Multi-Constrained QoS Routing: A Norm Approach
Guoliang Xue∗, Senior Member, IEEE and S. Kami Makki†, Member, IEEE
Abstract— A fundamental problem in quality-of-service (QoS) routing is the multi-constrained path (MCP) problem, where one seeks a source-destination path satisfying K ≥ 2 additive QoS constraints in a network with K additive QoS parameters.
The MCP problem is known to be NP-complete. One popular approach is to use the shortest path with respect to a single edge weighting function as an approximate solution toMCP. In a pioneering work, Jaffe showed that the shortest path with respect to a scaled 1-norm of the K edge weights is a 2-approximation to MCP in the sense that the sum of the larger of the path weight and its corresponding constraint is within a factor of 2 from minimum. In a recent paper, Xue et al. showed that the shortest path with respect to a scaled ∞-norm of the K edge weights is a K-approximation to MCP, in the sense that the largest ratio of the path weight over its corresponding constraint is within a factor of K from minimum. In this paper, we study the relationship between these two optimization criteria and present a class of provably good approximation algorithms toMCP. We first prove that a good approximation according to the second optimization criterion is also a good approximation according to the first optimization criterion, but not vice versa. We then present a class of very simple K-approximation algorithms according to the second optimization criterion, based on the computation of a shortest path with respect to a single edge weighting function.
Index Terms— QoS routing, multiple additive QoS parameters, approximation algorithms, scaled p-norm.
1. INTRODUCTION
A fundamental problem in quality-of-service (QoS) routing is the multi-constrained path (MCP) problem, where one seeks a source-destination path satisfying K ≥ 2 additive QoS constraints in a network with K additive QoS parameters, such as cost, delay, and reliability [2], [10], [18], [20], [25].
Commonly, the network is modeled by a directed graph where the n vertices represent computers or routers and the m edges represent links. To model multiple QoS parameters, each edge is associated with K edge weights, representing cost, delay, and reliability, etc., of the edge. Correspondingly, each path has multiple path weights associated with it, representing cost, delay, and reliability, etc., of the path. If an edge weight represents cost or delay of the edge, then the corresponding path weight is the sum of the weights associated with the edges on the path. For this reason, QoS parameters such as cost and delay are called additive parameters. If an edge weight represents the reliability of the edge, then the corresponding
∗Guoliang Xue is a Full Professor in the Department of Computer Science and Engineering at Arizona State University, Tempe, AZ 85287-8809. Email:
xue@asu.edu. The research of this author was supported in part by ARO grant W911NF-04-1-0385 and NSF grants CCF-0431167 and ANI-0312635.
†S. Kami Makki is an Assistant Professor with the Department of Electrical Engineering & Computer Science at The University of Toledo, Toledo, OH 43606-3390. Email:kmakki@eng.utoledo.edu.
‡The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S.
Government.
path weight is the product of the weights associated with the edges on the path. Since the logarithm of the product of N positive numbers is the sum of the logarithms of the N positive numbers, QoS parameters such as reliability are also called additive parameters. It is well known that the MCP problem is NP-complete as long as there are K ≥ 2 additive QoS parameters [25]. Another kind of QoS parameters are known as concave parameters (such as bandwidth) where the corresponding weight of a path is the smallest of the weights of the edges on the path [25]. Problems involving concave QoS parameters can be easily solved by considering a polynomial number of subgraphs each with only those edges whose weights are greater than or equal to a particular chosen value. Therefore the number of concave QoS parameters does not affect the computational complexity of theMCPproblem (whether it is in P or NP-complete). If all QoS parameters are concave, or all except one of the QoS parameters are concave, the MCP problem is polynomial time solvable. Therefore in this paper we restrict our attention to additive parameters only.
Due to its increasing important applications, the MCP problem has been studied by many researchers. Existing works for this problem can be classified into two broad classes:
sophisticated approximation schemes, and simple heuristic algorithms. An approximation scheme can compute, for any constant > 0, a path that is within a factor of (1 + ) of the optimal solution, with a running time that is bounded by a polynomial in the input size of the given instance, with treated as a constant. Heuristic algorithms are normally simple and fast, without providing a priori theoretical guarantees of the computed path.
