10.3 Approximation for the InterMCP problem
10.3.3 End-to-End Approximation
General Algorithm
We name aIdMcp an algorithm in which the domains Di from D|S| to D1 translate the link
weights, and then, run the algorithm InterMcpp. The algorithm takes a parameter θ ∈ R as input and changes the original link weights ~w(l) for every link l∈ Ei+ into:
~ wθ= w1, w 2· θ W2 + 1, . . . , w K· θ WK + 1 T (10.3)
and the path constraints ~W into: ~
Wθ= (W1,⌊θ⌋, . . . , ⌊θ⌋)T. (10.4)
We explain the role of θ in the next section.
Approximation
We show that aIdMcp returns solutions that are as close as desired to the optimal for an optimization version of the InterMCP problem, in which we try to optimize the value of the function c(p) = maxk∈[2..K]
w
k(p) Wk
. We can rephrase this sentence in a more intu- itive manner: consider the problem of finding the inter-domain feasible path that minimizes maxk∈[2..K]wk(p)
Wk
; the algorithm aIdMcp provides a solution that is as close as desired to the optimum of this problem.
We consider the original InterMCP problem and we denote as p∗ the feasible s-t path
that has the lowest value of the function c(p) = maxk∈[2..K]wk(p) Wk
among the feasible s-t paths for the InterMCP problem. Given a translation parameter θ, we denote as pθ the final
path that the algorithm aIdMcp returns. The s-t path pθ verifies w
1(pθ) ≤ D and has the
lowest value of the function cθ(p) = maxk∈[2..K]
wθ k(p)
C
among the s-t paths p′ that verify
w1(p′)≤ D. We show that the distance between the performance of p∗ and the performance
of pθ is bounded and can be made as small as requested.
Lemma 10.3.1. c(pθ)− c(p∗) ≤ n−1
θ , where n denotes the total number of nodes in the
considered network (i.e. in the graph that includes all the traversed domains). Proof. We provide the proof of the lemma in Appendix B.
The previous lemma proves that aIdMcp finds solutions that are as close as desired to the optimal solutions for the considered optimization version of the MCP problem, supposing that the value of θ is large enough. This is an interesting result, which enables the algorithm aIdMcp to guarantee that the computed MCPs for a given request have good performance.
116 CHAPTER 10. APPROXIMATION ALGORITHMS: GOOD PATHS, FASTER
We assume that we know a lower bound ζ > 0 on the optimal value c(p∗). We consider a
positive real number ǫ, and we define θ as θ = n−1
ǫ·ζ , where n is the total number of nodes in
the considered network. With these assumptions, (B.8) indicates that: c(pθ)
c(p∗) ≤
|p∗|
θ· c(p∗) + 1. (10.5)
We know that the length of any loop-free path p∗ is bounded: |p∗| ≤ n − 1. In addition, we
have defined ζ such that ζ ≤ c(p∗). Therefore, the inequality (10.5) leads to:
c(pθ)≤ nθ− 1
· ζ + 1 (10.6)
In (10.6), we can replace θ by its definition. Therefore, we obtain the final inequality below. c(pθ)≤ (1 + ǫ) · c(p∗) (10.7) The inequalities 10.5 to 10.7 show that, to provide a (1 + ǫ)-approximation algorithm stricto sensu for the considered optimization problem, it is sufficient to obtain a lower bound on the cost c(p∗) for the optimal path p∗ and to know an upper bound on the total number of nodes
in the network. We have not studied the methods to determine such bounds yet.
Worst-Case Time Complexity
We introduce the notations V = maxi∈[1..D](Vi) and E = maxi∈[1..D](Ei) to compute an
asymptotic bound on the worst-case time complexity of aIdMcp.
Theorem. The worst-case time complexity of aIdMcp is in OD· E · (⌊θ⌋ + 1)K−1. Proof. The weight translation operations take O (|Etot| · C) time, where |Etot| is the total num-
ber of edges in the graph that includes all the traversed domains and their inter-connections. Every domain runs InterMcpp, which takes O(|Vi| + |Bi| + |Bi+1| + |Ei+|) · (C + 1)K−1
time. The total time complexity of these operations for all the traversed domains is in O(3· |Vtot| + |Etot|) · (⌊θ⌋ + 1)K−1
, where |Vtot| denotes the total number of nodes in the
graph that includes all the traversed domains. With the notations E and V , the worst-case time complexity is in OD· (3 · V + E) · (⌊θ⌋ + 1)K−1. We simplify this expression into OD· E · (⌊θ⌋ + 1)K−1, which proves the theorem.
The algorithm aIdMcp ends in polynomial time. However, its complexity rises rapidly when the number K of constraints and the value of the parameter θ (the accuracy of the computations) increase. Therefore, we think that the cost of approximation schemes in terms of execution time hinders their application for inter-domain MCP computations.
10.4
Conclusion
A few approximation algorithms have been proposed for the MCP problem. The most common approximation technique consists in rounding and scaling the link weights to decrease their granularity. We have adapted this idea to solve the inter-domain routing problem more rapidly. We have described a per-domain problem that enables applying approximation techniques,
10.4. CONCLUSION 117
and an algorithm for solving this problem. Our inter-domain path computation method re- turns solutions that are quantifiably close to the optimum and its worst-case time complexity is polynomial, whereas the original problem is N P-complete. Despite this significant complex- ity reduction compared to exact methods, we believe that the complexity of approximation mechanisms is prohibitive in practice. Therefore, in the next chapter, we study heuristic solutions, which do not bring provable performance guarantees but determine good solutions rapidly for most problem instances.
Key points of Chapter 10
• We have proposed aId-Mcp, a solution to compute inter-domain MCPs, with good performance, in polynomial-time. This solution shows that it is possible to adapt MCP approximation techniques for efficient inter-domain computations. • The worst-case time complexity of aId-Mcp is polynomial, whereas the one of exact methods is significantly larger (N P-complete problem). Nevertheless, the complexity of approximation methods is prohibitive in some scenarios; therefore, faster solutions (heuristics) must be studied.
CHAPTER
11
A Second Proposition:
Finding Feasible Paths
Rapidly
11.1
Introduction
Because of the complexity of the problem MCP, the complexity of exact and approximation methods for solving the inter-domain MCP problem becomes rapidly prohibitive when the size of the problem instances rises. Therefore, we derive a novel and efficient heuristic from our exact algorithms.
We present our heuristic, which is named kID-MCP, in Section 11.2. It can be used in various technological contexts. For QoS and TE, we integrate it in the PCE framework and describe the required protocol enhancements. The analytical evaluation described in Section 11.3 and the simulation study provided in Section 11.4 show that kID-MCP computes inter-domain MCPs with a reasonable complexity and without degrading the QoS guarantees.