Multi-Constrained QoS Routing: A Norm Approach

Transcription

1 1 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. ne popular approach is to use the shortest path with respect to a single edge weighting function as an approximate solution to MCP. In a pioneering work, affe 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 to MCP. 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. INTRDUCTIN 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 xue@asu.edu. The research of this author was supported in part by AR grant W911NF and NSF grants CCF and ANI S. Kami Makki is an Assistant Professor with the Department of Electrical Engineering & Computer Science at The University of Toledo, Toledo, H 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 ffice 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 the MCP problem (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 ɛ > 0, a path that is within a factor of (1 + ɛ) of the optimal solution, with a running time that is bounded by a polynomial in the input size of the given instance, with ɛ 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. affe in [10] studied the MCP problem with K = 2 and 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 ɛ > 0, this heuristic algorithm has time complexity (mn( n ɛ )K 1 ) and can find a feasible path if there is a path whose last K 1 path weights are no more than (1 ɛ) 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

2 2 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 weighting 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 (mn(log log n + 1 ɛ )), where ɛ 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 ((m+n log n) H ɛ ), 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 (m( n ɛ )). rda and Sprintson in [19] studied disjoint QoS paths. Xue et al. in [31] studied the MCP 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 ((m( n ɛ )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 optimization 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 approximation algorithms for the MCP problem. 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. ur results provide a theoretical support of many heuristic algorithms for the MCP problem 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 affe [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 PRBLEM DEFINITINS AND CMPARISN F PTIMIZATIN CRITERIA 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 algorithm for 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 = (ω 1,..., ω K ) is an edge weight vector so that ω k (e) 0 is the k th weight of edge e, e E, 1 k K. For a path π in G, the k th weight of π, denoted by ω k (π), is the sum of the k th weights over the edges on π: ω k (π) = e π ω k(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 = (W 1,..., W K ) to denote a constraint vector where each is a positive constant. For a path π connecting s to t, we are interested in the following two path metrics. f AFFE (π) = max{ω k (π), }. (2.1) f MCP (π) = max 1 k K ω k (π). (2.2) f AFFE (π) measures the sum (over k {1,..., K}) of the larger of the kth path weight ω k (π) and the kth constraint. This path metric was introduced by affe [10] and has also been used in [1]. f MCP (π) measures the maximum (over k {1,..., K}) of the ratio of the kth path weight ω k (π) over the kth constraint. 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 ω k (e), 1 k K, associated with each edge e E; a constraint vector W = (W 1,..., W K ) where each is a positive constant; and a source-destination node pair (s, t). QUESTIN: is there an s t path π such that ω k (π), 1 k K? In the above definition, the inequality ω k (π) is called the k th 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 f MCP (π) 1. We say that DMCP(G, s, t, K, W, ω) is feasible if 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 to MCP and MCP defined in the sequel. The DMCP problem is known to be NP-complete [5], [25], for any integer K such that K 2. Therefore the following

