J Comb OptimDOI 10.1007/s10878-013-9693-x

Improved approximation algorithms for computingk disjoint paths subject to two constraints

Longkun Guo · Hong Shen · Kewen Liao

© Springer Science+Business Media New York 2013

Abstract For a given graph G with distinct vertices s and t , nonnegative integral costand delay on edges, and positive integral bound C and D on cost and delay respectively,the k bi-constraint path (kBCP) problem is to compute k disjoint st-paths subject to Cand D. This problem is known to be NP-hard, even when k = 1 (Garey and Johnson,Computers and Intractability, 1979 ). This paper first gives a simple approximationalgorithm with factor-(2, 2), i.e. the algorithm computes a solution with delay andcost bounded by 2 ∗ D and 2 ∗ C respectively. Later, a novel improved approxima-tion algorithm with ratio (1 + β, max{2, 1 + ln(1/β)}) is developed by constructinginteresting auxiliary graphs and employing the cycle cancellation method. As a conse-quence, we can obtain a factor-(1.369, 2) approximation algorithm immediately anda factor-(1.567, 1.567) algorithm by slightly modifying the algorithm. Besides, whenβ = 0, the algorithm is shown to be with ratio (1, O(ln n)), i.e. it is an algorithmwith only a single factor ratio O(ln n) on cost. To the best of our knowledge, this isthe first non-trivial approximation algorithm that strictly obeys the delay constraintfor the kBCP problem.

Keywords k-disjoint bi-constraint path · NP-hard · Bifactor approximationalgorithm · Auxiliary graph · Cycle cancellation

This research was partially supported by Natural Science Foundation of China under its Youth funding#61300025, Natural Science Foundation of Fujian Province under its Youth funding #2012J05115,Doctoral Funds of Ministry of Education of China for Young Scholars #20123514120013 and FuzhouUniversity Development Fund (2012-XQ-26).

L. Guo (B)School of Mathematics and Computer Science, Fuzhou University, Fuzhou, Chinae-mail: lkguo@fzu.edu.cn

H. Shen · K. LiaoSchool of Computer Science, University of Adelaide, Adelaide, SA, Australia

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1 Introduction

In real networks, there are many applications that require quality of service (QoS) andsome degree of robustness simultaneously. Typically, the QoS related problems requirerouting between the source node and the destination node to satisfy several constraintssimultaneously, such as bandwidth, delay, cost and energy consumption. Nevertheless,in networks, some time-critical applications also require routing to remain functioningwhile edge or vertex failures occur. A common solution is to compute k disjoint pathsthat satisfy the QoS constraints, and then to use some paths as active paths whilstthe other paths as backup paths. The routing traffic is carried on the active paths, andswitched to the disjoint backup paths while edge or vertex failures occur on the activepaths. However, for some time-critical applications even the time to discover failuresof routing and restore data transmission in backup paths is too long for them. Forsuch applications, packages are routed via k paths simultaneously, and the traffic isswitched from failed paths to functioning paths if edge or vertex failures occur, suchthat routing can tolerate k − 1 edge (vertex) failures. Therefore, given cost and delayas the QoS constraints, the disjoint QoS Path problem arises as below:

Definition 1 For a graph G = (V, E) and a pair of distinct vertices s, t ∈ V , a costfunction c : E → Z

+0 , a delay function d : E → Z

+0 , a cost bound C ∈ Z

+ and a delaybound D ∈ Z

+, the k-disjoint QoS Paths problem is to compute k disjoint st-pathsP1, . . . , Pk , such that

∑i=1,...,k c(Pi ) ≤ C and d(Pi ) ≤ D for every i = 1, . . . , k.

This problem is N P-hard even when all edges of G are with cost 0 (Li et al. 1989). Soit is impossible to approximate the k-disjoint QoS Paths problem within a non-trivialsingle factor ratio unless P = N P . An alternative method is to compute k disjointpaths with total cost bounded by C and delay bounded by D (equal to k D when D is asin Definition 1), and then route the packages via the paths according to their urgencypriority, i.e., route urgent packages via paths of low delay whilst other packages viaother paths of higher delay of the k disjoint paths. Therefore, the disjoint bi-constraintpath problem arises as in the following:

Definition 2 (The k disjoint bi-constraint path problem, kBCP) For a graph G =(V, E) with a pair of distinct vertices s, t ∈ V , a cost function c : E → Z

+0 , a delay

function d : E → Z+0 , a cost bound C ∈ Z

+ and a delay bound D ∈ Z+, the k-

disjoint bi-constraint path problem is to calculate k disjoint st-paths P1, . . . , Pk , suchthat

∑i=1,...,k c(Pi ) ≤ C and

∑i=1,...,k d(Pi ) ≤ D.

