Availability-Based Path Selection and Network Vulnerability Assessment

Avalablty-Based Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl Abstract In data-communcaton networks, network relablty s of great concern to network operators and customers, snce (1) the connecton requested by the customer should obey the agreed-upon avalablty, otherwse the network operator or servce provder may face lablty costs as stpulated n the Servce Level Agreement (SLA), and (2) t s mportant to determne the most vulnerable parts of a network to gan nsght nto where and how the network operator can ncrease the network relablty. In ths paper, we frst study the problem of establshng a connecton over at most k (partally) lnk-dsjont paths and for whch the avalablty s no less than δ (0 < δ 1). We analyze the complexty of ths problem n generc networks, Shared-Rsk Lnk Group (SRLG) networks and mult-layer networks. We subsequently propose a polynomal-tme heurstc algorthm and an exact Integer Non-Lnear Program (INLP) for avalabltybased path selecton. The proposed algorthms and two exstng heurstc algorthms are compared n terms of acceptance rato and runnng tme. Subsequently, n three aforementoned types of networks, we study the problem of fndng a (set of) network cut(s) for whch the falure probablty of ts lnks s bggest. Index Terms Avalablty, Routng, Survvablty, SRLG networks, Mult-layer networks, Mn-cut. I. INTRODUCTION Due to the mportance of data-communcaton networks, even short servce dsruptons may result n sgnfcant economc loss. Hence, survvablty mechansms to protect connectons are called for. For nstance, by allocatng a par of lnk-dsjont paths (nstead of only one unprotected path), data are transported by the prmary path, and upon lnk falure, can be swtched to the backup path. Ideally, a survvablty mechansm should also take nto account the relablty of lnks. For nstance, f both prmary and backup paths contan lnks that have a hgh probablty to become unavalable, then proper protecton cannot be provded. Connecton avalablty, a value between 0 and 1, s therefore mportant and refers to the probablty that a connecton (ncludng ts survvablty mechansm) s n the operatng state durng the requested lfe-tme of the connecton. However, a survvablty mechansm that does not allow for more than 2 lnk-dsjont paths for each connecton may stll fal to satsfy the customer s avalablty requrement and k > 2 lnk-dsjont paths may be needed. Obvously, the bgger k s, the greater the avalablty of the connecton could be, but also the greater the resource consumpton (e.g., bandwdth) and hence prce. Ths paper frst deals wth the Avalablty- Based Path Selecton (ABPS) problem, whch s to establsh a connecton over at most k > 0 (fully or partally) lnk-dsjont paths, for whch the avalablty s at least δ (0 < δ 1). Apart from consderng how to provde a relable connecton to customers, t s also mportant for network operators to determne the most vulnerable part of the network,.e., a subset of lnks wth hghest falure probablty whose removal wll dsconnect the network. The network operator could then replace/strengthen those lnks n order to ncrease the network relablty. Hence, ths paper also tackles ths so-called Network Vulnerablty Assessment (NVA) problem, whch s to fnd a set of network cuts for whch the falure probablty of the lnks n a cut belongng to that set s hghest. Our key contrbutons are as follows: We consder the Avalablty-Based Path Selecton (ABPS) problem n generc networks, Shared-Rsk Lnk Group (SRLG) networks and mult-layer networks. We prove that, n general, the ABPS problem cannot be approxmated n polynomal tme. We propose a polynomal-tme heurstc algorthm and an exact Integer Non-Lnear Program (INLP) to solve the ABPS problem. We compare, va smulatons, the proposed algorthms wth two exstng algorthms n terms of performance and runnng tme. We consder the Network Vulnerablty Assessment (NVA) problem n generc networks, SRLG networks and mult-layer networks. The remander of ths paper s organzed as follows. Related work s presented n Secton II. Secton III explans the calculaton of avalablty for dfferent path types: unprotected path, k fully lnk-dsjont and k partally lnk-dsjont. In Secton IV, we formally defne the Avalablty-Based Path Selecton (ABPS) problem n generc networks and analyze ts complexty. In Secton V and VI, we consder the ABPS problem n SRLG networks and mult-layer networks, respectvely. Secton VII presents our heurstc routng algorthm and an exact INLP. Secton VIII provdes our smulaton results. In Secton IX, we study the Network Vulnerablty Assessment problem n the three aforementoned types of networks. We conclude n Secton X. II. RELATED WORK Avalablty-aware routng under both statc and dynamc traffc demands has been extensvely nvestgated [1], [2], [3], [4], [5], [6]. When the traffc matrx s gven n advance (statc