3 3 optimization version of this problem has been studied in the literature [30], [31]. Definition 2.2 (MCP(G, s, t, K, W, ω)): INSTANCE: a K-edge weighted directed graph G = (V, E, ω), with K nonnegative real-valued edge weights ω k (e), 1 k K, associated with each edge e E; a constraint vector W = (W 1,..., W K ) where each is a positive constant; and a source-destination node pair (s, t). PRBLEM: find an s t path π such that f MCP (π ) f MCP (π) for any s t path π. We call π an optimal path or an optimal solution of MCP(G, s, t, K, W, ω), and call f MCP (π ) the optimal value of MCP(G, s, t, K, W, ω). It is clear that 1 if and only if DMCP(G, s, t, K, W, ω) is feasible. In [10], affe studied the following optimization version of the multi-constrained path problem. Definition 2.3 (MCP(G, s, t, K, W, ω)): INSTANCE: a K-edge weighted directed graph G = (V, E, ω), with K nonnegative real-valued edge weights ω k (e), 1 k K, associated with each edge e E; a constraint vector W = (W 1,..., W K ) where each is a positive constant; and a source-destination node pair (s, t). PRBLEM: find an s t path π such that f AFFE (π ) f AFFE (π) for any s t path π. We call π an optimal path or an optimal solution of MCP(G, s, t, K, W, ω), and call f AFFE (π ) the optimal value of MCP(G, s, t, K, W, ω). Clearly, = K when DMCP is feasible and > K otherwise. affe proposed to design efficient approximation algorithms for MCP, based on the shortest path with respect to a single edge weighting function. Indeed, affe 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 α 1,..., α K, the shortest s t path π α with respect to the edge weighting function guarantees that ω α (e) = α k ωk(e), e E (2.3) f AFFE (π α ) (2 1 K ) ζopt. (2.4) In addition, with α1 = = αk = 1, the s t path π α with respect to ω α guarantees that f AFFE (π α ) 2. (2.5) In other words, one can obtain a 2-approximation to the MCP 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 MCP problem has been studied by many researchers [2], [16], [22], [23], [27], [28], [29], [30], [31]. In a recent paper, Xue et al. [31] studied the MCP 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. ω max (e) = ω k (e) max, e E. (2.6) 1 k K We call the edge weighting function ω α (e) the scaled 1- norm of the K edge weighting functions ω 1 (e),..., ω K (e), and call the edge weighting function ω max (e) the scaled - norm of the K edge weighting functions ω 1 (e),..., ω K (e), formally defined in the next section. n one hand, we know that the shortest path with respect to the scaled 1-norm is a 2-approximation to the MCP problem. n the other hand, we know that the shortest path with respect to the scaled - norm is a K-approximation to the MCP problem. A natural question to ask is the following. Should we strive to compute a path π which is a good approximation to MCP or should we strive to compute a path π which is a good approximation to MCP? The following theorem shows that a good approximation to MCP is also a good approximation to MCP, but a good approximation to MCP could be a very poor approximation to MCP. Theorem 2.1: Let β 1 be any given constant. 1) If s t path π is a β-approximation to MCP(G, s, t, K, W, ω), then π is also a max{1, β }-approximation to MCP(G, s, t, K, W, ω), where is the optimal value of MCP(G, s, t, K, W, ω). 2) Let η > 0 be an arbitrarily large constant and ɛ > 0 be an arbitrarily small constant. There exist a K- edge weighted graph G(V, E, ω 1,..., ω K ), QoS constraints W = (W 1,..., W K ), source-destination node pair (s, t), and an s t path π such that π is a (1 + ɛ) approximation to MCP(G, s, t, K, W, ω), but is not a η -approximation to MCP(G, s, t, K, W, ω) for any η < η, where [1, 2] is the optimal value of the corresponding MCP(G, s, t, K, W, ω) instance. PRF. Let π be a β-approximation to MCP. We have ω k (π) β, 1 k K. Therefore max{ω k (π), } max{β, 1}, 1 k K. (2.7) As a result, we have f AFFE (π) = KX max{ω k (π), } max{β K, 1} X. (2.8) It follows from the definition of f AFFE ( ) that for any s t path π we must have f AFFE (π) = max{ω k (π), }. (2.9) Inequalities (2.8) and (2.9) together imply that π is a max{1, β }-approximation to MCP. 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 ω k so that ω 1 (s, x) = = ω K 1 (s, x) = 1 + ɛ,