This paper will focus on the kBCP problem’s bifactor approximation algorithms,whose definition is roughly as in the following:

Definition 3 An algorithm A is a bifactor (α, β)-approximation for the kBCP prob-lem, if and only if for every instance of kBCP A computes k disjoint st-paths, whosetotal delay and total cost are bounded by α ∗ D and β ∗ C respectively.

Since a β-approximation with the single factor ratio on cost is identical to a bifactor(1, β)-approximation, we use them interchangeably in the text.

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1.1 Related work

This kBCP problem is NP-hard even when k = 1 (Garey and Johnson 1979). To thebest of our knowledge, this paper presents the first non-trivial approximation algorithmfor the kBCP problem. However, a number of papers have addressed problems closelyrelated to kBCP, in particular the k restricted shortest path problem (kRSP), whichis to calculate k disjoint st-paths of minimum cost-sum under the delay constraint∑

i=1,...,k d(Pi ) ≤ D. An algorithm with bifactor approximation ratio (2, 2) has beendeveloped in Guo and Shen (2012) for general k, while no approximation solution thatstrictly obeys the delay (or cost) constraint is known even when k = 2. Bifactor ratioof (1+ 1

r , r(1+ 2(log r+1)r )(1+ε)) and (1+ 1

r , 1+r) have been achieved respectivelyin Chao and Hong (2007), Orda and Sprintson (2004) for the case k = 2, under theassumption that the delay of each path in the optimal solution of kRSP is bounded by D

k .Special cases of the kBCP problem have been well studied. When the delay con-

straint is removed, this problem is reduced to the min-sum problem, which is to calcu-late k disjoint paths with the total cost minimized. This problem is known polynomiallysolvable (Suurballe 1974). Moreover, when k = 1, the problem reduces to the singlebi-constraint path (BCP) problem, which is known as the basic QoS routing problem(Garey and Johnson 1979) and admits full polynomial time approximation scheme(FPTAS) (Garey and Johnson 1979; Lorenz and Raz 2001). Recently, the MCP prob-lem (the Multi-Constraint Path problem, including the single BCP path problem as aspecial case) is still attracting considerable interests of the researchers. The strongestresult known is a (1 + ε)-approximation due to Xue et al. (2008).

Additionally, special cases of the disjoint QoS path problem have also been inves-tigated. When the cost constraint is removed, the disjoint QoS problem reduces to thelength bounded disjoint path problem of finding two disjoint paths with the length ofeach path constrained by a given bound. This problem is a variant of the min-Maxproblem of finding two disjoint paths with the length of the longer path minimized.Both of the two problems are known to be NP-complete, and with a best possibleapproximation ratio of 2 in digraphs Li et al. (1989), which can be achieved by apply-ing the algorithm for the min-sum problem in Suurballe (1974), Suurballe and Tarjan(1984). Contrastingly, the min-min problem of finding two paths with the length ofthe shorter path minimized is NP-complete and does not admit K -approximation forany K ≥ 1 (Bhatia et al. 2006; Guo and Shen 2013; Xu et al. 2006). The problemremains NP-complete and admits no polynomial time approximation scheme even inplanar digraphs (Guo and Shen 2012).

1.2 Our technique and results

The main result of this paper is a factor-(1 + β, max{2, 1 + ln 1β}) approximation

algorithm for any 0 < β ≤ 1 for the kBCP problem. The main idea of the algorithmis firstly to compute k-disjoint paths, such that there exists a real number 0 ≤ α ≤ 2that the total delay of the computed paths is bounded by αD and total cost boundedby (2 − α) ∗ C . Secondly, the key idea is to improve the computed k paths by novellycombining cycle cancellation method (Orda and Sprintson 2004) and auxiliary layergraph technique (Xue et al. 2008). The improving phase (i.e. the second step) would

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decrease the delay of the computed k paths at the price of cost increasing in the case thatthe delay of the k paths is large. The approximation ratio of our algorithm is shown to be(1+β, max{2, 1+ln 1

β}) after rebalancing the delay and cost of the k computed paths.