traffc), Zhang et al. [3] present a mathematcal model to compute avalablty for dfferent protecton types (unprotected, dedcated protecton and shared protecton) for a gven statc traffc matrx. Furthermore, an Integer Lnear Program (ILP) and a heurstc algorthm are proposed to fnd avalabltyaware paths. Tornatore et al. [4] address the avalablty desgn problem: to accommodate a gven traffc matrx by usng shared/dedcated protecton paths. Song et al. [5] propose an avalablty-guaranteed routng algorthm, where dfferent protecton types are allowed. They defne a new cost functon for computng a backup path when the unprotected path fals to satsfy the avalablty requrement. She et al. [1] prove that for dedcated protecton, fndng two lnk-dsjont paths wth maxmal relablty (avalablty) s NP-hard. They also propose two heurstcs for that problem. Luo et al. [6] analyze the problem of protecton wth dfferent relablty, whch s to fnd one unprotected path or dedcated protecton path such that the cost of the whole path s mnmzed and the relablty requrement s satsfed. They subsequently propose an exact ILP as well as two approxmaton algorthms. However, the relablty (avalablty) calculaton n [6] s dfferent from the aforementoned papers, and assumes a sngle-lnk falure model. Assumng each lnk n the network has a falure probablty (=1-avalablty), Lee and Modano [2] mnmze the total falure probablty of unprotected, partally lnkdsjont and fully lnk-dsjont paths by establshng INLPs. They further transform the proposed INLPs to ILPs by usng lnear approxmatons. Dfferent from the aforementoned artcles, we target a more general problem, whch s to fnd at most k (fully or partally) lnk-dsjont paths for whch the avalablty requrement s satsfed. From the perspectve of network relablty calculaton, assumng each lnk n the network s assocated wth a falure probablty value (=1-avalablty), Provan and Ball [7] prove the problem of computng the probablty that the network stays connected s #P-complete 1. Karger [9] proposes a Fully Polynomal Randomzed Approxmaton Scheme (FPRAS) to solve ths problem. There s also work focusng on how to mathematcally model the avalablty of varous network topologes or dfferent protecton segments/paths. Zou et al. [10] nvestgate how to mathematcally calculate the avalablty of dfferent types of network topologes, e.g., tree topology, double star, crown or trple star. Tornatore et al. [11] mathematcally model the avalablty of segment protecton (SP). In segment protecton, a workng path (WP) can be parttoned nto several workng segments (WSs) and each WS s protected by a backup segment (BS). Moreover, they consder two SP cases, namely (1) overlap SP, where dfferent WSs can share the same lnk, and (2) no-overlap SP, where WSs are fully lnk-dsjont. By expressng the duallnk falure va a contnuous tme Markov chan, Mello et al. [12] approxmately estmate the (un)avalablty of the 1 Valant [8] shows that problems n ths class are at least as hard as NPcomplete problems. shared protecton path. Regardng SRLG networks, Hu [13] proves that the problem of fndng 2 SRLG-dsjont paths s NP-hard. To solve t, Hu [13] presents an exact ILP and Xu et al. [14] propose a trapavodance heurstc algorthm. However, the SRLG-dsjont routng problem s not the same as the one studed n ths paper, due to Eq. (11) n Secton V. Hence, the algorthms n [13] [14] cannot be used to effectvely solve our problem. In generc networks, the (s, t) Mn-Cut problem refers to parttonng the network nto two dsjont subsets such that nodes s and t are n dfferent subsets and the total weght of the cut lnks s mnmzed. Accordng to [15], ths problem can be solved by fndng the maxmum flow from s to t. There s also a lot of work on the Mn-Cut problem wth no specfed node pars (s, t). An enumeraton of classcal algorthms to solve the Mn-Cut problem can be found n [16]. III. CONNECTION AVAILABILITY The avalablty of a system s the fracton of tme the system s operatonal durng the entre servce tme. Lke [1], [2], [3], we frst assume that, n generc networks, the lnks avalabltes are uncorrelated/ndependent. If a connecton s carred by a sngle (unprotected) path, ts avalablty s equal to the path avalablty; f t s protected by k 2 dsjont paths, the avalablty wll be determned by these k protecton paths. The avalablty A j of a network component j can be calculated as [10]: A j = MT T F MT T F + MT T R where MT T F represents Mean Tme To Falure and MT T R denotes Mean Tme To Repar. We assume that the lnk avalablty s equal to the product of avalabltes of all ts components (e.g., amplfers). A. Lnk Falure Scenaros For smplcty, suppose there are two (fully) lnk-dsjont paths p 1 and p 2, and the avalablty of lnk l s denoted as A l (= 1 falure probablty), where 0 < A l 1, then ther total avalablty A 2 F D can be computed based on the followng scenaros: Sngle-lnk falure: Here t s assumed that all the lnks n the network have very low falure probablty. In ths context, a path p s avalablty (denoted by A p ) s equal to ts lowest traversed lnk avalablty (hghest falure probablty),.e., A p = mn l p A l. Usng two dsjont paths (whch s a conventonal survvablty mechansm) wll therefore lead to a total connecton avalablty of 1. However, ths approach only works when all the lnks are hghly relable. In Appendx A, we wll address the ABPS problem under the sngle-lnk falure scenaro. Multple lnk falures: Ths s a more general scenaro where at one certan pont n tme, several lnks n the network may fal smultaneously. Hence, for a path p, ts avalablty A p should take nto account all ts lnks avalabltes,.e., A p = l p A l. Consequently, A 2 F D = (1)

