Design of Survivable Networks by Mechthild Stoer (auth.)

By Mechthild Stoer (auth.)

The challenge of designing a inexpensive community that survives the failure of 1 or extra nodes or edges of the community is important to fashionable telecommunications engineering. the strategy built during this publication is designed to unravel such difficulties to optimality. specifically, a slicing aircraft method is defined, according to polyhedral combinatorics, that's ableto clear up real-world difficulties of this kind briefly computation time. those effects are of curiosity for practitioners within the zone of communique community layout. The publication is addressed particularly to the combinatorial optimization neighborhood, but in addition to people who are looking to research polyhedral tools. furthermore, fascinating new learn problemsare formulated.

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0, 3. 2_>ru>_ 1 a n d 2 > r v _ > 1. A characteristic of these decompositions is that the subproblems must be solved in a certain order, not independently of each other, since the output of several of these subproblems determines the input to the last one. Both Nodes in t h e A r t i c u l a t i o n Set Are of Type 0 Consider a k E C O N problem, given by a graph G = (V, E) and a node type vector r E ~ v . ) Let { u , v } be an articulation set with ru = rv = 0, such that G - {u,v} has two components G1 = (V1,/)I) and G2 = (1)2,/)2) with v ( G ) = 1 and r(V2) >_ 1.

5: Gadget 40 No C H A P T E R 4. D E C O M P O S I T I O N Node in the Articulation Set Is of Type 0 If both nodes u and v in the articulation set are of type 1 or 2, we need only distinguish between partial solutions of type (a) or (d), so we only compute the values a and d and solve the kECON problem on the graph G2 with an additional uv-edge of cost d - a. The additive constant z is a. ) If the edge uv is used by an optimal solution C, we replace uv in C by the optimal partial tree of type (d), otherwise we augment C by the optimal partial tree of type (a).

If there exist edges uv, we add t h e m either to G1 or to G2 but not to both. If C C_ E is a feasible solution to the kECON problem on G, the set T := C C~E1 (called a "partial tree") may have four different forms, according to whether u and v are connected in T, or whether u or v are used at all. More exactly, T is a feasible solution to one of the four following Steiner tree problems (P~), (Pb), (Pc), (Pd). These subproblems are defined on G1 and use the same costs and the same node types for all nodes in V~ \ {u,v} as in the original problem; only the node types of u and v vary as follows: (P~): r~ = 1 and rv = 1 and an artificial edge uv with cost 0 is added; (Pb): r~ = 0 and r~ = 1; (P~): r~ = 1 and r~ = O; (Pd): r~ = 1 and r~ = 1.

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