As a consequence of the main result, we obtain a factor-(1.369, 2) approximationalgorithm by setting 1 + ln 1

β= 2, and a factor-(1.567, 1.567) algorithm by setting

1 + β = 1 + ln 1β

and slightly modifying our algorithm (to improve either cost ordelay that is with worse ratio). Nevertheless, by slightly modifying our ratio proof, weshow that when β = 0 the approximation algorithm is with ratio (1, O(ln n)), i.e. it iswithin a single factor ratio of O(ln n) on cost. To the best of our knowledge, this is thefirst non-trivial approximation algorithm, which strictly obeys the delay constraint,for the kBCP problem.

We note that our algorithms are with pseudo-polynomial time complexity, sincethe auxiliary graph we construct is of size O(C ∗ n). However, by using the classicpolynomial time approximation scheme design technique (Garey and Johnson 1979),

i.e. for any small ε > 0 setting the cost of every edge to

⌊c(e)εCn

⌋

in G before the con-

struction of auxiliary graph, we can immediately obtain a polynomial time algorithmwith ratio ((1 + β) ∗ (1 + ε), max{2, (1 + ln 1

β) ∗ (1 + ε)}).

2 An improved approximation algorithm for computing k disjointbi-constraint paths

This section will firstly present a simple approximation method for computing k-disjoint paths. We show that there exists a real number 0 ≤ α ≤ 2, such that thek computed paths are with delay-sum bounded by αD and cost-sum bounded by(2−α)∗C . Later, we improve the k paths by balancing the cost and delay according tothe value of α, via constructing auxiliary layer graphs and using the cycle cancellationmethod therein. Though the presented simple algorithm itself is not with better ratiothan that of the algorithm for k = 2 in Orda and Sprintson (2004), it suits the improvingphase better.

2.1 A basic approximation algorithm

Observing that the difficulty of computing k-disjoint bi-constraint paths mainly comesfrom the two given constraints, the key idea of our algorithm is to deal with one newconstraint B instead of the two given constraints C and D. Our algorithm firstly assignsa new mixed cost b(e) = c(e)

C + d(e)D to every edge in graph, and secondly computes

k disjoint paths with the new cost sum minimized. Note that the second step can beaccomplished in polynomial time by employing the SPP algorithm due to Suurballe(1974), Suurballe and Tarjan (1984). The detailed algorithm is as in Algorithm 1.

The time complexity and performance guarantee of Algorithm 1 is given by thefollowing theorem:

Theorem 4 Algorithm 1 runs in O(km log1+ mn

n) time. There exists a real number0 ≤ α ≤ 2, such that the k-disjoint paths computed in Algorithm 1 are with delay-sumbounded by αD and cost-sum bounded by (2 − α) ∗ C.

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Algorithm 1 A basic approximation algorithm for the k-BCP problemInput: A graph G = (V, E), each edge e with cost c(e) and delay d(e), a given cost constraint C ∈ Z

+and delay constraint D ∈ Z

+;Output: k disjoint paths P1, P2 . . . , Pk .

1. Set the new cost of edge e as b(e) = c(e)C + d(e)

D ;2. Compute the k disjoint paths P1, P2 . . . , Pk in G by using Suurballe and Tarjan’s algorithm (Suurballe

1974; Suurballe and Tarjan 1984), such that∑k

i=1∑

e∈Pib(e) is minimized;

3. Return P1, P2 . . . , Pk .

Proof The main part of Algorithm 1 takes O(km log1+ mn

n) time to compute k-disjointpaths by using Surrballe and Tarjan’s algorithm (Suurballe 1974; Suurballe and Tarjan1984), and other parts of the algorithm take trivial time. Hence the time complexityof the algorithm is O(km log1+ m

nn).

