On the Approximability of the Minimum Congestion Unsplittable Shortest Path Routing Problem

Abstract

We are given an undirected simple graph G = (V,E) with edge capacities $c_{e} \epsilon \mathcal{Z}+$, e ∈ E, and a set K ⊆ V ² of commodities with demand values d _(s,t) εℤ, (s, t) ∈ K. An unsplittable shortest path routing (USPR) of the commodities K is a set of flow paths Φ_(s,t), (s, t) ∈ K, such that each Φ_(s,t) is the unique shortest (s, t)-path for commodity (s, t) with respect to a common edge length function $\lambda = (\lambda_e) \epsilon \mathbb{Z}^{E}_{+}$. The minimum congestion unsplittable shortest path routing problem (Min-Con-USPR) is to find an USPR that minimizes the maximum congestion (i.e., the flow to capacity ratio) over all edges. We show that it is $\mathcal{NP}$-hard to approximate Min-Con-USPR within a factor of $\mathcal{O}(|V|^1-\epsilon)$ for any ε > 0. We also present a simple approximation algorithm that achieves an approximation guarantee of $\mathcal{O}(|E|)$ in the general case and of 2 in the special case where the underlying graph G is a cycle. Finally, we construct examples where the minimum congestion that can be obtained with an USPR is a factor of Ω(|V|²) larger than the congestion of an optimal unsplittable flow routing or an optimal shortest multi-path routing, and a factor of Ω(|V|) larger than the congestion of an optimal unsplittable source-invariant routing. This indicates that unsplittable shortest path routing problems are indeed harder than their corresponding unsplittable flow, shortest multi-path, and unsplittable source-invariant routing problems.

The Min-Con-USPR problem is of great practical interest in the planning of telecommunication networks that are based on shortest path routing protocols.

Mathematical Subject Classification (2000): 68Q25, 90C60, 90C27, 05C38, 90B18.