1 (1 A p1 )(1 A p2 ), whch ndcates the probablty that at least one path s avalable. In ths paper, we assume multple lnk falures may occur. B. End-to-End Path Avalablty If a path p contans the lnks l 1, l 2, l 3,..., l m, and ther correspondng avalabltes are denoted by A l1, A l2, A l3,..., A lm, then the avalablty of ths path (represented by A p ) s equal to 2 : A p = A l1 A l2 A l3 A lm (2) If we take the log of the lnk avalabltes, fndng the path wth the hghest avalablty turns nto a shortest path problem. When, for a sngle connecton, there are k 2 lnk-dsjont paths p 1, p 2,..., p k wth avalabltes represented by A p1, A p2,..., A pk, the connecton avalablty calculaton can be dvded nto two cases, namely: (1) fully lnk-dsjont paths: these k paths have no lnks n common, and (2) partally lnkdsjont paths: at least two of these k paths traverse at least one same lnk. In case (1), the avalablty (represented by A k F D ) s: A k F D = 1 + 0<<j<u k k (1 A p ) = =1 k A p =1 0<<j k A p A pj A pu + + ( 1) k 1 A u A u 1 a A v A w s b t A p A pj k =1 A p (3) Fg. 1: Network wth dfferent lnk avalablty values. If we use Eq. (3) to calculate the avalablty for the partally lnk dsjont case, the avalablty of the overlappng lnks wll be counted more than once. To amend ths, we use a new operator, whch s defned as follows: X 1 X 2 X k Y = { k =1 X f X = Y k =1 X Y otherwse where X 1, X 2,..., X k and Y represent dfferent lnk avalabltes. Therefore, the avalablty (represented by A k P D ) of k partally lnk-dsjont paths can be calculated as follows: 2 A network havng node and lnk avalabltes can be transformed to a drected network wth only lnk avalabltes, as done n [17]. Therefore, we assume the nodes have avalablty 1 n ths paper. (4) A k P D = 1 k (1 A p ) =1 = 1 (1 A p1 ) (1 A p2 ) (1 A pk ) (5) k = A p A p A pj + =1 0<<j<u k 0<<j k A p A pj A pu + + ( 1) k 1 k A p =1 where s used to denote the operatons of dfferent sets. Let us use an example to descrbe the dfference between case (1) and case (2), where k s set to 2 for smplcty. In Fg. 1 where the lnk avalablty s labeled on each lnk, paths s a t and s b t are fully lnk dsjont. Accordng to Eq. (3), ther avalablty s equal to: 1 (1 A u A w ) (1 A u A v ) =A u A w + A u A v A 2 u A w A v (6) On the other hand, paths s a t and s a b t are only partally lnk dsjont. Accordng to Eq. (5), the connecton avalablty can be calculated as follows: 1 (1 A u A w ) (1 A u A v ) =A u A w + A u A v A u A w A v (7) The followng theorem wll formalze the ntutve noton that f a set of paths p wth avalabltes A p have overlappng lnks that ther total avalablty s less than when those paths would have been fully lnk dsjont. Theorem 1: For gven A p, where 1 k, A k F D A k P D Ṗroof: A proof by mathematcal nducton: When k = 2, A 2 F D = A p 1 + A p2 A p1 A p2, and A 2 P D = A p1 +A p2 A p1 A p2. Snce A p1 A p2 A p1 A p2 accordng to Eq. (4) when 0 A p 1, the theorem s correct for k = 2. Assume when k = m the theorem s correct: m m A m F D = 1 (1 A p ) A m P D = 1 (1 A p ) (8) =1 =1 When k = m + 1, A m+1 F D = 1 (1 A p 1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ) and A m+1 P D = 1 (1 A p 1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ). Accordng to Eq. (8), we have: (1 A p1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ) (1 A p1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ) (9) Snce (1 A pm ) (1 A pm+1 ) (1 A pm ) (1 A pm+1 ), we have: (1 A p1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ) (1 A p1 ) (1 A p2 ) (1 A pm ) (1 A pm+1 ) (10) By mergng Eq. (9) and Eq. (10), we ascertan that A m+1 F D A m+1 P D.

IV. ABPS PROBLEM AND COMPLEXITY A. Problem Defnton Defnton 1: Gven a network represented by G(N, L) where N represents the set of N nodes and L denotes the set of L lnks, and each lnk l has ts own avalablty value A l. For a request represented by r(s, t, δ), where s and t denote the source and destnaton, respectvely, and δ (0 < δ 1) represents the avalablty requrement, establsh a connecton over at most k (partally) lnk-dsjont paths for whch the avalablty s at least δ. A varant, called the Avalablty-Based Backup Path Selecton (ABBPS) problem, s defned as: Defnton 2: Gven an exstng prmary path p from s to t and a requested avalablty δ, fnd at most k 1 paths that are fully or partally lnk-dsjont wth p, such that the avalablty of these k paths s no less than δ. B. Complexty Analyss In ths secton, we study the