It remains to show the approximation ratio. To make the proof concise, we denoteby O PT an optimal solution for the k-disjoint BCP paths problem, and SO L the

solution of Algorithm 1. Obviously∑

e∈O PT b(e) =∑

e∈O PT c(e)C +

∑e∈O PT d(e)

D ≤ 2holds. Then since the new cost of k disjoint paths attains minimum, we have

∑

e∈SO L

b(e) ≤∑

e∈O PT

b(e) ≤ 2. (1)

From Inequality (1) and the definition of cost b(e), there exists a real number 0 ≤α ≤ 2, such that the delay-sum of the algorithm is α times of d(O PT ). Then we have∑

e∈SO L b(e) = ∑ki=1

∑e∈Pi

b(e) = α+ c(SO L)c(O PT )

. From Inequality (1), α+ c(SO L)c(O PT )

≤2 holds. That is, c(SO L) ≤ (2 − α)c(O PT ) ≤ (2 − α)C . This completes the proof.

Note that α differs for different instances, i.e. Algorithm 1 may return a solution withcost 2 ∗ c(O PT ) and delay 0 for some instances, while a solution with cost 0 anddelay 2 ∗ d(O PT ) for other instances. Hence, the bifactor approximation ratio forAlgorithm 1 is actually (2, 2).

In real networks, the two given constraints may not be of equal importance, say,delay is far more important comparing to cost. In this case, applications require thatthe delay of the resulting solution is bounded by (1+β)D, where 0 < β < 1 is a givenpositive real number. Apparently, we could get an algorithm similar to Algorithm 1excepting setting the new cost as b(e) = c(e)

C + βd(e)

D . The ratio of the new algorithmis given as below:

Corollary 5 For any given real number 0 < β < 1, there exists a real number0 ≤ α ≤ 1 + β, such that by setting the new cost as b(e) = β

c(e)C + d(e)

D , Algorithm 1

returns k paths with delay-sum bounded by αD and cost-sum bounded by 1+β−αβ

∗ C.

That is, in the worst case, the ratio of Algorithm 1 is (1 + β, 1 + 1β).

The proof of Corollary 5 is omitted here, since it is similar to the proof of Theorem1. According to Corollary 5, our algorithm can bound the delay-sum of the k-disjointpath by (1 + β)D for any 0 < β < 1, by relaxing the cost constraint to (1 + 1

β) ∗ C .

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For example, if β = 0.01, then the bifactor approximation ratio of the algorithm is(1.01, 101). Thus, the algorithm decrease the delay of the k-disjoint paths at a highprice. In the next subsection, we shall develop an improved method that pays less tomake delay-sum of the k-disjoint paths bounded by (1 + β)D.

2.2 The improving phase

To make the delay of the solution resulting from Algorithm 1 bounded by (1+β)D, ourimproving phase is, basically a greedy method, using the so-called cycle cancellationto improve the disjoint paths in iterations until a solution with the best possible ratio(1+β, max{2, 1+ ln 1

β}) is obtained. The cycle cancellation method is an approach of

using cycles to change the edges of the disjoint paths, which first appears in Orda andSprintson (2004) and is derived from the following proposition that can be immediatelyobtained from flow theory (Ahuja et al. 1993):

Proposition 6 Let P1, P2 . . . , Pk and Q1, Q2 . . . , Qk be two sets of k disjoint st-paths in digraph G, G be G excepting that all edges of P1, P2 . . . , Pk are reversed,and O be a cycle in G. Then

1. The edges of P1, P2 . . . , Pk and O, excepting the pairs of parallel edges withopposite direction, contain k-disjoint paths;

2. There exist a set of edge disjoint cycles O1, . . . , Oh in G, such that the edgesof P1, P2 . . . , Pk and O1, . . . , Oh, excepting the pairs of parallel edges withopposite direction, contain Q1, Q2 . . . , Qk.

From Clause 2 of the proposition above, it is obvious that there exists a set of cyclesO1, . . . , Oh that can improve the k computed disjoint QoS paths P1, P2 . . . , Pk toan optimal solution. However, it is hard to compute the cycles O1, . . . , Oh , so weemploy a greedy approach to compute a set of cycles to obtain an approximationapproach. The improving phase is composed by iterations, each of which computesa cycle and then uses it to improve P1, P2 . . . , Pk . More precisely, to obtain a good

ratio, the algorithm computes in iteration j a cycle O j withd(O j )

c(O j )minimized among

the cycles in G. The layout of the algorithm is given as in Algorithm 2.Following clause 1 of Proposition 6, Algorithm 2 will correctly return k disjoint

paths. It remains to show the cost and delay of the k disjoint paths are constrained asbelow:

Theorem 7 The approximation ratio of Algorithm 2 is (1 + β, max{2, 1 + ln 1β}).

