7.3.1 The Intra-Domain MCP Problem is N P-Complete
Determining MCPs requires hard computations: Wang and Crowcroft have proved that the MCP problem with two or more additive metrics is N P-complete [180]. To establish the N P- completeness of the MCP problem, they proceed by induction. They exhibit two polynomial transformations: first, from the well-known N P-complete PARTITION problem [82] to the MCP problem with two constraints, and then, from the MCP problem with n constraints to the MCP problem with n + 1 constraints. With this proof and related lemmas, Wang and
7.3. TRACTABILITY OF CONSTRAINED-PATH COMPUTATIONS 79
Figure 7.1: Absence of total order relationship for multi-dimensional weight spaces and con- sequences for the path computation algorithms
Crowcroft demonstrate that any combination of two metrics among delay, delay jitter, cost, and loss probability, leads to an N P-complete MCP problem [180]. The N P-completeness implies that some instances of the MCP problem cannot be solved in practice because the time complexity of the computation is prohibitive [82, 185].
7.3.2 The Inter-Domain MCP Problem is N P-Complete
We investigate the computational complexity of the inter-domain MCP problem. We know that the problem MCP is N P-complete. In addition, we show that the MCP problem is a special case of the problem InterMCP. Thus, the problem InterMCP is N P-complete [82]. Theorem. The problem InterMCP is N P-complete.
Proof. We consider an instance of the problem MCP with a source s, a target t, and an integer number K ≥ 2 of bound constraints Wk with k in [1..K] in a graph G(V, E). The
problem InterMCP with the same source, target, and constraints, and with the loop-free domain sequence S = (V ) is equivalent3 to the previous instance of the problem MCP.
As the problem InterMCP is N P-complete, some of its instances cannot be solved exactly in polynomial time by a deterministic Turing machine.4 Nevertheless, many instances of the
inter-domain MCP problem are solvable in polynomial time, as explained in Section 7.3.4.
7.3.3 Origin of the Large Complexity
In this section, we explain that the absence of a total order relationship [36] on the set of the path weights is one of the causes for the N P-completeness of the MCP problem. To clarify our explanations, we compare the MCP problem to the problem with a single constraint.
The problem of computing a path subject to a single bound constraint on an additive metric is solvable in polynomial time with Dijkstra’s or Bellman-Ford algorithm [72]. These algorithms maintain a single shortest path from the considered source node s to every in- termediate node a. We consider the example in Figure 7.1: in this figure, there are several candidate paths between s and a, with path weights 11, 6, and 14. It is clear that the inter- mediate path with weight 6 leads to better end-to-end paths between s and t than the two
3
Both problems have exactly the same solutions.
4
80 CHAPTER 7. RESOURCE PROVISIONING FOR INTER-DOMAIN TRAFFIC
alternative paths, because its weight is the lowest. Thus, the path computation algorithms need to memorize only this intermediate path. Usually, determining the path that the algo- rithm must memorize is straightforward because the paths weights take their values in (R, ≤), which is a totally ordered set [36].
By contrast, the MCP problem does not define a method to compare the weight vectors of two paths. Given two paths p1and p2between two nodes s and a, it is not always possible to determine if the weights ~w(p1) are better than the weights ~w(p2). We illustrate this problem on the example in Figure 7.1. There are three possible intermediate paths p1, p2, and p3 between two nodes s and a, with ~w(p1) = (4, 3)T, ~w(p2) = (8, 2)T, and ~w(p3) = (3, 3)T. All the weights of p3are lower than the ones of p1, so it is clear that the best end-to-end s-t-path does not include p1. However, p3 is better than p2 for the weight w1 but worse than p2 for
w2. Therefore, it is hard to predict which of these paths shall lead to the best end-to-end solutions from s to t and we must consider both paths during MCP computation operations. This simple example shows that to solve certain instances of the MCP problem, an algorithm must consider many sub-paths to intermediate nodes and must combine them to the possible links toward the target, which drastically affects the computational complexity.
Kuipers and Van Mieghem [111] assert that the N P-complete behavior of the MCP prob- lem emerges only in particular network configurations that occur rarely in practice. In Refer- ence [112], they investigate the conditions that affect the complexity of the MCP problem. In particular, they consider the number of paths to intermediate nodes that must be considered to solve the MCP problem exactly. They show that this number depends on the network topology, the granularity of the link weights, the correlation of these weights, and the con- straints. In the next section, we detail MCP problem instances that are important in practice and that can be solved in polynomial time.
7.3.4 Problem Instances Solvable in Polynomial Time
Influence of the Metrics on the Complexity
The MCP problem can be simplified when the considered link metrics are bound by inter- dependency relationships. In particular, Ma and Steenkiste [120] show that for a broad class of scheduling algorithms the problem of finding a path subject to constraints on bandwidth, delay, jitter, and buffer space is solvable in polynomial time, because delay, jitter, and buffer space become functions of the bandwidth. Orda [134] describes the metric relationships for networks that use rate-based schedulers. In addition, he exploits the typical hierarchical struc- ture of large-scale networks to derive an efficient algorithm for the minimum-cost bounded- delay problem. The work of Ma, Steenkiste, and Orda enables operators to solve the routing problems that involve delay and jitter constraints in a simple manner in networks that use rate-based scheduling. They first translate the constraints into a bandwidth requirement, and then they solve the bandwidth-constrained problem with usual methods (e.g., link-pruning algorithm). Nevertheless, in the general case, some metrics, such as the propagation delay, cannot be formulated as a function of bandwidth. Thus, these methods do not solve the general MCP problem.
In addition to the inter-dependency of the link metrics, the granularity of the metric values affects the complexity of the MCP problem. Yuan [188] and Chen et alii [46] explain that when all considered metrics except one take bounded integer values, the MCP problem is solvable in polynomial time. In particular, Chen and Nahrstedt propose extended versions of the Bellman-Ford and the Dijkstra’s algorithm that can solve this problem [46]. The MCP