complexty of the ABPS problem n generc networks. For the case k = 1, by takng the log of the lnk avalabltes, the ABPS problem turns nto a shortest path problem, whch s polynomally solvable. Theorem 2: The ABPS problem s NP-hard for k 2. Proof: The case for partally lnk-dsjont paths can be reduced to the case of fully lnk-dsjont paths by a transformaton such as n Fg. 2. More specfcally, f we assume that all lnks n Fg. 2, except for (s, s ) and (t, t), have avalablty less than δ, then no lnk, except for (s, s ) and (t, t), can be an unprotected lnk n the soluton of the ABPS problem for the partally lnk dsjont case from s to t. Hence, solvng the fully lnk-dsjont ABPS problem from s to t s equvalent to solvng the partally lnk-dsjont ABPS problem from s to t. We therefore proceed to prove that the fully lnk dsjont varant for k = 2 s NP-hard. The proof for k > 2 follows analogously from the proof for k = 2. x a u 1 u1 v 1 v1 u 2 u 2 v 2 v 2 Fg. 3: A lobe for each x. 2 3 n u q u q 1 1 2 2 m m Fg. 4: Lobes for all clauses. C two nodes y and z are created and a lnk connects z and y +1 wth avalablty of 1, where 0 < < m. We assume that s = x 1 and t = x n+1. Moreover, we draw a lnk (s, y 1 ) wth avalablty a and a lnk (z m, t) wth avalablty 1, where 0 < b < a 2 < 1. Fg. 4 depcts ths process. To relate the clause and varables n the constructed graph, we add the followng lnks: () lnks (y j, u k ) and (v k, z j) are added f the k-th occurrence of varable x exsts n clause C j ; or () lnks (y j, u k ) and (v k, z j) are added f the k-th occurrence of varable x exsts wth a negaton n the clause C j. For nstance, a network correspondng to 3SAT nstance (x 1 x 2 x 4 ) (x 1 x 2 x 3 ) (x 2 x 3 x 4 ) (x 1 x 3 x 4 ) s shown n Fg. 5. Based on the constructed graph, whch 2 3 4 v q v q x +1 a 1 1 2 2 3 3 4 4 Fg. 2: Reducton of ABPS problem from partally lnk dsjont to fully lnk dsjont. We frst ntroduce the NP-hard 3SAT problem [18] and then reduce the ABPS problem to t. The 3SAT problem s defned as: There s a boolean formula C 1 C 2...C m, where C denotes the -th clause. Each clause contans 3 varables wth an OR relaton. The queston s whether there s a truth assgnment to the varables that smultaneously satsfes all m clauses. Gven a 3SAT nstance, the graph constructon follows smlarly to [1]. Assume there are n varables n the 3SAT nstance. Frst, we create a lobe for each varable x, whch s shown n Fg. 3, where q represents the number of occurrences of varable x n all the clauses. The avalablty value for each lnk s also shown n Fg. 3, where 0 < b < 1. For each clause Fg. 5: Constructed graph that corresponds to (x 1 x 2 x 4 ) (x 1 x 2 x 3 ) (x 2 x 3 x 4 ) (x 1 x 3 x 4 ). corresponds to a gven 3SAT nstance, we are asked to solve the ABPS problem for k = 2 and δ = a + b q ab q, where q s the sum of occurrences for each varable n all the clauses,.e., q = n =1 q. Because one shortest path can at most have avalablty a, whch s less than δ, we have to fnd 2 lnkdsjont paths. Next, we wll prove that the fully lnk dsjont varant of the ABPS problem s NP-hard. 3SAT to ABPS: If there exsts a truth assgnment that satsfes all the clauses, then each clause j has (at least) one varable wth true or (negated) false assgnment to make ths clause true. Therefore, an upper subpath y j u k v k z j y j+1 or a lower subpath y j u k v k z j y j+1 wll be selected.

By concatenatng these m subpaths wth s y 1 and z m t we obtan one path (denoted by p 1 ) wth avalablty a. Snce each varable only has one truth assgnment, p 1 cannot traverse both the upper subpath and lower subpath n the same lobe. Subsequently, we can get another fully lnk-dsjont path p 2 : For each lobe (correspondng to varable x ), p 2 traverses the upper (lower) subpath wth avalablty of b q f p 1 goes through the lnk of lower (upper) subpath. The avalablty of p 2 s b q = b n =1 q, therefore p 1 and p 2 together have avalablty of a + b q ab q, whch satsfes the requrement δ. ABPS to 3SAT: If there are two fully lnk-dsjont paths from s to t wth avalablty no less than a + b q ab q, then one path must have avalablty a. To understand ths, assume that none of the two paths has avalablty a; wthout loss of generalty, we denote one path has avalablty of a c b e, where c can be ether 0 or 1 ndcatng whether lnk (s, y 1 ) has been traversed, and e > 0 s the number of lnks that have avalablty b. Snce there exsts only one lnk wth avalablty a, the other lnk-dsjont path has avalablty a c b f, where c s ether 0 or 1 meanng whether lnk (s, y 1 ) has been traversed and c + c 1, and f > 0 s the number of lnks whch have avalablty b. Hence, the avalablty of these two paths s a c b e + a c b f a c+c b e+f < b + b < a < δ, when b < a 2. Based on ths analyss, there must exst one path p 1 from s to t wth avalablty a, whch goes through (s, y 1 ) and (z m, t) and the other lnks wth avalablty of 1. To satsfy the avalablty requrement, there must also exst another fully lnk-dsjont path p 2 from s to t