Proof If∑k

i=1 d(Pi ) ≤ (1 + β)D before the execution of Algorithm 2, the approx-imation ratio is obviously (1 + β, 2). So we need only to show that the ratio of thealgorithm is (1+β, 1+ln 1

β) for the case

∑ki=1 d(Pi ) > (1+β)D before the execution

of Algorithm 2. Assume that Algorithm 2 runs in h iterations. Note that Algorithm 2terminates when d(SO L)+∑h

i=1 d(Oi ) ≤ (1+β)D. So for the former h−1 iterations(i.e. all iterations but excluding the last one), d(SO L) + ∑h−1

i=1 d(Oi ) > (1 + β)Dholds. That is,

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Algorithm 2 An improved algorithm based on cycle cancellation.Input: A graph G = (V, E) with nonnegative integral cost c(e) and delay d(e) on each edge e, a givencost constraint C ∈ Z

+ and delay constraint D ∈ Z+, disjoint QoS paths P1, P2 . . . , Pk computed by

Algorithm 1;Output: Improved disjoint QoS paths Q1, Q2 . . . , Qk .

1. If∑k

i=1 d(Pi ) ≤ (1 + β)D ;then return P1, P2 . . . , Pk as Q1, Q2 . . . , Qk , terminate;

2. Reverse direction of the edges of P1, P2 . . . , Pk in G, set their cost to a small positive real number0 <

ε < 1mnD , and negative their delay;

3. Compute cycle O j with c(Oi ) ≤ C , d(O j ) < 0 andd(O j )

c(O j )attaining minimum, by the method given

in next section;/* Following clause 2 of Proposition 6, if

∑ki=1 d(Pi ) ≥ d(O PT ) and

∑ki=1 c(Pi ) ≥ 0, there always

exist cycle O j with c(Oi ) ≤ C and d(O j ) < 0. */4. Improve P1, P2 . . . , Pk by adding the edges of O j and removing the pairs of parallel edges in opposite

direction;5. Go to Step 1.

d(SO L) − D +h−1∑

i=1

d(Oi ) > βD. (2)

Based on the inequality above, the key idea of our proof is to sum up the cost incrementof the former h−1 iterations, and then show that the sum has a promising upper bound.Let �D = d(O PT )−d(SO L), �C = c(O PT )− c(SO L), and the cycle computed

in the j th iteration be O j . Then sinced(O j )

c(O j )attains minimum according to Step 3 of

Algorithm 2, we haved(O j )

c(O j )≤ �D−∑ j−1

i=1 d(Oi )

C . That is,

c(O j ) ≤ d(O j )

�D − ∑ j−1i=1 d(Oi )

C.

By summing up c(O j ) in all the former h − 1 iterations, we have:

h−1∑

j=1

c(O j ) ≤ Ch−1∑

j=1

d(O j )

�D − ∑ j−1i=1 d(Oi )

. (3)

It remains to show that the coefficient of C in the right part of Inequality (3) is bounded.Following the definition of Definite Integral, we have:

h−1∑

j=1

d(O j )

�D − ∑ j−1i=1 d(Oi )

=h−1∑

j=1

1

�D − ∑ j−1i=1 d(Oi )

d(O j ),

≤�D−∑h−1

i=1 d(Oi )∫

�D

1

xdx, (4)

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where the maximum is attained when d(O j ) = −1 for every j . Now we are to givea promising upper bound for the Definite Integral above. According to Inequality (2),−�D + ∑h−1

i=1 d(Oi ) > βD > 0 holds. Therefore, the total cost increment for theformer h − 1 iterations is bounded as below:

�D−∑h−1i=1 d(Oi )∫

�D

1

xdx =

−�D∫

−�D+∑h−1i=1 d(Oi ))

1

xdx ≤

−�D∫

βD

1

xdx . (5)

