Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints
aa r X i v : . [ c s . D S ] A p r Improved Approximation Algorithms forComputing k Disjoint Paths Subject to TwoConstraints
Longkun Guo ⋆ , Hong Shen , , Kewen Liao College of Mathematics and Computer Science, Fuzhou University, China School of Information Science and Technology, Sun Yat-Sen University, China School of Computer Science, University of Adelaide, Australia
Abstract.
For a given graph G with positive integral cost and delay onedges, distinct vertices s and t , cost bound C ∈ Z + and delay bound D ∈ Z + , the k bi-constraint path ( k BCP) problem is to compute k disjoint st -paths subject to C and D . This problem is known NP-hard,even when k = 1 [4]. This paper first gives a simple approximation al-gorithm with factor- (2 , , i.e. the algorithm computes a solution withdelay and cost bounded by ∗ D and ∗ C respectively. Later, a novelimproved approximation algorithm with ratio (1 + β, max { , β } ) is developed by constructing interesting auxiliary graphs and employ-ing the cycle cancellation method. As a consequence, we can obtain afactor- (1 . , approximation algorithm by setting β = 2 and afactor- (1 . , . algorithm by setting β = 1 + ln β . Besides, bysetting β = 0 , an approximation algorithm with ratio (1 , O (ln n )) , i.e. analgorithm with only a single factor ratio O (ln n ) on cost, can be immedi-ately obtained. To the best of our knowledge, this is the first non-trivialapproximation algorithm for the k BCP problem that strictly obeys thedelay constraint.
Keywords: k -disjoint bi-constraint path, NP-hard, bifactor approximation al-gorithm, auxiliary graph, cycle cancellation. In real networks, there are many applications that require quality of service andsome degree of robustness simultaneously. Typically, the quality of service (QoS)related problem requires routing between the source node and the destinationnode to satisfy several constraints simultaneously, such as bandwidth, delay,cost and energy consumption. Nevertheless, in networks, some time-critical ap-plications also require routing to remain functioning while edge or vertex failure ⋆ This project was supported by the Natural Science Foundation of Fujian Province(2012J05115), Doctoral Fund of Ministry of Education of China for Young Schol-ars (20123514120013) and Fuzhou University Development Fund (2012-XQ-26).Longkun Guo ([email protected]) is the corresponding author. ccurs. A common solution is to compute k disjoint paths that satisfy the QoSconstraints, and use one path as an active path whilst the other paths as backup paths. The routing traffic is carried on the active path, and switched to thedisjoint backup paths while an edge or vertex failure occurs on the active path.However, for some time-critical applications even the time to discover failures ofrouting and restore data transmission in backup paths is too long for them. Forsuch applications, packages are routed via k paths simultaneously, and the trafficis switched from failed paths to functioning paths if edge or vertex failures occur,such that routing can tolerate k − edge (vertex) failures. Therefore, given costand delay as 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 cost function c : E → Z + , a delay function d : E → Z + , a cost bound C ∈ Z + and a delay bound D ∈ Z + , the k -disjoint QoS Paths problem is to compute k disjoint st -paths P , . . . , P k , such that P i =1 ,...,k c ( P i ) ≤ C and d ( P i ) ≤ D for every i = 1 , . . . , k . This problem is NP-hard even when all edges of G are with cost 0 [8], whichresults in the difficulty to approximate the k -disjoint QoS Paths problem. Analternative method is to compute k disjoint with total cost bounded by C anddelay bounded by D ( equal to kD in Definition 1), and then route the packagesvia the paths according to their urgency priority, i.e., route urgent packagesvia paths of low delay whilst deferrable ones via paths of high delay of the k disjoint paths. Therefore, The disjoint bi-constraint path problem arises as inthe following: Definition 2 (The k disjoint bi-constraint path problem, k BCP) For a graph G = ( V, E ) with a pair of distinct vertices s, t ∈ V , a cost function c : E → R + ,a delay function d : E → R + , a cost bound C ∈ Z + and a delay bound D ∈ R + , the k -disjoint bi-constraint path problem is to calculate k disjoint st -paths P , . . . , P k , such that P i =1 ,...,k c ( P i ) ≤ C and P i =1 ,...,k d ( P i ) ≤ D . This paper will focus on bifactor approximation algorithms for the k BCP prob-lem, which are introduced as below:
Definition 3
An algorithm A is a bifactor ( α, β ) -approximation for the k BCPproblem, if and only if for every instance of k BCP , A computes k disjoint st -paths of which the delay sum and the cost sum are bounded by α ∗ D and β ∗ C respectively. Since a β -approximation with the single factor ratio on cost is identical to abifactor (1 , β ) -approximation, we use them interchangeably in the text. This k BCP problem is NP-hard even when k = 1 [4]. To the best of our knowl-edge, this paper is the first one that presents non-trivial approximation algo-ithms for the k BCP problem formally. However, a number of papers have ad-dressed problems closely related to k BCP, in particular the k restricted shortestpath problem ( k RSP), which is to calculate k disjoint st -paths of minimumcost-sum under the delay constraint P i =1 ,...,k d ( P i ) ≤ D . An algorithm with bifac-tor approximation ratio (2 , has been developed in [6] for general k , whileno approximation solution that strictly obeys the delay (or cost) constraintis known even when k = 2 . For a positive real number r , bifactor ratio of (1 + r , r (1 + r +1) r )(1 + ǫ )) and (1 + r , r (1 + r +1) r )) have been achievedrespectively in [10,3] for the case k = 2 and under the assumption that the delayof each path in the optimal solution of k RSP is bounded by Dk .Special cases of this problem have been studied. When the delay constraint isremoved, this problem is reduced to the min-sum problem, which is to calculate k disjoint paths with the total cost minimized. This problem is known polyno-mially solvable [11]. Moreover, when k = 1 , the problem reduces to the singlebi-constraint path (BCP) problem, which is known as the basic QoS routingproblem [4] and admits full polynomial time approximation scheme (FPTAS)[4,9]. Recently, the single BCP problem is still attracting considerable interestsof the researchers. The strongest result known is a ( ǫ )-approximation due toXue et al [14].Additionally, when the cost constraint is removed, the disjoint QoS problemreduces to the length bounded disjoint path problem of finding two disjointpaths with the length of each path constrained by a given bound. This problemis a variant of the min-Max problem of finding two disjoint paths with thelength of the longer path minimized . Both of the two problems are knownto be NP- complete [8], and with the best possible approximation ratio of 2 indigraphs [8], which can be achieved by applying the algorithm for the min-sumproblem in [11,12]. Contrastingly, the min-min problem of finding two paths withthe length of the shorter path minimized is NP-complete and doesn’t admit K approximation for any K ≥ [5,13,2]. The problem remains NP-complete andadmits no polynomial time approximation scheme in planar digraphs [7]. The main result of this paper is a factor- (1 + β, max { , β } ) approximationalgorithm for any < β ≤ for the k BCP problem. The main idea of the algo-rithm is firstly to compute k -disjoint paths with delay-sum bounded by αD andcost-sum bounded by (2 − α ) ∗ C , where ≤ α ≤ is a real number, and secondlyto improve the computed k paths by novelly combining cycle cancellation [10]and cost-bounded auxiliary graph construction [14]. The key technique to provethe algorithm’s approximation ratio is using definite integral to compute a closeform for the sum of the cost increment during the improving phase.As a consequence of the main result, we can obtain a factor- (1 . , approx-imation algorithm by setting β = 2 , and a factor- (1 . , . algorithmby setting β = 1 + ln β and slightly modifying our algorithm (to improveeither cost or delay that is with worse ratio). Nevertheless, by slightly modifying lgorithm 1 A basic approximation algorithm for the k -BCP problem Input:
A graph G = ( V, E ) , each edge e with cost c ( e ) and delay d ( e ) , a given costconstraint C ∈ Z + and delay constraint D ∈ Z + ; Output: k disjoint paths P , P . . . , P k .1. Set the new cost of edge e as b ( e ) = c ( e ) C + d ( e ) D ;2. Compute the k disjoint paths P , P . . . , P k in G by using Suurballe and Tarjan’salgorithm [11,12], such that P ki =1 P e ∈ P i b ( e ) is minimized;3. Return P , P . . . , P k . our ratio proof, we show that an approximation algorithm with ratio (1 , O (ln n )) ,i.e. an algorithm with single factor ratio of O (ln n ) on cost, can be immediatelyobtained by setting β = 0 . To the best of our knowledge, this is the first non-trivial approximation algorithm for the k BCP problem that strictly obeys thedelay constraint.We note that our algorithms are with pseudo-polynomial time complexity,since the auxiliary graph we construct is of size O ( C ∗ n ) . However, by using theclassic polynomial time approximation scheme design technique [4], i.e. for anysmall ǫ > setting the cost of every edge to j c ( e ) ǫCn k in G before the constructionof auxiliary graph, we can immediately obtain a polynomial time algorithm withratio ((1 + β ) ∗ (1 + ǫ ) , max { , β } ∗ (1 + ǫ )) . We shall omit the details dueto the paper length limitation. k disjoint bi-constraint paths This section will first present a simple approximation method for computing k -disjoint paths with delay-sum bounded by αD and cost-sum bounded by (2 − α ) ∗ C , where ≤ α ≤ is a real number, and secondly improve the computed k paths by balancing the value of α and − α . Though the presented simplealgorithm is with worse ratio than that of the algorithm for k = 2 in [10], it suitsthe improving phase better. Observing that the difficulty of computing k -disjoint bi-constraint paths mainlycomes from the two given constraints, the key idea of our algorithm is to dealwith one new constraint B instead of the two given constraints C and D . Ouralgorithm firstly assigns a new mixed cost b ( e ) = c ( e ) C + d ( e ) D to every edge ingraph, and secondly computes k disjoint paths with the new cost sum boundedby B = CC + DD = 2 . Note that the second step can be accomplished in polynomialtime by employing the SPP algorithm due to Suurballe and Tarjan [11,12]. Thedetailed algorithm is as in Algorithm 1.he time complexity and performance guarantee of Algorithm 1 is given bythe following theorem: Theorem 4
Algorithm 1 runs in O ( km log mn n ) time, and computes k -disjointpaths with delay-sum bounded by αD and cost-sum bounded by (2 − α ) ∗ C , where ≤ α ≤ is a real number.Proof. The main part of Algorithm 1 takes O ( km log mn n ) to compute k -disjoint paths by using Surrballe and Tarjan’s algorithm [11,12], and other partsof the algorithm take trivial time. Hence the time complexity of the algorithmis O ( km log mn n ) .It remains to show the approximation ratio. To make the proof concise, wedenote by OP T an optimal solution for the k -disjoint BCP paths problem, and SOL the solution of Algorithm 1. Obviously P e ∈ OP T b ( e ) ≤ holds. Then sincethe k disjoint paths is with minimum new cost, we have X e ∈ SOL b ( e ) ≤ X e ∈ OP T b ( e ) ≤ . (1)Assume the delay-sum of the algorithm is α times of d ( OP T ) , then following Al-gorithm 1 ≤ α ≤ holds. Therefore, we have P e ∈ SOL b ( e ) = P ki =1 P e ∈ P i b ( e ) = α + c ( SOL ) c ( OP T ) . From Inequality (1), α + c ( SOL ) c ( OP T ) ≤ holds. That is, c ( SOL ) ≤ (2 − α ) c ( OP T ) ≤ (2 − α ) C . This completes the proof.Note that α differs for different instances, i.e. Algorithm 1 may return a solutionwith cost ∗ c ( OP T ) and delay for some instances, while a solution with cost0 and delay ∗ d ( OP T ) for other instances. Hence, the bifactor approximationratio for Algorithm 1 is actually (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, applicationsrequire that the delay of the resulting solution is bounded by (1 + β ) D , where < β < is a positive real number. Apparently, we could get an algorithmsimilar to Algorithm 1 excepting setting the new cost as b ( e ) = β c ( e ) C + d ( e ) D . Theratio of the new algorithm is given as below: Corollary 5
By setting the new cost as b ( e ) = β c ( e ) C + d ( e ) D for a given realnumber < β < , Algorithm 1 returns k paths with delay-sum bounded by αD and cost-sum bounded by β − αβ ∗ C , where ≤ α ≤ β is a real number.Therefore the ratio of the algorithm is (1 + β, β ) . The proof of Corollary 5 is omitted here, since it is very similar to the proofof Theorem 1. According to Corollary 5, our algorithm can bound the delay-sum of the k -disjoint path by (1 + β ) D for any < β < , by relaxing thecost constraint to (1 + β ) ∗ C . For example, if β = 0 . , then the bifactorapproximation ratio of the algorithm is (1 . , . Thus, the algorithm decreasethe delay of the k -disjoint paths at a high price. In the next subsection, we shalldevelop an improved method that pays less to make delay-sum of the k -disjointpaths bounded by (1 + β ) D . lgorithm 2 An improved algorithm based on cycle cancellation.