wth avalablty of no less than b q. For each lobe, p 2 should traverse ether the upper subpath or the lower subpath, otherwse p 1 and p 2 cannot be fully lnk dsjont. Therefore, p 2 wll traverse the (entre) lower subpath f p 1 goes through lnk (u k, v k ) n the upper subpath, and traverse the (entre) upper subpath f p 1 goes through lnk (u k, v k ) n the lower subpath for each lobe x. That s to say, p 1 cannot smultaneously traverse one lnk n the upper subpath and another lnk n the lower subpath for each same lobe. Consequently, p 1 ether goes va an upper subpath y j u k v k z j y j+1 to set varable x to true or va a lower subpath y j u k v k z j y j+1 to set varable x to false for clause j, where = 1, 2,..., n and j = 1, 2,..., m. Hence, all the m clauses can be smultaneously satsfed. Theorem 3: The ABBPS problem s NP-hard for k 2. Proof: For k 3, the ABBPS problem s equvalent to the ABPS problem for k 1 fully or partally lnk-dsjont paths, and hence NP-hard. In Appendx B, we prove that the ABBPS problem s also NP-hard for k = 2. We proceed to study the approxmablty of the ABPS problem. Theorem 4: The ABPS problem for k 2 cannot be approxmated to arbtrary degree n polynomal tme, unless P=NP. Proof: We can check n polynomal tme whether a sngle path can accommodate the requested avalablty. Hence, the theorem s equvalent to: for a request r(s, t, δ) and any constant number d > 1, there s no polynomal-tme algorthm that can fnd at least 2, but at most k, fully or partally lnkdsjont paths from s to t wth avalablty at least δ d. We prove the theorem for the fully lnk dsjont varant 3 of the ABPS problem for k = 2. We wll use a proof by contradcton and assume a polynomal-tme approxmaton algorthm A exsts for any d > 1. In the constructed graph based on the gven 3SAT nstance n Fg. 5 (also usng the same notaton and condtons), assume δ = a + b q ab q, so algorthm A can fnd two a+b fully lnk-dsjont paths wth avalablty at least q ab q d. Next, we prove that when 0 < b < a 2d, except for an exact soluton, there exsts no soluton wth avalablty no less than a+b q ab q d. If the exact soluton s not acheved by algorthm A, accordng to our prevous analyss, then one path must have avalablty of a c b e and the other path has avalablty of a c b f. Therefore, the avalablty of these two paths s equal to a c b e + a c b f a c+c b e+f. For a gven d, we have a c b e +a c b f a c+c b e+f < b+b = 2b < a d, when 0 < b < a 2d and 0 < a < 1. Therefore, under 0 < b <, except for an exact soluton, any two fully lnk-dsjont paths cannot have avalablty less than a+bq ab q d. To fulfll the assumpton, algorthm A has to fnd two lnk-dsjont paths wth avalablty a+b q ab q. In ths context, the fully lnk dsjont varant of the ABPS problem for k = 2 can be solved exactly n polynomal tme, whch s a contradcton. V. SHARED-RISK LINK GROUPS In ths secton, we assume two types of falures/avalabltes, namely Shared-Rsk Lnk Group (SRLG) falures and sngle lnk falures/avalabltes. A Shared-Rsk Lnk Group (SRLG) [16] reflects that a certan set/group of lnks n a network wll fal smultaneously. For nstance, n optcal networks, several fbers may resde n the same duct and a cut of the duct would cut all fbers n t. One duct n ths context corresponds to one dstnct SRLG. If each lnk s a sngle member of an SRLG, then no SRLGs exst. Hence the ABPS problem n SRLG networks ncludes as a specal case the ABPS problem n generc networks as dscussed n the prevous secton. Each lnk can belong to one or more SRLGs, and the lnks n the same SRLG wll smultaneously fal when the correspondng SRLG fals. The probablty of ths happenng (or not) s the SRLG falure (avalablty) probablty. We assume there are g SRLGs n the network G(N, L), and that the falure probablty of the -th SRLG (represented by srlg ) s denoted by π, for 1 g. For a partcular lnk l L, we denote by SR l the set of all SRLGs to whch l belongs. Dfferent from [2], where all SRLG events are assumed to be mutually exclusve, we assume that multple SRLG events may occur smultaneously. The avalablty of a sngle path should ncorporate the SRLG avalabltes as well as the lnk avalabltes. Consequently, the avalablty of path p can be calculated as: ;srlg p (1 π ) l p 3 The partally lnk dsjont varant follows analogously. a 2d A l (11)