Many heuristic algorithms for MCP have been proposed for both the special case of K = 2 and the general case of K ≥ 2. Jaffe in [10] studied the MCP problem with K= 2and presented simple and provably good approximation algorithms, using an optimization criterion that is formally defined in Section 2. This work was generalized to the general case of K ≥ 2 by Andrew and Kusuma in [1]. Chen et al.
in [2] studied the MCP problem with K = 2 and proposed a polynomial time heuristic algorithm based on scaling and rounding of the second parameter so that the second parameter of each edge is approximated by a bounded integer. In [32], Yuan generalized the heuristic of [2] to the case of K ≥ 2 and proposed a limited granularity heuristic. For any given > 0, this heuristic algorithm has time complexity O(mn(n)K−1) and can find a feasible path if there is a path whose last K −1 path weights are no more than (1 − ) of the corresponding constraints, but may fail to find a path when this condition does not hold. Korkmaz et al. in [14] proposed a randomized heuristic for the MCP problem, which may find a feasible path quickly, but may fail even when there is a feasible path.
Liu et al. in [16] proposed a selection function approach to
study the tradeoff between the quality of the path and the time required to compute the path. Related works can be found in [11], [14], [20], [21], [22], [23] and the references therein.
Many researchers have proposed to obtain a new edge weight- ing function using the K edge weights and use the shortest path with respect to this new edge weighting function as an approximation to the MCP problem. Works along this line can be found in [4], [12], [13], [27], [28], [29]. Researchers have also investigated integrated path metrics using a linear or nonlinear combination of the K path weights [15], [23], [24].
Most of the existing approximation schemes concentrate on the MCP problem with two additive QoS parameters.
Warburton in [26] first developed a fully polynomial time approximation scheme (FPTAS) [5] for the MCP problem on an acyclic graph. Hassin in [9] presented an improved FPTAS. Lorenz and Raz in [17] presented an even better FPTAS with a time complexity of O(mn(log log n + 1)), where is the approximation parameter. Their FPTAS applies to general graphs, rather than just acyclic graphs. Goel et al.
in [6] presented an approximation algorithm for the single source all destinations delay sensitive least cost path problem with a time complexity of O((m+n log n)H), where H is the hop count of the longest computed path. Ergun et al. in [3]
presented an FPTAS for the case of acyclic graphs with a time complexity of O(m(n)). Orda and Sprintson in [19]
studied disjoint QoS paths. Xue et al. in [31] studied theMCP problem with K being any integer constant greater than or equal to 2. They presented an FPTAS for general graphs with a time complexity of O((m(n)K−1) and a very simple K- approximation algorithm by computing a shortest path with respect to a scaled ∞-norm of the K edge weights.
In this paper, we study the relationship between the op- timization criterion used in [1], [10] and the optimization criterion used in [2], [16], [22], [23], [24], [27], [28], [29], [30], [31], [32], and present a class of provably good approx- imation algorithms for the MCPproblem. We first prove that a good approximation according to the second optimization criterion is also a good approximation according to the first optimization criterion, but not vice versa. We then present a class of very simple K-approximation algorithms according to the second optimization criterion, based on the computation of a shortest path with respect to a single edge weighting function. Our results provide a theoretical support of many heuristic algorithms for theMCPproblem that are based on a shortest path with respect to a single edge weighting function which is a linear combination of the K edge weights. The class of K-approximation algorithms presented in this paper contains, as special cases, the K-approximation algorithm of Xue et al. [31], the algorithm of Jaffe [10], the algorithm of Andrew and Kusuma [1], along with many other algorithms which are not previously known.
The rest of this paper is organized as follows. In Section 2, we define the problems and some notations. We also study the relationship between two optimization criteria used for solving the MCP problem. In Section 3, we present a class of K- approximation algorithms for MCP which are based on the computation of a shortest path with respect to a single edge weighting function. We conclude this paper in Section 4.
2. PROBLEMDEFINITIONS ANDCOMPARISON OF
OPTIMIZATIONCRITERIA
We use an integer constant K ≥ 2 to denote the number of additive QoS parameters. Unless specified otherwise, all other constants, functions, and variables are assumed to have real values. All logarithms are based-2 logarithms. A polynomial time β-approximation algorithmfor a minimization problem is an algorithm A that, for any instance of the problem, computes a solution that is at most β times the optimal solution of the instance, in time bounded by a polynomial in the input size of the instance [5].
