a new $4$-approximation algorithm for the Combinatorial Motion Planning problem which runs in $mathcal{O}(n^2alpha(n^2,n))$ time, where $alpha$ is the functional inverse of the Ackermann function, and a fully distributed version for the same in asynchronous message passing systems, which runs in $mathcal{O}(nlog_2n)$ time with a message complexity of $mathcal{O}(n^2)$. This also includes the first fully distributed algorithm in asynchronous message passing systems to perform shortcut operations on paths, a procedure which is important in approximation algorithms for the vehicle routing problem and its variants. We also show that our algorithm gives feasible solutions to the $k$-TSP problem with an approximation factor of $2$ in both centralized and distributed environments. The broad idea of the algorithm is to distribute the set of vertices into two subsets and construct paths for each salesman over each of the two subsets. Finally we combine these pairwise disjoint paths for each salesman to obtain a set of paths that span the entire graph. This is similar to the algorithm by Yadlapalli et. al. cite{3.66} but differs in respect to the fact that it does not require us to use minimum cost matching as a subroutine, and hence can be easily distributed.