Input:
A graph G = ( V, E ) , each edge e with cost c ( e ) and delay d ( e ) , a given costconstraint C ∈ Z + and delay constraint D ∈ Z + , disjoint QoS paths P , P . . . , P k computed by Algorithm 1; Output:
Improved disjoint QoS paths Q , Q . . . , Q k .1. If P ki =1 d ( P i ) ≤ (1 + β ) D ; then return P , P . . . , P k as Q , Q . . . , Q k , terminate;2. Reverse direction of the edges of P , P . . . , P k in G , set their cost to a smallpositive real number < ǫ < mnD , and negative their delay;3. Compute cycle O j with c ( O j ) ≤ C , d ( O j ) < and d ( O j ) c ( O j ) attaining minimum, bythe method given in next section;/* Following clause 2 of Proposition 6, if P ki =1 d ( P i ) ≥ d ( OP T ) and P ki =1 c ( P i ) ≥ , there always exist cycle O j with c ( O j ) ≤ C and d ( O j ) < . */4. Improve P , P . . . , P k by adding the edges of O j and removing the pairs of paralleledges in opposite direction;5. Go to Step 1. To make the delay of the solution resulting from Algorithm 1 bounded by (1 + β ) D, our improving phase is, basically a greedy method, using the so-called cyclecancellation to improve the disjoint paths in iterations until a solution with thebest possible ratio (1 + β, max { , β } ) is obtained. The cycle cancellationmethod is an approach of using cycles to change the edges of the disjoint paths,which first appears in [10] and is derived from the following proposition that canbe immediately obtained from flow theory [1]: Proposition 6
Let P , P . . . , P k and Q , Q . . . , Q k be two sets of k disjoint st -paths in G , G be G excepting that all edges of P , P . . . , P k are reversed, and O be a cycle in G . Then1. The edges of P , P . . . , P k and O , excepting the pairs of parallel edges withopposite direction, compose k -disjoint paths;2. There exist a set of edge disjoint cycles O , . . . , O h in G , such that the edgesof P , P . . . , P k and O , . . . , O h , excepting the pairs of parallel edges withopposite direction, compose Q , Q . . . , Q k . From the proposition above, it is obvious that there exists a set of cycles O , . . . , O h that can improve k disjoint QoS paths P , P . . . , P k to an optimal solution.However, it is hard to identify all the cycles O , . . . , O h , so we employ a greedyapproach to compute a set of cycles to obtain an approximation approach. Theimproving phase is composed by iterations, each of which computes a cycle andthen uses it to improve P , P . . . , P k . More precisely, to obtain a good ratio,the algorithm computes in iteration j a cycle O j with d ( O j ) c ( O j ) minimized amongthe cycles in G . The layout of the algorithm is as given in Algorithm 2.ollowing clause 1 of Proposition 6, Algorithm 2 will correctly return k dis-joint paths. It remains to show the cost and delay of the k disjoint paths isconstrained as below: Theorem 7
The approximation ratio of Algorithm 2 is (1+ β, max { , β } ) .Proof. For the case that P ki =1 d ( P i ) ≤ (1 + β ) D holds before the improvingphase, the approximation ratio of Algorithm 2 is obviously (1 + β, .It remains to show the ratio of the algorithm is (1 + β, β ) for the casethat P ki =1 d ( P i ) > (1 + β ) D . Assume that Algorithm 2 runs in h iterations, thekey idea of the proof is to sum up the cost increment while using the cycle toimprove the k disjoint paths in iterations, and show that the cost sum is bounded(by giving the cost sum a close form).Note that in the case, we have αD ≥ P ki =1 d ( P i ) > (1 + β ) D , so α > β holds. Let ∆D = d ( OP T ) − d ( SOL ) ≥ (1 − α ) d ( OP T ) and ∆C = c ( OP T ) − c ( SOL ) ≤ ( α − c ( OP T ) . Clearly, ∆D < and ∆C > hold. Let the cyclecomputed in the j th iteration be O j , then since d ( O j ) c ( O j ) attains minimum in Step3 of Algorithm 2, we have d ( O j ) c ( O j ) ≤ ∆D − P j − i =1 d ( O i ) C . That is, c ( O j ) ≤ d ( O j ) ∆D − P j − i =1 d ( O i ) C. By summing up c ( O j ) in h − iterations (excluding the last iteration), we have: h − X j =1 c ( O j ) ≤ C h − X j =1 d ( O j ) ∆D − P j − i =1 d ( O i ) . Following the definition of Definite Integral, we have: h − X j =1 d ( O j ) ∆D − P j − i =1 d ( O i ) = h − X j =1 ∆D − P j − i =1 d ( O i ) d ( O j ) ≤ Z ∆D − P h − i =1 d ( O i ) ∆D x dx, (2)where the maximum is attained when d ( O j ) = − for every j .Algorithm 2 terminates when d ( SOL ) + P hi =1 d ( O i ) ≤ (1 + β ) D , so in the h − iterations d ( SOL ) + P h − i =1 d ( O i ) > (1 + β ) D holds. That is d ( SOL ) − D + P h − i =1 d ( O i ) > βD , and hence − ∆D + P h − i =1 d ( O i ) > βD > holds. So weobtain a close form for the cost sum of the h − iterations: Z ∆D − P h − i =1 d ( O i ) ∆D x dx = Z − ∆D − ∆D + P h − i =1 d ( O i )) x dx ≤ Z − ∆DβD x dx = ln | ∆D | βD = ln α − β . (3)t last, the cost increment in the h th iteration is bounded by c ( OP T ) . Sothe final cost is c ( SOL ) ≤ (2 − α ) C + C ln α − β + C = C (3 − α + ln α − β ) , where SOL is the solution resulting from Algorithm 2.Let f ( α ) = 3 − α + ln α − β . Remind that α ≤ , so f ′ ( α ) = α − − > , f ( α ) is monotonous increasing on α , and attains maximum while α = 2 . So we have c ( SOL ) ≤ (1 + ln β ) c ( OP T ) .Therefore, the cost of the output of Algorithm 2 is bounded by (1+ln β ) c ( OP T ) ,and delay bounded by (1 + β ) d ( OP T ) . This completes the proof.From Theorem 7, by setting β = 2 , we can immediately obtain an improvedalgorithm with best possible delay ratio under the same cost bound C . Thatis: Corollary 8
By setting β = 2 , we have β = e , and hence Algorithm 2 isnow with a bifactor approximation ratio of (1 + e ,
2) = (1 . , . For those applications in which delay and cost are of equal importance, by setting β = 1 + β and slightly modifying Algorithm 2 to improve either cost ordelay that is of worse ratio, we can obtain an improved algorithm with ratio asin the following corollary: Corollary 9 If β = 1 + β , Algorithm 2 is with a bifactor approximationratio of (1 . , . . Now we consider the case that β = 0 , i.e. the delay constraint is strictly satisfied.In this case, Inequality (3) in the proof of Theorem 7 will become P h − j =1 d ( O j ) ∆D − P j − i =1 d ( O i ) ≤ R | ∆D || βD =0 | x 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 strictlyis very high, i.e. it requires extra O (ln n ) times of cost. However, this is the firstalgorithm with logarithmic factor approximation ratio for the k -BCP problemwith strict delay constraint. O j with minimum d ( O j ) c ( O j ) Let G = ( V, E ) be G , excepting that the edges of P , P . . . , P k are with direc-tion reversed, cost sat to 0, and delay negatived. This section will show how tocompute a cycle O with cost bounded by C and d ( O ) c ( O ) minimized in G . The keyidea is firstly to construct an auxiliary graphs H ( v ) for each v where every cycleis with cost at most 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 withminimum d ( O ( v )) c ( O ( v )) in G according to O ′ . lgorithm 3 Construction of auxiliary graph H . Input : Graph G = ( V, E ) , two distinct vertices s, t ∈ V , a cost c : e → Z +0 and adelay d : e → Z +0 on every edge e ∈ E , a cost constraint C and a delay constraint D ; Output : Auxiliary graph H ( v ) .1. For every vertex v l of V , add to H ( v ) vertices v l , . . . , v Cl ;2. For every edge e = h v j , v l i ∈ E , add to H ( v ) the edges D v j , v c ( e )+1 l E , . . . , D v C − c ( e ) j , v Cl E , 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 = 2 , . . . , C , add to H ( v ) backward edge (cid:10) v i , v (cid:11) with delay 0 and cost 0,where a backward edge is an edge (cid:10) v i , v j (cid:11) where i > j ./* H ( v ) contains backward edges, and hence cycles, only after adding the edges ofStep 3.