We model a computer network using a K-edge weighted directed graph G = (V, E, ~ω), where V is the set of n vertices, E is the set of m edges, and ~ω = (ω1, . . . , ωK) is an edge weight vectorso that ωk(e) ≥ 0 is the kth weightof edge e,
∀ e ∈ E, ∀ 1 ≤ k ≤ K. For a path π in G, the kth weightof π, denoted by ωk(π), is the sum of the kth weights over the edges on π: ωk(π) =P
e∈πωk(e).
Throughout this paper, we will use s ∈ V and t ∈ V to denote the source node and the destination node. We will use W~ = (W1, . . . , WK)to denote a constraint vector where each Wk is a positive constant. For a path π connecting s to t, we are interested in the following two path metrics.
fJAFFE(π) =
K
X
k=1
max{ωk(π), Wk}. (2.1)
fOMCP(π) = max
1≤k≤K
ωk(π) Wk
. (2.2)
fJAFFE(π)measures the sum (over k ∈ {1, . . . , K}) of the larger of the kth path weight ωk(π)and the kth constraint Wk. This path metric was introduced by Jaffe [10] and has also been used in [1]. fOMCP(π) measures the maximum (over k ∈ {1, . . . , K}) of the ratio of the kth path weight ωk(π) over the kth constraint Wk. This path metric has been used by researchers in [2], [16], [22], [23], [24], [27], [29], [30], [31], [32].
The decision version of the multi-constrained path (DMCP) problem is defined in the following.
Definition 2.1 (DMCP(G, s, t, K, ~W , ~ω)): INSTANCE: a K-edge weighted directed graph G = (V, E, ~ω), with K nonnegative real-valued edge weights ωk(e), 1 ≤ k ≤ K, associated with each edge e ∈ E; a constraint vector W~ = (W1, . . . , WK) where each Wk is a positive constant;
and a source-destination node pair (s, t). QUESTION: is there an s–t path π such that ωk(π) ≤ Wk, ∀ 1 ≤ k ≤ K?
In the above definition, the inequality ωk(π) ≤ Wk is called the kth QoS constraint. A path π satisfying all K QoS constraints is called a feasible path or a feasible solution of DMCP(G, s, t, K, ~W , ~ω). It is clear that an s–t path π is a feasible solution to DMCP(G, s, t, K, ~W , ~ω) if and only if fOMCP(π) ≤ 1. We say that DMCP(G, s, t, K, ~W , ~ω) is feasibleif it has a feasible path, and infeasible otherwise. We may simply use DMCP to denote DMCP(G, s, t, K, ~W , ~ω), without any confusion, the same rule can be applied toOMCP andJMCPdefined in the sequel.
TheDMCPproblem is known to be NP-complete [5], [25], for any integer K such that K ≥ 2. Therefore the following
optimization version of this problem has been studied in the literature [30], [31].
Definition 2.2 (OMCP(G, s, t, K, ~W , ~ω)): INSTANCE: a K-edge weighted directed graph G = (V, E, ~ω), with K nonnegative real-valued edge weights ωk(e), 1 ≤ k ≤ K, associated with each edge e ∈ E; a constraint vector W~ = (W1, . . . , WK) where each Wk is a positive constant;
and a source-destination node pair (s, t). PROBLEM: find an s–t path πO such that fOMCP(πO) ≤ fOMCP(π)for any s–t path π.
We call πO an optimal path or an optimal solution of OMCP(G, s, t, K, ~W , ~ω), and call ζOopt , fOMCP(πO) the optimal value of OMCP(G, s, t, K, ~W , ~ω). It is clear that ζOopt≤ 1 if and only ifDMCP(G, s, t, K, ~W , ~ω) is feasible.
In [10], Jaffe studied the following optimization version of the multi-constrained path problem.
Definition 2.3 (JMCP(G, s, t, K, ~W , ~ω)): INSTANCE: a K-edge weighted directed graph G = (V, E, ~ω), with K nonnegative real-valued edge weights ωk(e), 1 ≤ k ≤ K, associated with each edge e ∈ E; a constraint vector W~ = (W1, . . . , WK) where each Wk is a positive constant;
and a source-destination node pair (s, t). PROBLEM: find an s–t path πJ such that fJAFFE(πJ) ≤ fJAFFE(π)for any s–t path π.