4 + 4 s x t s & %&'() '()* x.0/213/215476! "#$ # t (0, 0, 0) y (0, 0, 0) y %& & ) ), - + (a) DMCP is infeasible (b) DMCP is feasible Fig. 1. A (1 + ɛ) approximation to MCP could be a very poor approximation to MCP. ω K (s, x) = 0; ω 1 (x, t) = = ω K 1 (x, t) = 0, ω K (x, t) = (1 + ɛ)δ; ω 1 (s, y) = = ω K (s, y) = 0; ω 1 (y, t) = = ω K 1 (y, t) = 1, ω K (y, t) = ηδ. We also assume (W 1,..., W K 1, W K ) = (1,..., 1, δ). The parameters η, ɛ and δ are chosen in the following way: η > 0 is a given arbitrarily large constant (compared with K and ɛ) and ɛ (0, 1] is an arbitrarily small constant such that η is much larger than 1+ɛ. After η and ɛ are fixed, we choose δ > 0 sufficiently small so that (1+ɛ)(K 1+δ) > K 1+ηδ. This is possible because (1 + ɛ)(k 1 + δ) lim = 1 + ɛ > 1. (2.10) δ 0 K 1 + ηδ There are exactly two s t paths in G: π 1 = s x t and π 2 = s y t. The path weights of π 1 are ω 1 (π 1 ) = = ω K 1 (π 1 ) = 1 + ɛ, ω K (π 1 ) = (1 + ɛ)δ. The path weights of π 2 are ω 1 (π 2 ) = = ω K 1 (π 2 ) = 1, ω K (π 2 ) = ηδ. Therefore we have f MCP (π 1 ) = 1 + ɛ, f AFFE (π 1 ) = (1 + ɛ)(k 1 + δ), f MCP (π 2 ) = η; f AFFE (π 2 ) = K 1 + ηδ. By our choice of η, ɛ and δ, π 1 is the optimal solution to MCP, and π 2 is the optimal solution to MCP. = 1+ɛ, = K 1+ηδ. However, π 2 is a very poor approximation to MCP, as f MCP (π 2) η 2 and η is a very large number. In particular, for any η < η, π 2 is not a η -approximation to MCP. 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 ω 1 (s, t) = = ω K 1 (s, t) = 1, and ω K (s, t) = δ (illustrated in Figure 1(b) for K = 3), where the corresponding DMCP instance is feasible. In this case, the path π 3 = s t is the optimal path for both MCP and MCP. = 1, ζopt = K 1 + δ. The path π 2 is a (1 + ɛ)-approximation to MCP. However, = f MCP (π 2 ) = η, which is a very large number. f MCP (π 2) 3. A CLASS F K-APPRXIMATIN ALGRITHMS In this section, we will present a class of approximation algorithms for MCP(G, s, t, K, W, ω) each of which can find an s t path π in (Km + n log n) time such that ω k (π) K, for 1 k K. In other words, each algorithm in this class is a K-approximation algorithm for MCP(G, s, t, K, W, ω). ur class of K-approximation algorithms is based on the notion of scaled p-norm (defined in the following) in K dimensional Euclidean space R K. 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 = (x 1,..., x K ) be any point in R K, the K-dimensional Euclidean space. Let W = (W 1,..., W K ) be an ordered sequence of K positive real numbers. We define the W -scaled p-norm of x (denoted by x p, W ) by the following equation: [ K ( ) ] 1 p p xk x p, W =. (3.1) By letting p in (3.1), we obtain x, W = x k max. (3.2) 1 k K By letting p = 1 in (3.1), we have x 1, W = K x k. 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 for MCP In this section, we present a very general K-approximation algorithm for MCP(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 π R in (Km + n log n) time. Moreover, π R is a K-approximation to MCP(G, s, t, K, W, ω). In other words, ω k (π R ) K, 1 k K, where is the optimal value of MCP(G, s, t, K, W, ω).

