Approximability of unsplittable shortest path routing problems

In this paper, we discuss the relation of unsplittable shortest path routing (USPR) to other routing schemes and study the approximability of three USPR network planning problems. Given a digraph D = (V,A) and a set K of directed commodities, an USPR is a set of flow paths Φ(s,t), (s, t) ∈ K, such that there exists a metric λ = (λa) ∈ Z+ with respect to which each Φ(s,t) is the unique shortest (s, t)-path. In the Min-Con-USPR problem, we seek for an USPR that minimizes the maximum congestion over all arcs. We show that this problem is hard to approximate within a factor of O(|V |1−ǫ), but easily approximable within min(|A|, |K|) in general and within O(1) if the underlying graph is an undirected cycle or a bidirected ring. We also construct examples where the minimum congestion that can be obtained by USPR is a factor of Ω(|V |2) larger than that achievable by unsplittable flow routing or by shortest multi-path routing, and a factor of Ω(|V |) larger than by unsplittable source-invariant routing. In the Cap-USPR problem, we seek for a minimum cost installation of integer arc capacities that admit an USPR of the given commodities. We prove that this problem is NP-hard to approximate within 2 − ǫ (even in the undirected case), and we devise approximation algorithms for various special cases. The fixed charge network design problem FC-USPR, where the task is to find a minimum cost subgraph of D whose fixed arc capacities admit an USPR of the commodities, is shown to be NPO-complete. All three problems are of great practical interest in the planning of telecommunication networks that are based on shortest path routing protocols. Our results indicate that they are harder than the corresponding unsplittable flow or shortest multi-path routing problems.