We call πJ an optimal path or an optimal solution of JMCP(G, s, t, K, ~W , ~ω), and call ζJopt , fJAFFE(πJ) the optimal value of JMCP(G, s, t, K, ~W , ~ω). Clearly, ζJopt = PK
k=1Wk when DMCP is feasible and ζJopt > PK k=1Wk
otherwise.
Jaffe proposed to design efficient approximation algorithms for JMCP, based on the shortest path with respect to a single edge weighting function. Indeed, Jaffe in [10] (for the case of K = 2) and Andrew and Kusuma in [1] (for the case of K > 2) showed that for carefully chosen positive numbers α1, . . . , αK, the shortest s–t path παwith respect to the edge weighting function
ωα(e) =
K
X
k=1
αk·ωk(e)
Wk , ∀ e ∈ E (2.3) guarantees that
fJAFFE(πα) ≤ (2 − 1
K) × ζJopt. (2.4) In addition, with α∗1= · · · = α∗K = 1, the s–t path πα∗ with respect to ωα∗ guarantees that
fJAFFE(πα∗) ≤ 2 × ζJopt. (2.5) In other words, one can obtain a 2-approximation to theJMCP problem by computing a shortest path with respect to a single edge weighting function that is a linear combination of the K edge weights.
The OMCP problem has been studied by many re- searchers [2], [16], [22], [23], [27], [28], [29], [30], [31]. In a recent paper, Xue et al. [31] studied the OMCP problem and presented a K-approximation algorithm based on the computation of a shortest path with respect to a single edge
weighting function defined in the following.
ωmax(e) = max
1≤k≤K
ωk(e) Wk
, ∀ e ∈ E. (2.6) We call the edge weighting function ωα∗(e)the scaled 1- norm of the K edge weighting functions ω1(e), . . . , ωK(e), and call the edge weighting function ωmax(e)the scaled ∞- norm of the K edge weighting functions ω1(e), . . . , ωK(e), formally defined in the next section. On one hand, we know that the shortest path with respect to the scaled 1-norm is a 2-approximation to the JMCP problem. On the other hand, we know that the shortest path with respect to the scaled ∞- norm is a K-approximation to theOMCPproblem. A natural question to ask is the following.
Should we strive to compute a path π which is a good approximation toJMCPor should we strive to compute a path π which is a good approximation to OMCP? The following theorem shows that a good approximation to OMCP is also a good approximation toJMCP, but a good approximation to JMCPcould be a very poor approximation toOMCP.
Theorem 2.1: Let β ≥ 1 be any given constant.
1) If s–t path π is a β-approximation to OMCP(G, s, t, K, ~W , ~ω), then π is also a max{1, βζOopt}-approximation to JMCP(G, s, t, K, ~W , ~ω), where ζOopt is the optimal value ofOMCP(G, s, t, K, ~W , ~ω).
2) Let η > 0 be an arbitrarily large constant and >
0 be an arbitrarily small constant. There exist a K- edge weighted graph G(V, E, ω1, . . . , ωK), QoS con- straints ~W = (W1, . . . , WK), source-destination node pair (s, t), and an s–t path π0 such that π0 is a (1 + ) approximation to JMCP(G, s, t, K, ~W , ~ω), but is not a
η0
ζoptO -approximation toOMCP(G, s, t, K, ~W , ~ω)for any η0 < η, where ζOopt∈ [1, 2] is the optimal value of the corresponding OMCP(G, s, t, K, ~W , ~ω)instance.
PROOF. Let π be a β-approximation to OMCP. We have ωk(π) ≤ β · ζOopt· Wk, ∀ 1 ≤ k ≤ K. Therefore
max{ωk(π), Wk} ≤ max{β · ζOopt,1} · Wk, ∀ 1 ≤ k ≤ K. (2.7) As a result, we have
fJAFFE(π) =
K
X
k=1
max{ωk(π), Wk} ≤ max{β · ζOopt,1} ·
K
X
k=1
Wk. (2.8) It follows from the definition of fJAFFE(·) that for any s–t path π we must have
fJAFFE(π) =
K
X
k=1
max{ωk(π), Wk} ≥
K
X
k=1
Wk. (2.9) Inequalities (2.8) and (2.9) together imply that π is a max{1, βζOopt}-approximation to JMCP. This proves our first claim. Next, we will construct an example to validate our second claim.