where ;srlg p (1 π ) n Eq. (11) s the contrbuton of all the traversed SRLGs, whle l p A l s the avalablty of path p under the condton that all ts traversed SRLGs do not fal. For example, n Fg. 6, suppose there are three SRLGs n the network wth falure probabltes 0.1, 0.4 and 0.2, respectvely, and all the lnks have avalablty 0.9. We calculate the avalablty of path s a b t, whch traverses 2 SRLGs (srlg 1 and srlg 3 ): The probablty that both srlg 1 and srlg 3 do not fal s (1 0.1) (1 0.2). Under ths condton, all the lnks on path s a b t have avalablty 0.9 and therefore path s a b t has a total avalablty of (1 0.1) (1 0.2) (0.9) 3 = 0.52488. relable path s avalablty s 0.72 0.4 = 0.288. However, the optmal soluton s path s-b-t wth avalablty 0.45 n the physcal layer. The reason s that (s, a) and (a, t) n the logcal layer share the same lnk (s, b), whch leads to a lower avalablty value. Fg. 7(b) shows a smlar example wth Fg. 7(a), except that each lnk n the physcal layer has one addtonal wavelength number. In the absence of wavelength converson, t s requred that the lghtpath occupes the same wavelength on all lnks t traverses, whch s referred to as the wavelength-contnuty constrant n WDM-enabled networks. Now, suppose we want to fnd the most relable lghtpath from s to t, that obeys the wavelength-contnuty constrant. Clearly, f we are only aware of the lnks n the logcal layer, the result s path s-a-t wth avalablty 0.8 0.45 = 0.36. However, snce ths path s mapped to the path s-a-b-t n the physcal layer, t volates the wavelength-contnuty constrant. The optmal soluton s path s-a-t n the physcal layer va wavelength λ 1. Its avalablty s 0.8 0.3 = 0.24. Fg. 6: Avalablty calculaton n an SRLG network. Next, we wll prove that the sngle path varant of the ABPS problem n SRLG networks s NP-hard. To that end, we frst ntroduce the Mnmum Color Sngle-Path (MCSP) problem, whch s NP-hard [19]. Gven a network G(N, L), and gven the set of colors C = {c 1, c 2,..., c g } where g s the total number of colors n G, and gven the color {c l } of every lnk l L, the Mnmum Color Sngle-Path (MCSP) problem s to fnd a path from source node s to destnaton node t that uses the least amount of colors. Theorem 5: The ABPS problem s NP-hard n SRLG networks even for k = 1. Proof: Assume we have a network where all the lnks have avalablty 1 when ther SRLGs do not fal, and that there are g SRLGs wth the same falure probablty 1 g. Hence, a path s avalablty s only determned by the number of SRLGs t traverses. If we denote one SRLG by one partcular color, then the sngle-path ABPS problem n SRLG networks can be reduced to the MSCP problem. VI. MULTI-LAYER NETWORKS In mult-layer (e.g., IP-over-WDM) networks or overlay networks, the abstract lnks n the logcal layer are mapped to dfferent physcal lnks n the physcal layer. In ths context, two or more abstract lnks that contan the same physcal lnks may have correlated avalabltes or falure probabltes. Moreover, usually only the lnks n the logcal layer are known n mult-layer networks. Let us frst consder the example of mult-layer networks shown n Fg. 7. In Fg. 7(a), the avalablty s labeled on each lnk n the physcal layer, and the lnks n the logcal layer are mapped to the lnks n the physcal layer wth the greatest avalablty. Suppose we want to fnd a most relable unprotected path from s to t n Fg. 7(a). Snce we are only aware of the lnks n the logcal layer, we fnd that the most (a) Increasng jont avalablty 1 2 1 2 (b) Decreasng jont avalablty Fg. 7: A mult-layer network example. For any two lnks l and m n the logcal layer of a multlayer network, we denote ther avalabltes by A l and A m, respectvely. Let us use as the actual jont avalablty value of these two lnks, and then we derve that A l A m can be greater than A l A m (Fg. 7(a)), or less than A l A m (Fg. 7(b)), or equal to A l A m. For the latter case, we say that l and m are uncorrelated, otherwse we say that l and m are correlated: f A l A m > A l A m, l and m are ncreasng correlated, else l and m are decreasng correlated. Analogously, the operator 1

can be used for more than two lnks. Next, we wll prove the NP-hardness of the ABPS problem for k = 1 n mult-layer networks. Theorem 6: The ABPS problem s NP-hard n mult-layer networks even for k = 1. Proof: When all the lnks are uncorrelated n multlayer networks, the ABPS wth k = 1 problem s solvable polynomal tme. SRLG networks can be regarded as a specal case of ncreasng correlaton n mult-layer networks, snce the lnks that share at least one common SRLG group (denote ths ( lnk set by L ) wll have a greater avalablty than (1 π ) A l ). Snce the sngle-path ABPS l L ;srlg l problem n SRLG networks s NP-hard, as we proved n Theorem 5, the sngle-path ABPS problem n mult-layer networks s also NP-hard. VII. HEURISTIC AND EXACT ALGORITHMS A. Heurstc Algorthm Algorthm 1 (G, s, t, δ, k, I) 1: Fnd one shortest path p 1, return t f the avalablty requrement s satsfed, otherwse go to Step 2. 2: ps 1, H p 1, P H, P b and Q 3: Whle ps k 4: P H 5: For each path ap P 6: P b P and counter 0 7: Whle counter I do 8: Randomly remove one lnk (u, v) ap and fnd one shortest path ψ u v from u to v. 9: If t succeeds then 10: Replace (u, v) wth ψ u v n ap, denote t as ap 11: P b. Remove(ap), P b. Add(ap ), ap ap 12: Fnd another lnk-dsjont path p 2 wth P b. 13: Return {p 2 } P b f δ s met. 14: For each lnk (u, v) ap 15: If ts avalablty s at least δ then 16: Q. Add((u, v)) 17: whle (Q ) do 18: (u, v) EXCT-MIN(Q) 19: Fnd a path p 3 whch shares (u, v) wth ap. 20: If (p 3 / P b ) and {p 3 } P b satsfy δ then 21: Return {p 3 } P b 22: else H Max Avalablty{H, {p 3 } P b } 23: counter counter + 1. 