The subdivision-constrained routing requests problem - Li, Jianping; Li, Weidong; Lichen, Junran (Journal of combinatorial optimization, 2014)
We are given a digraph D =( V , A ; w ), a length (delay) function w : A → R + , a positive integer d and a set of k requests, where s i ∈ V is called as the i th source node, t i ∈ V is called the i th sink node and B i is called as the i th length constraint. For a given positive integer d , the subdivision-constrained routing requests problem (SCRR, for short) is to find a directed subgraph D ′=( V ′, A ′) of D , satisfying the two constraints: (1) Each request ( s i , t i ; B i ) has a path P i from s i to t i in D ′ with length no more than B i ; (2) Insert some nodes uniformly on each arc e ∈ A ′ to ensure that each new arc has length no more than d . The objective is to minimize the total number of the nodes inserted on the arcs in A ′. We obtain the following three main results: (1) The SCRR problem is at least as hard as the set cover problem even if each request has the same source s , i.e. , s i = s for each i =1,2,…, k ; (2) For each request ( s , t ; B ), we design a dynamic programming algorithm to find a path from s to t with length no more than B such that the number of the nodes inserted on such a path is minimized, and as a corollary, we present a k -approximation algorithm to solve the SCRR problem for any k requests; (3) We finally present an optimal algorithm for the case where contains all possible requests ( s i , t i ) in V × V and B i is equal to the length of the shortest path in D from s i to t i . To the best of our knowledge, this is the first time that the dynamic programming algorithm within polynomial time in (2) is designed for a weighted optimization problem while previous optimal algorithms run in pseudo-polynomial time.