The graph G (illustrated in Figure 1(a) for K = 3) has vertex set V = {s, t, x, y}, edge set E = {(s, x), (s, y), (x, t), (y, t)}, with edge weighting functions ωk so that ω1(s, x) = · · · = ωK−1(s, x) = 1 + ,
x
s t
y
(0, 0, 0)
(a) DMCP is infeasible
x
s t
y
(0, 0, 0)
! " # $
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& '()*
+
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)&
), - + .0/213/215476
(b) DMCP is feasible Fig. 1. A (1 + ) approximation to JMCP could be a very poor approximation to OMCP.
ωK(s, x) = 0; ω1(x, t) = · · · = ωK−1(x, t) = 0, ωK(x, t) = (1 + )δ; ω1(s, y) = · · · = ωK(s, y) = 0; ω1(y, t) =
· · · = ωK−1(y, t) = 1, ωK(y, t) = ηδ. We also assume (W1, . . . , WK−1, WK) = (1, . . . , 1, δ). The parameters η, and δ are chosen in the following way: η > 0 is a given arbitrarily large constant (compared with K and ) and ∈ (0, 1]is an arbitrarily small constant such that η is much larger than 1+. After η and are fixed, we choose δ > 0 sufficiently small so that (1+)(K −1+δ) > K −1+ηδ. This is possible because
δ→0lim
(1 + )(K − 1 + δ)
K− 1 + ηδ = 1 + > 1. (2.10) There are exactly two s–t paths in G: π1= s → x → tand π2= s → y → t. The path weights of π1 are ω1(π1) = · · · = ωK−1(π1) = 1 + , ωK(π1) = (1 + )δ. The path weights of π2 are ω1(π2) = · · · = ωK−1(π2) = 1, ωK(π2) = ηδ.
Therefore we have
fOMCP(π1) = 1 + , fJAFFE(π1) = (1 + )(K − 1 + δ), fOMCP(π2) = η; fJAFFE(π2) = K − 1 + ηδ.
By our choice of η, and δ, π1 is the optimal solution to OMCP, and π2is the optimal solution toJMCP. ζOopt= 1+, ζJopt= K − 1 + ηδ. However, π2 is a very poor approximation to OMCP, as fOMCPζopt(π2)
O
≥ η2 and η is a very large number.
In particular, for any η0 < η, π2 is not a ζηopt0 O
-approximation toOMCP.
In the above example, the DMCP instance is infeasible.
Similar conclusions can be drawn if we add to graph G an edge (s, t) with weights ω1(s, t) = · · · = ωK−1(s, t) = 1, and ωK(s, t) = δ (illustrated in Figure 1(b) for K = 3), where the corresponding DMCP instance is feasible. In this case, the path π3= s → tis the optimal path for bothOMCP and JMCP. ζOopt = 1, ζJopt = K − 1 + δ. The path π2 is a (1 + )-approximation to JMCP. However, fOMCPζopt(π2)
O
= fOMCP(π2) = η, which is a very large number. 2
3. A CLASS OFK-APPROXIMATIONALGORITHMS
In this section, we will present a class of approximation algorithms for OMCP(G, s, t, K, ~W , ~ω) each of which can find an s–t path π in O(Km + n log n) time such that
ωk(π) ≤ K · ζOoptWk, for 1 ≤ k ≤ K. In other words, each algorithm in this class is a K-approximation algorithm for OMCP(G, s, t, K, ~W , ~ω).
Our class of K-approximation algorithms is based on the notion of scaled p-norm (defined in the following) in K dimensional Euclidean space RK. We first define scaled p- norm in Section 3-A. We then present a very general K- approximation algorithm in Section 3-B. We discuss special cases of this K-approximation algorithm in Section 3-C.
