Routing with congestion in acyclic digraphs

Abstract

We study the version of the k-disjoint paths problem where k demand pairs (s 1 , t 1),. .. , (s k , t k) are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than c paths. We show that on directed acyclic graphs the problem is solvable in time n O(d) if we allow congestion k − d for k paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time f (k)n o(d/ log d) for any computable function f .

A relaxation of the Directed Disjoint Paths problem: A global congestion metric helps

“…As discussed in the introduction, the Directed Disjoint Paths problem is NP-complete for fixed k = 2 [13] and W[1]-hard with parameter k in DAGs [25]. Allowing for vertex congestion does not improve the tractability of the problem: Disjoint Paths with Congestion parameterized by number of requests is also W [1]-hard in DAGs for every fixed congestion c ≥ 1, as observed in [1]. When c = 0 and s ≥ 1, DEDP is equivalent to the Directed Disjoint Paths with Congestion problem and thus the aforementioned bounds also apply to it.…”

“…In the following theorem we complete this picture by showing that DEDP is NP-complete for fixed k ≥ 3 and s ≥ 1, even if c is quite large with respect to n (note that if c = n all instances are trivially positive), namely for c as large as n − n α with α being any fixed real number such that 0 < α ≤ 1. The same reduction also allows to prove W [1]-hardness in DAGs with parameter k. The idea is, given the instance of DDPC, build an instance of DEDP where the "disjoint" part corresponds to the original instance, and the "congested" part consists of c new vertices that are necessarily used by s + 1 paths. This is why we restrict the value of d to be of the form n α , but not smaller; otherwise, the "disjoint" part, which corresponds to the instance of DDPC, would be too small compared to the total size of the graph, and a brute-force algorithm could solve the problem in polynomial time.…”

“…3 and a large range of values of c. The proof does not hold, for example, when c = n − d for a constant d. We also showed that considering only d Relaxation of the Directed Disjoint Paths Problem as a parameter is still not enough to improve the tractability of the problem: Theorem 4 shows that (d)-DEDP is W[1]-hard in DAGs even if all requests share the same source. The situation changes when we consider stronger parameterizations including d: we show that the problem is XP with parameters d and s (cf.…”

“…By a simple reduction from the Directed Disjoint Paths with Congestion problem, it is easy to prove that DEDP is NP-complete for fixed k ≥ 3 and s ≥ 1, even if c is large with respect to n, namely at most n − n α for some real value 0 < α ≤ 1, and W [1]-hard in DAGs with parameter k. By applying the framework of Johnson et al [14], we give an n O(k+w) algorithm to solve DEDP in digraphs with directed tree-width at most w.…”

“…Hardness results for DEDPIn this section we provide hardness results for the DEDP problem. Namely, we first provide in Theorem 3 a simple reduction from Disjoint Paths with Congestion, implying NPcompleteness for fixed values of k, c, d and W[1]-hardness in DAGs with parameter k. We then prove in Theorem 4 that DEDP is W[1]-hard in DAGs with parameter d.…”