We investigate the problem of designing survivable broadband virtual private networks that employ the Open Shortest Path First (OSPF) routing protocol to route the packages. The capacities available for the links of the network are a minimal capacity plus multiples of a unit capacity. Given the directed communication demands between all pairs of nodes, we wish to select the capacities in a such way, that even in case of a single node or a single link failure a specified percentage of each demand can be satisfied and the costs for these capacities are minimal. We present a mixed-integer linear programming formulation of this problem and several heuristics for its solution. Furthermore, we report on computational results with real-world data.
Let G = (V, E) be a simple graph and s and t be two distinct vertices of G. A path in G is called -bounded for some ∈ N if it does not contain more than edges. We prove that computing the maximum number of vertex-disjoint -bounded s, t-paths is APXcomplete for any ≥ 5. This implies that the problem of finding k vertex-disjoint -bounded s, t-paths with minimal total weight for a given number k ∈ N, 1 ≤ k ≤ |V | − 1, and nonnegative weights on the edges of G is N PO-complete for any length bound ≥ 5. Furthermore, we show that these results are tight in the sense that for ≤ 4 both problems are polynomially solvable, assuming that the weights satisfy a generalized triangle inequality in the weighted problem. Similar results are obtained for the analogous problems with path lengths equal to instead of at most and with edge-disjointness instead of vertex-disjointness.
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.
We consider a knapsack problem with precedence constraints imposed on pairs of items, known as the precedence constrained knapsack problem (PCKP). This problem has applications in management and machine scheduling, and also appears as a subproblem in decomposition techniques for network design and other related problems. We present a new approach for determining facets of the PCKP polyhedron based on clique inequalities. A comparison with existing techniques, that lift knapsack cover inequalities for the PCKP, is also presented. It is shown that the clique-based approach generates facets that cannot be found through the existing cover-based approaches, and that the addition of clique-based inequalities for the PCKP can be computationally beneficial.
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