A. Scaled p-Norm
Let p ≥ 1 be any real number. Let x = (x1, . . . , xK) be any point in RK, the K-dimensional Euclidean space. Let W~ = (W1, . . . , WK) be an ordered sequence of K positive real numbers. We define the ~W -scaled p-normof x (denoted by ||x||p, ~W) by the following equation:
||x||p, ~W =
"K X
k=1
|xk| Wk
p#
1 p
. (3.1)
By letting p → ∞ in (3.1), we obtain
||x||∞, ~W = max
1≤k≤K
|xk| Wk
. (3.2)
By letting p = 1 in (3.1), we have ||x||1, ~W =PK k=1
|xk| Wk. We call ||x||1, ~W the ~W -scaled 1-norm of x and call ||x||∞, ~W the ~W -scaled ∞-norm of x. It is a simple mathematical exercise [7], [8] that
||x||∞, ~W ≤ ||x||p, ~W ≤ ||x||1, ~W, ∀ 1 ≤ p ≤ ∞. (3.3)
B. A General K-Approximation Algorithm forOMCP In this section, we present a very general K-approximation algorithm forOMCP(G, s, t, K, ~W , ~ω). The algorithm is listed in the following as Algorithm 1. We will discuss important special cases of this algorithm in next section.
Theorem 3.1: Algorithm 1 finds an s–t path πR in O(Km + n log n)time. Moreover, πR is a K-approximation to OMCP(G, s, t, K, ~W , ~ω). In other words, ωk(πR) ≤ K · ζOoptWk, ∀ 1 ≤ k ≤ K, where ζOopt is the optimal value of OMCP(G, s, t, K, ~W , ~ω).
Algorithm 1 General-OMCP(G, s, t, K, ~W , ~ω)
1: Construct an auxiliary graph GR = (V, E, ωR) where V and E are the same as in G and ωRis a new edge weight- ing function such that for each e ∈ E, ωR(e) satisfies the following inequality: ||(ω1(e), . . . , ωK(e))||∞, ~W ≤ ωR(e) ≤ ||(ω1(e), . . . , ωK(e))||1, ~W.
2: Compute a shortest s–t path πR in GR, with respect to the edge weighting function ωR. Output πR.
PROOF. The graph GR can be constructed in O(n + Km) time, as we need to spend O(K) time to compute ωR(e) for each edge e ∈ E, leading to O(Km) time for all edges.
Using Dijkstra’s shortest path algorithm, πRcan be computed in O(m + n log n) additional time. This proves the time complexity of the algorithm.
Recall that ζOopt is the optimal value of OMCP(G, s, t, K, ~W , ~ω). Therefore there exists an s–t path πopt such that ωk(πopt) ≤ ζOoptWk, ∀ 1 ≤ k ≤ K. This implies
X
e∈πopt
ωk(e) ≤ ζOoptWk, ∀ 1 ≤ k ≤ K. (3.4) We can rewrite (3.4) as
X
e∈πopt
ωk(e) Wk
≤ ζOopt, ∀ 1 ≤ k ≤ K. (3.5) Summing (3.5) over all K possible values of k, we have
X
e∈πopt K
X
k=1
ωk(e)
Wk ≤ K · ζOopt. (3.6) It follows from the definition of ωR that
ωR(e) ≤ ||(ω1(e), . . . , ωK(e))||1, ~W =
K
X
k=1
ωk(e) Wk
, ∀ e ∈ E. (3.7) Therefore (3.6) and (3.7) imply
ωR(πopt) = X
e∈πopt
ωR(e) ≤ X
e∈πopt K
X
k=1
ωk(e) Wk
≤ K · ζOopt. (3.8) Since πR is a shortest s–t path in GR, we have (using inequality (3.8))
ωR(πR) ≤ ωR(πopt) ≤ K · ζOopt. (3.9) Using the definition of ωR(·), we have
1≤k≤Kmax P
e∈πRωk(e)
Wk = max
1≤k≤K
X
e∈πR
ωk(e) Wk
≤ X
e∈πR 1≤k≤Kmax
ωk(e)
Wk ≤ X
e∈πR
ωR(e)
= ωR(πR) ≤ K · ζOopt. (3.10) This implies
ωk(πR) = X
e∈πR
ωk(e) ≤ K · ζOoptWk, ∀ 1 ≤ k ≤ K. (3.11) In other words, πR is a K-approximation to
OMCP(G, s, t, K, ~W , ~ω). 2
C. Special Cases of the General K-Approximation Algorithm In Section 3-B, we have presented a general K- approximation algorithm for OMCP(G, s, t, K, ~W , ~ω).