24: ps ps + 1 Our heurstc, called Mn-Mns Algorthm () to solve the ABPS problem n generc networks, SRLG networks and mult-layer networks, s presented n Algorthm 1. Snce we want to use as least (and no more than k) lnk-dsjont paths to satsfy the requested avalablty, we gradually ncrease the number of paths. The pseudo code to solve the ABPS problem n mult-layer networks s smlar to the one for generc networks, except for the path avalablty calculaton. Also, the pseudo code to solve the ABPS problem n SRLG networks s smlar to the one n generc networks, and we wll specfy the dfferences later. In what follows, we explan each step of the heurstc algorthm. We assgn lnk l L wth the weght of log(a l ) ( log( SR l(1 π ) A l ) for SRLG networks) n. If a shortest path (represented by p 1 ) n Step 1 fals to satsfy the avalablty requrement, we keep t as the ntal path flow. In Step 2, we use ps to record the number of already found lnk-dsjont paths. Intally ps s set to 1. H stores the already found ps lnk-dsjont paths, and t s ntally assgned p 1. Whle ps s no greater than k, Steps 3-24 contnue fndng a soluton. In Step 4, we assgn to P the already found paths H. Based on P, from Step 5 to Step 23, we each tme select one path ap from path set P. We also use a varable, denoted by counter n Algorthm 1, to record the number of teratons. Intally, counter s set to 0. As long as the number of teratons s less than an nput value I, Steps 7-23 proceed fndng a soluton based on ap and path set P b. The (sub)path from u to v found by the algorthm s denoted by ψ u v. In Step 8, we randomly remove one lnk (u, v) from ap, and we apply a shortest path algorthm from u to v to obtan a path ψ u v. By concatenatng subpath ψ u v and the lnks of path ap except for (u, v), we obtan a new path ap. Further, by substtutng ap wth ap n P b, we have a new path set P b. After that, the algorthm tres to fnd P b s fully lnkdsjont path n Step 12. When solvng the ABPS problem n SRLG networks, snce each SRLG only contrbutes once to the path avalablty calculaton, the lnk l s weght s set to log( {SR l \SR c } (1 π ) A l ) before runnng a shortest path algorthm n Step 12 (also the same for Step 19), where SR c are the common traversed SRLGs between lnk l and path set P b. If t fals to fnd p 2 or {p 2 } P b cannot satsfy the avalablty requrement, the algorthm tres to fnd a path whch s partally lnk dsjont wth ap (n Steps 13-22). The general dea s that we frst use a queue Q to store the lnks n ap whose avalablty s no less than δ n Steps 14-16. After that, as long as Q s not empty n Steps 17-22, each tme the lnk wth the greatest avalablty n Q s extracted as the unprotected lnk (represented by (u, v)), and then we remove all the lnks traversed by ap except for (u, v). Subsequently, we fnd one shortest path ψ s u from s to u (f t exsts), and fnd another shortest path ψ v t from v to t (f t exsts). By concatenatng ψ s u, (u, v) and ψ v t, we can get a new path p 3, whch s partally lnk dsjont wth ap. If a and b denote dfferent sets of k > 1 lnk-dsjont paths, the functon Max Avalablty(a, b) n Step 22 returns the one wth greater avalablty. The tme complexty of can be computed as follows. Step 1 has a tme complexty of O(N log N + L). From Step 3 to Step 24, there are at most O(I) + O(2I) + + O(kI) = O(k 2 I) teratons before the algorthm termnates. Steps 14-16 have a tme complexty of O(N) snce a path contans at most N 1 lnks and therefore Steps 17-22 consume

O(N(N log N + L)) tme. Fnally, the whole tme complexty of s O(k 2 IN(N log N + L)). B. Exact INLP Formulaton In ths subsecton, we present an exact Integer Non-Lnear Program (INLP) to solve the ABPS problem n generc, SRLG and mult-layer networks. We frst solve the ABPS problem n generc networks and start by explanng the requred notaton and varables. INLP notaton: r(s, t, δ): Traffc request, wth source s, destnaton t and requested avalablty δ. A,j : Avalablty of lnk (, j). g : The total number of SRLGs. π,j m : The falure probablty of the m-th SRLG f lnk (, j) belongs to t, otherwse t s 0. INLP varable: P r,u,j : Boolean varable equal to 1 f lnk (, j) s traversed by path u (1 u k) for request r; 0 otherwse. Flow conservaton constrants: (,j) L N P r,u,j (j,) L 1 u k Avalablty constrant: k u=1 (,j) L P r,u j, = ( 1 P r,u,j + P r,j A,j ) ( mn 1 P r,u,j + P r,u,j 1, 1, 0, = s = t otherwse ) r,v A,j, 1 P,j + P r,v,j A,j (12) 1 u<v k (,j) L + + ( 1) k 1 r,u mn (1 P,j + P r,u,j A,j) δ (13) 1 u k (,j) L When both the flow conservaton constrant (Eq. (12)) and the avalablty constrant (Eq. (13)) are satsfed, an optmal soluton s found by the INLP, otherwse there s no soluton. There s no objectve (needed) n the proposed INLP, but one could nclude the objectve of mnmzng the number of paths (or lnks) used. Eq. (12) accounts for the flow conservaton for each of the at most k paths. For a partcular u th path (1 u k), t ensures that () for the source node s of request r, the outgong traffc