Combining Inequality (3), (4) and (5), we have∑h−1

j=1 c(O j ) ≤ ∫ −�DβD

1x dx =

ln |�D|βD = ln α−1

βfor the former h − 1 iterations. For the last iteration, the cost

increment is apparently bounded by c(O PT ) according to the definition of the cycleO j in Step 3 of Algorithm 2. So the total cost of the resulting solution of Algorithm2 is c(SO L ′) ≤ (2 − α)C + C ln α−1

β+ C = C(3 − α + ln α−1

β), where SO L ′ is the

solution resulting from Algorithm 2.Let f (α) = 3 − α + ln α−1

β. Then the maximum of f (α) is the ratio on cost for

the algorithm. Remind that α ≤ 2, so f ′(α) = 1α−1 − 1 > 0, f (α) is monotonous

increasing on α, and attains maximum while α = 2. So we have c(SO L ′) ≤ (1 +ln 1

β)c(O PT ).

Therefore, the cost of the output of Algorithm 2 is bounded by (1 + ln 1β)c(O PT ),

and delay bounded by (1 + β)d(O PT ). This completes the proof.

From Theorem 7, by setting 1 + ln 1β

= 2, we can immediately obtain an improvedalgorithm with best possible delay ratio under the same cost bound 2C . That is:

Corollary 8 By setting 1 + ln 1β

= 2, we have β = 1e , and hence Algorithm 2 is now

with a bifactor approximation ratio of (1 + 1e , 2) = (1.369, 2).

For those applications in which delay and cost are of equal importance, by setting1 + ln 1

β= 1 + β and slightly modifying Algorithm 2 to improve either cost or

delay that is of worse ratio, we can obtain an improved algorithm with ratio as in thefollowing corollary:

Corollary 9 If 1 + ln 1β

= 1 + β, Algorithm 2 is with a bifactor approximation ratioof (1.567, 1.567).

Now we consider the case that β = 0, i.e. the delay constraint is strictly satisfied. In this

case, Inequality (5) in the proof of Theorem 7 will become∑h−1

j=1d(O j )

�D−∑ j−1i=1 d(Oi )

≤∫ |�D||βD=0|

1x dx = ln |�D| ≤ ln D. So we have:

Corollary 10 When β = 0, Algorithm 2 is with a ratio of (1, O(ln n)).

From Corollary 10, we can see that the price of obeying one constraint strictly is veryhigh, i.e. it requires extra O(ln n) times of cost. However, this is the first algorithmwith logarithmic factor approximation ratio for the k-BCP problem with strict delayconstraint.

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(b)

(a)

Fig. 1 Construction of auxiliary graph H(v = s) with cost constraint C = 6: (a) graph G; (b) auxiliarygraph H(v = s)

3 Computing cycle O j with minimumd(O j )

c(O j )

Let G = (V, E) be G, excepting that the edges of P1, P2 . . . , Pk are with directionreversed, cost sat to 0, and delay negatived. This section will show how to computea cycle O with cost bounded by C and d(O)

c(O)minimized in G. The key idea is firstly

to construct an auxiliary graphs H(v) for each v where every cycle is with cost atmost C , secondly to compute the cycle O ′ with minimum d(O ′)

c(O ′) among all cycles in

all H(v)s for each v ∈ G, and thirdly to obtain cycle O with minimum d(O(v))c(O(v))

in Gaccording to O ′.

3.1 Construction of auxiliary graph H(v)

The algorithm of constructing the auxiliary graph H(v) is inspired by the method ofcomputing a single path subject to multiple constraints (Xue et al. 2008). The fulllayout of the algorithm is as shown in Algorithm 3 (An example of such constructionis as depicted in Fig. 1).

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Algorithm 3 Construction of auxiliary graph H .

Input: Graph G = (V, E), two distinct vertices s, t ∈ V , a cost c : e → Z+0 and a delay d : e → Z

+0 on

every edge e ∈ E , a cost constraint C ∈ Z+ and a delay constraint D ∈ Z

+;Output: Auxiliary graph H(v).