That algorithm is based on computing a shortest s–
t path with respect to a new edge weighting function ωR(e), where ||(ω1(e), . . . , ωK(e))||∞, ~W ≤ ωR(e) ≤
||(ω1(e), . . . , ωK(e))||1, ~W, ∀ e ∈ E. Here we discuss some important special choices of ωR(·).
Example 3.1: As our first special case, we choose ωR(e) =
||(ω1(e), . . . , ωK(e))||∞, ~W. Since this choice of ωR(·) satis- fies the condition of Algorithm 1, we have
Corollary 3.1: A K-approximation toOMCP can be ob- tained by computing a shortest s–t path with respect to the edge weighting function ωR(e) = ||(ω1(e), . . . , ωK(e))||∞, ~W. 2This is the edge weighting function used by Xue et al.
in [31]. This example shows that Algorithm 1 contains the K-approximation algorithm forOMCPpresented in [31] as a special case.
Example 3.2: As our second special case, we choose ωR(e) = ||(ω1(e), . . . , ωK(e))||1, ~W. Since this choice of ωR(·)satisfies the condition of Algorithm 1, we have
Corollary 3.2: A K-approximation toOMCP can be ob- tained by computing a shortest s–t path with respect to the edge weighting function ωR(e) = ||(ω1(e), . . . , ωK(e))||1, ~W. 2This is the edge weighting function used by Jaffe in [10]
for K = 2 and by Andrew and Kusuma in [1] for K ≥ 2.
This is representative of the algorithms forMCPbased on the computation of a shortest path with respect to a single edge weighting function which is a linear or nonlinear combination of the K edge weights [1], [4], [10], [12], [13], [23], [27], [24], [29]. It has been shown in [1], [10] that the shortest path with respect to ||(ω1(e), . . . , ωK(e))||1, ~W is a 2-approximation to JMCP. According to Theorem 2.1, a 2-approximation to JMCP could be a very poor approximation to OMCP.
Corollary 3.2 asserts that the shortest s–t path with respect to
||(ω1(e), . . . , ωK(e))||1, ~W is a good approximation toOMCP as well. Therefore Corollary 3.2 reveals new properties of a known algorithm.
Example 3.3: As our third special case, we choose ωR(e) = ||(ω1(e), . . . , ωK(e))||p, ~W, where p ∈ (1, ∞) is a fixed real number. According to (3.3), this choice of ωR(·) satisfies the condition of Algorithm 1. Therefore we have
Corollary 3.3: Let p ∈ (1, ∞) is a fixed real number. A K-approximation toOMCPcan be obtained by computing a shortest s–t path with respect to the edge weighting function ωR(e) = ||(ω1(e), . . . , ωK(e))||p, ~W. 2 Corollary 3.3 reveals a class of K-approximation algorithms for the OMCP problem that we are not aware of before.
Together with Corollaries 3.1 and 3.2, it says that a K- approximation to the OMCP problem can be obtained by computing a shortest s–t path with respect to the scaled p-norm of the K edge weights, for any given value of p in the interval [1, ∞]. We want to emphasize that it is not our goal to include as many algorithms as possible. It is our goal to reveal some intrinsic properties of the OMCP
problem. We have proved that every scaled p-norm leads to a simple K-approximation algorithm for the OMCP problem.
It is interesting to find out which value of p leads to the best algorithm in this class, in terms of theoretical guarantees and practical implementations. This forms an important future research topic.
4. CONCLUSIONS
In this paper, we have studied the multi-constrained QoS routing problem where each edge has K ≥ 2 additive QoS parameters. We studied the relationship between two well- known optimization criteria used in the literature. We then presented a class of K-approximation algorithms that are based on a shortest path with respect to a single edge weighting function. We have demonstrated that our class of algorithms contains some well-known algorithms as special cases. Our results provide a strong theoretical foundation for fast heuristic algorithms for theMCPproblem based on the computation of a shortest path with respect to a single edge weighting function which is a linear combination of the K edge weights.
ACKNOWLEDGMENT
The authors wish to thank the associate editor and three anonymous reviewers whose comments on an earlier version of this paper have helped to significantly improve the presentation of this paper.
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