*/ H ( v ) The algorithm of constructing the auxiliary graph H ( v ) is inspired by the methodof computing a single path subject to multiple constraints [14]. The full layoutof the algorithm is as shown in Algorithm 3 (An example of such constructionis as depicted in Figure 1).Following Algorithm 3, every backward edge in the constructed auxiliarygraph H ( v ) must contain vertex v . Hence every cycle in H ( v ) contains at mostone backward edge. On the other hand, following Algorithm 3 a cycle in H ( v ) contains at least one backward edge. Therefore, there exist exactly one backwardedge in any cycle of H ( v ) . Because H ( v ) \ { (cid:10) v , v (cid:11) , . . . , (cid:10) v C , v (cid:11) } is an acyclicgraph where any path is with cost at most C , 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 ) apparently corresponds to a set of cycles in G . Conversely, every cycle containing v in G corresponds to a cycle in H ( v ) . Based on the observation, the followinglemma gives the key idea of computing a cycle O of G with d ( O ) c ( O ) minimized andcost bounded by C : Lemma 12
Let O ( v i ) be a cycle with minimum d ( O ( v i )) c ( O ( v i )) in H ( v i ) , and O ( v ) bethe cycle with minimum d ( O ( v )) c ( O ( v )) among the n cycles O ( v ) , . . . , O ( v n ) . 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 this 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 ) thatcorresponds to O ′ is also with d ( O ′ ( v )) c ( O ′ ( v )) = d ( O ′ ) c ( O ′ ) < d ( O ) c ( O ) ≤ d ( O ( v )) c ( O ( v )) , contradictingwith the minimality of O ( v ) in H ( v ) . This completes the proof. x y z t s x x x x x x x y y y y y y y z z z z z z z t t t t t t t (1 ,
4) (1,1)(a) (b) edge with delay and cost equal to its corresponding edge in G edge with delay 0 and cost ǫ s s s s s s (2,1)(1,2) (2,1)( − , ǫ ) (2 , Fig. 1.
Construction of auxiliary graph H ( v = s ) with cost constraint C = 6 : (a) graph G ; (b) auxiliary graph H ( v = s ) . The cycle O = syts in G is exclude in the auxiliarygraph H ( s ) as shown in (b), keeping the cost of k disjoint paths constrained by C = 6 . .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 ′ with minimum d ( O ′ ) c ( O ′ ) among all cycles in all H ( v ) s foreach v ∈ G . Following Lemma 12, the cycle O in G is the cycle with minimum d ( O ) c ( O ) among the cycles in G corresponding to all the computed O ′ s. The detailedsteps are as below:1. For i = 1 to n (a) Construct H ( v i ) for v i ∈ G by Algorithm 3;(b) Compute cycle O ( v i ) with minimum d ( O ( v i )) c ( O ( v i )) in H ( v i ) by employing theminimum cost-to-time ratio cycle algorithm in [1];(c) Select O ( v ) with minimum d ( O ( v )) c ( O ( v )) from the n computed cycles O ( v ) , . . . , O ( v n ) ;2. Select the cycle O with minimum d ( O ) c ( O ) among the cycles in G that correspondto O ( v ) .Clearly, the cycle O attains minimum d ( O ) c ( O ) in G . Besides, following Lemma 11we have c ( O ) ≤ C . Therefore the cycle O is correctly the promised cycle. Thiscompletes the proof of the approximation ratio. This paper gave a novel approximation algorithm with ratio (1 + β, max { , β } ) for the k BCP problem based on improving a simple ( α, − α ) -approximationalgorithm by constructing interesting auxiliary graphs and employing the cyclecancellation method. By setting β = 0 , an approximation algorithm with bifac-tor ratio (1 , O (ln n )) , i.e. an O (ln n ) -approximation algorithm can be obtainedimmediately. To the best of our knowledge, it is the first non-trivial approxima-tion algorithm for this problem that obeys the delay constraint strictly. We arenow investigating whether any constant factor approximation algorithm existsfor computing a solution that strictly obey the delay constraint. References
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