5 5 Algorithm 1 General-MCP(G, s, t, K, W, ω) 1: Construct an auxiliary graph G R = (V, E, ω R ) where V and E are the same as in G and ω R is a new edge weighting function such that for each e E, ω R (e) satisfies the following inequality: (ω 1 (e),..., ω K (e)), W ω R (e) (ω 1 (e),..., ω K (e)) 1, W. 2: Compute a shortest s t path π R in G R, with respect to the edge weighting function ω R. utput π R. PRF. The graph G R can be constructed in (n + Km) time, as we need to spend (K) time to compute ω R (e) for each edge e E, leading to (Km) time for all edges. Using Dijkstra s shortest path algorithm, π R can be computed in (m + n log n) additional time. This proves the time complexity of the algorithm. Recall that is the optimal value of MCP(G, s, t, K, W, ω). Therefore there exists an s t path π opt such that ω k (π opt ), 1 k K. This implies e π opt ω k (e), 1 k K. (3.4) We can rewrite (3.4) as ω k (e) e π opt e π opt, 1 k K. (3.5) Summing (3.5) over all K possible values of k, we have ω k (e) K W. (3.6) k It follows from the definition of ω R that KX ω R ω k (e) (e) (ω 1(e),..., ω K(e)) 1, W =, e E. (3.7) Therefore (3.6) and (3.7) imply ω R (π opt ) = X e π opt ω R (e) X KX e π opt ω k (e) K. (3.8) Since π R is a shortest s t path in G R, we have (using inequality (3.8)) ω R (π R ) ω R (π opt ) K. (3.9) Using the definition of ω R ( ), we have e π max ω k(e) R = max 1 k K 1 k K ω k (e) ω k (e) max ω R (e) 1 k K = ω R (π R ) K. (3.10) This implies ω k (π R ) = ω k (e) K, 1 k K. (3.11) In other words, π R is a K-approximation to MCP(G, s, t, K, W, ω). C. Special Cases of the General K-Approximation Algorithm In Section 3-B, we have presented a general K- approximation algorithm for MCP(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 ω R (e), where (ω 1 (e),..., ω K (e)), W ω R (e) (ω 1 (e),..., ω K (e)) 1, W, e E. Here we discuss some important special choices of ω R ( ). Example 3.1: As our first special case, we choose ω R (e) = (ω 1 (e),..., ω K (e)), W. Since this choice of ω R ( ) satisfies the condition of Algorithm 1, we have Corollary 3.1: A K-approximation to MCP can be obtained by computing a shortest s t path with respect to the edge weighting function ω R (e) = (ω 1 (e),..., ω K (e)), W. This is the edge weighting function used by Xue et al. in [31]. This example shows that Algorithm 1 contains the K-approximation algorithm for MCP presented in [31] as a special case. Example 3.2: As our second special case, we choose ω R (e) = (ω 1 (e),..., ω K (e)) 1, W. Since this choice of ω R ( ) satisfies the condition of Algorithm 1, we have Corollary 3.2: A K-approximation to MCP can be obtained by computing a shortest s t path with respect to the edge weighting function ω R (e) = (ω 1 (e),..., ω K (e)) 1, W. This is the edge weighting function used by affe in [10] for K = 2 and by Andrew and Kusuma in [1] for K 2. This is representative of the algorithms for MCP based 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 (ω 1 (e),..., ω K (e)) 1, W is a 2-approximation to MCP. According to Theorem 2.1, a 2-approximation to MCP could be a very poor approximation to MCP. Corollary 3.2 asserts that the shortest s t path with respect to (ω 1 (e),..., ω K (e)) 1, W is a good approximation to MCP as well. Therefore Corollary 3.2 reveals new properties of a known algorithm. Example 3.3: As our third special case, we choose ω R (e) = (ω 1 (e),..., ω K (e)) p, W, where p (1, ) is a fixed real number. According to (3.3), this choice of ω R ( ) satisfies the condition of Algorithm 1. Therefore we have Corollary 3.3: Let p (1, ) is a fixed real number. A K-approximation to MCP can be obtained by computing a shortest s t path with respect to the edge weighting function ω R (e) = (ω 1 (e),..., ω K (e)) p, W. Corollary 3.3 reveals a class of K-approximation algorithms for the MCP problem that we are not aware of before. Together with Corollaries 3.1 and 3.2, it says that a K- approximation to the MCP 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 MCP

6 6 problem. We have proved that every scaled p-norm leads to a simple K-approximation algorithm for the MCP 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. CNCLUSINS 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 wellknown 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. ur results provide a strong theoretical foundation for fast heuristic algorithms for the MCP problem 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. ACKNWLEDGMENT 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. REFERENCES [1] L.L.H. Andrew and A.A.N.A. Kusuma; Generalised analysis of a QoSaware routing algorithm; IEEE Globecom 1998; pp [2] S. Chen and K. Nahrstedt; n finding multi-constrained paths; IEEE ICC 1998; pp [3] F. Ergun, R. Sinha and L. Zhang; An improved FPTAS for restricted shortest path; Information Processing Letters; Vol. 83(2002), pp [4] G. Feng, C. Douligeris, K. Makki, N. 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