1. For every vertex vl of V , add to H(v) vertices v0l , . . . , vC

l ;

2. For every edge e = ⟨v j , vl

⟩ ∈ E , add to H(v) the edges⟨v0

j , vc(e)l

⟩, . . . ,

⟨v

C−c(e)j , vC

l

⟩, each of

which is with cost c(e) and delay d(e);/*Note that d(e) can be negative in G = (V, E).*/

3. For all i = 1, . . . , C , add to H(v) backward edge⟨vi , v0

⟩with delay 0 and cost 0, where a backward

edge is an edge⟨vi , v j

⟩where i > j .

/*H(v) contains backward edges and cycles only after adding the edges of Step 3.*/

A backward edge in layer graph H(v) is an edge⟨vi , v j

⟩where i > j . Following

Algorithm 3, every backward edge in the constructed auxiliary graph H(v) must con-tain vertex v0. Hence every cycle in H(v) contains at most one backward edge. Onthe other hand, following Algorithm 3 a cycle in H(v) contains at least one backwardedge. Therefore, there exist exactly one backward edge in any cycle of H(v). BecauseH(v)\{⟨v1, v0

⟩, . . . ,

⟨vC , v0

⟩} is an acyclic graph where any path is with cost at mostC , we have:

Lemma 11 Any cycle in H(v) is with cost at most C.

Let O(v) be a cycle in H(v), then following the construction of H(v), O(v) appar-ently corresponds to a set of cycles in G. Conversely, every cycle containing v in Gcorresponds to a cycle in H(v). Based on this observation, the following lemma givesthe key idea of computing a cycle O of G with d(O)

c(O)minimized and cost bounded by C :

Lemma 12 Let O(vi ) be a cycle with minimum d(O(vi ))c(O(vi ))

in H(vi ), and O(v) be the

cycle with minimum d(O(v))c(O(v))

among the n cycles O(v1), . . . , O(vn). Assume O is a

cycle with minimum d(O)c(O)

in the set of cycles in G that correspond to O(v). Then for

any cycle O ′ in G with c(O ′) ≤ C, d(O)c(O)

≤ d(O ′)c(O ′) holds.

Proof Suppose the lemma is not true, then there must exist in G a cycle, say O ′,such that d(O)

c(O)>

d(O ′)c(O ′) and c(O ′) ≤ C hold. Then the cycle O ′(v) in H(v), which

corresponds to O ′, is with d(O ′(v))c(O ′(v))

= d(O ′)c(O ′) <

d(O)c(O)

≤ d(O(v))c(O(v))

. This contradicts with

the minimality of O(v) in H(v), and completes the proof.

3.2 Computing the cycle O with minimum d(O)c(O)

The main idea of the algorithm to compute a cycle O with d(O)c(O)

minimized in G, is to

compute the cycle O(v) with minimum d(O(v))c(O(v))

among all cycles in all H(vi )s for each

vi ∈ G. Following Lemma 12, any cycle O in the set of cycles in G corresponding toO(v) ∈ H(v) is with minimum d(O)

c(O). The detailed steps are as below:

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1. For i = 1 to n

(a) Construct H(vi ) for vi ∈ G by Algorithm 3;

(b) Compute cycle O(vi ) with minimum d(O(vi ))c(O(vi ))

in H(vi ) by employing theminimum cost-to-time ratio cycle algorithm in Ahuja et al. (1993);

(c) Select O(v) with minimum d(O(v))c(O(v))

from the n computed cycles O(v1), . . . ,

O(vn);

2. Select the cycle O with minimum d(O)c(O)

among the cycles in G that correspond toO(v).

Clearly, the cycle O attains minimum d(O)c(O)

in G. Besides, following Lemma 11 wehave c(O) ≤ C . Therefore the cycle O is correctly the promised cycle. This completesthe proof of the approximation ratio of Algorithm 2.

4 Conclusion

Based on improving a simple (α, 2 − α)-approximation algorithm via constructingauxiliary layer graphs and employing the cycle cancellation method, this paper gavea novel approximation algorithm with ratio (1 + β, max{2, 1 + ln 1

β}) for the kBCP

problem. By setting β = 0, an approximation algorithm with (1, O(ln n)), i.e. anO(ln n)-approximation algorithm can be obtained immediately. To the best of ourknowledge, it is the first non-trivial approximation algorithm which obeys the delayconstraint strictly for the kBCP problem. We are now investigating whether any con-stant factor approximation algorithm exists for computing a solution of kBCP thatstrictly obey the delay constraint.

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