Approximability of unsplittable shortest path routing problems

Abstract

In this article, 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 P * (s,t) , (s, t) ∈ K , such that there exists a metric λ = (λ a ) ∈ Z Z Z A + with respect to which each P * (s,t) is the unique shortest (s, t)-path. In the MinCon-USPR problem, we seek an USPR that minimizes the maximum congestion over all arcs. We show that this problem is N P-hard to approximate within a factor of O(|V | 1− ), but polynomially 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 (|V | 2 ) larger than that achievable by unsplittable flow routing or by shortest multipath routing, and a factor of (|V |) larger than that achievable by unsplittable source-invariant routing. In the CAP-USPR problem, we seek a minimum cost installation of integer arc capacities that admit an USPR of the given commodities. We prove that this problem is N P-hard to approximate within 2 − 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 N POcomplete. 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.

Fuzzy particle swarm optimization algorithms for the open shortest path first weight setting problem

“…Notable research in optimizing OSPF weights has been reported in the literature [2,4,6,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. The pioneering work on the OPSF weight setting problem was done by Fortz and Thorup [8,10,29] who used maximum utilization as the optimization objective.…”

“…These weights are in the range [1,20]. A solution for this topology can be (18,1,7,15,3,17,14,19,13,18,4,16,16).…”

“…Buriol et al [18] extended the genetic algorithm proposed in [13] to a memetic algorithm by adding a local search procedure while using the same cost function as that of Fortz and Thorup. Bley [19,20] proposed unsplittable shortest path routing (UPSR) and claimed that the proposed approach can be applied to other routing schemes such as OSPF, while considering minimization of maximum congestion over all arcs. Zagozdzon et al [14] proposed a two-phase algorithm for resolving the OSPF weight setting problem while considering the residual capacity as the optimization objective.…”

“…Other researchers [6,22,17] also assumed minimization of maximum link utilization. Other objectives considered in the optimization process were maximization of unused link bandwidth [15], minimization of network congestion [16,19,20], residual capacity of link [14], minimization of total link utilization [36], maximization of network throughput [23,24], and minimization of sum of energy consumption of all links [28]. Exceptions from these single-objectives optimization approaches were Nucci et al [25] where link failure and link no-failure states were used as the optimization objectives, and Sqalli et al [11,38] who used minimization of maximum 9 link utilization as well as minimization of congested links as the optimization objectives while using a simulated annealing (SA) algorithm [39].…”

Routing and Capacity Optimization for IP Networks

“…This problem cannot be approximated within a factor of O(2 log 1−ǫ |V | ) in the directed and within a factor of 2 − ǫ in the undirected case, unless P = N P. For various special cases, however, better approximation algorithms can be derived. For the case where the underlying network is an undirected cycle or a bidirected ring, for example, Min-Con-USPR and Cap-USPR are approximable within constant factors [5].…”