for each request s 1; () for the destnaton node t of request r, the ncomng traffc s 1; and () for an ntermedate node whch s nether source nor destnaton, ts ncomng traffc s equal to the outgong traffc. Eq. (13) ensures that ether the found sngle unprotected path or the (partally) lnk-dsjont paths should have avalablty no less than δ, accordng to the avalablty calculaton of k lnkdsjont paths n Eqs. (3) and (5). Snce the overlapped lnk s avalablty n the partally lnk-dsjont calculaton accordng to Eq. (5) can only be counted once, we take the mnmum value of the varables P r,u,j for each lnk and then take the product over all the lnks for (partally) lnk-dsjont paths. We also note that Eq. (13) can smultaneously calculate the avalablty of the fully lnk dsjont varant, partally lnk dsjont varant and the unprotected varant. For nstance when k = 2, Eq. (13) becomes: or (1 P r,1,j + P r,1,j A,j) + (1 P r,2,j + P r,2,j A,j) (,j) L (,j) L mn(1 P r,1,j + P r,1,j A,j, 1 P r,2,j + P r,2,j A,j) δ (14) (,j) L When P r,1,j = P r,2,j for all (, j) L, Eq. (14) s equal to (1 P r,1,j + P r,1,j A,j) δ (,j) L (,j) L (1 P r,2,j + P r,2,j A,j) δ whch s the avalablty constrant for a sngle unprotected path. To solve the ABPS problem n SRLG networks, we need to slghtly modfy Eq. (13) (and keep flow conversaton constrants Eq. (12) the same) by usng 1 m g k mn u=1 ( ) r,u 1 mn (1 P,j + P r,u,j (,j) L πm,j ) to multply the left sde of Eq. (13), whch s the non-falure probablty of the SRLGs whch at most k lnk-dsjont paths have traversed. In mult-layer networks, the avalablty of a subset of lnks may not be equal to the product of ther avalabltes. Therefore, we need one more functon f( L ), whch can return the jont avalablty of a subset of lnks L n mult-layer networks. The parameter of ths functon s a 0/1 lnk vector whch contans L elements, where 1 denotes the lnk s present to be calculated and 0 means t s not. Consequently, to solve the ABPS problem n multlayer networks exactly, we could replace the operator wth f() n Eq. (13), and keep all the other notaton and constrants the same as for the generc networks case. A. Smulaton Setup VIII. SIMULATION RESULTS Fg. 8: USA carrer backbone network. We conduct smulatons on two networks, one s USANet, dsplayed n Fg. 8, whch s a realstc carrer backbone network consstng of 24 nodes and 43 lnks, and the other s GÉANT, shown n Fg. 9, whch s the pan-european communcatons nfrastructure servng Europe s research and

1 8 31 11 32 16 12 15 2 23 30 17 38 7 18 29 4 22 19 14 28 21 3 5 6 10 24 33 27 37 Fg. 9: GÉANT pan-european research network. educaton communty consstng of 40 nodes and 63 lnks. The smulaton deals wth the ABPS problem n generc, SRLG and mult-layer networks. For generc networks, we assume the avalablty of fber lnks s dstrbuted among the set {0.99, 0.999, 0.9999}, wth a proporton of 1:1:2. Based on the same lnk avalabltes, n SRLG networks we assume that there are n total 5 SRLG events wth the falure probabltes 0.001, 0.002, 0.003, 0.004 and 0.005, respectvely. Each lnk has randomly been assgned to at most 3 SRLG events. Based on the same (ndvdual) lnk avalabltes as for generc 1 networks, n mult-layer networks, 3 of lnks are ncreasng correlated, and t follows that A l A m A n = max(a l, A m,, A n ); 1 3 of lnks are decreasng correlated, and t follows that A l A m A n = (A l A m A n ) 2 ; and the other 1 3 of lnks are uncorrelated. For all these three networks, snce we want to compare the ablty of fndng paths for the algorthms, the capacty s set to nfnty. We vary the number of traffc requests from 100 to 1000. The source and destnaton of each request are randomly selected, and each request has nfnte holdng tme. The requested avalablty ncludes two cases: () general avalablty requrement case: the avalablty s randomly dstrbuted among the set {0.98, 0.99, 0.995, 0.997, 0.999}; () hgh avalablty requrement case: the avalablty s randomly dstrbuted among the set {0.9995, 0.9996, 0.9997, 0.9998, 0.9999}, by whch we want to challenge the algorthm to fnd feasble paths under more dffcult condtons. Consderng the practcal tme complexty and the exstng proposed algorthms that only focus on fndng two lnk-dsjont paths, we choose k = 2. We set I n to be logn n these two networks (5 n USANet and 6 n GÉANT, respectvely). Under the same weght allocaton wth our algorthm, we compare the proposed heurstc and exact INLP wth two heurstcs: Two-step Relablty Algorthm () and Maxmal-Relablty Algorthm (), whch are proposed n [1]. frst calculates a shortest path, and then calculates (f t exsts) another shortest path after removng the lnks traversed by the frst path. apples 20 13 35 26 34 25 39 9 36 40 Suurballe s algorthm [20] to calculate a par of two lnkdsjont paths that have mnmum weght. Both algorthms frst apply a shortest path algorthm to check whether an unprotected path soluton exts. The smulaton s run on a desktop PC wth 3.00 GHz CPU and 4 GB memory. We use IBM ILOG CPLEX 12.6 to mplement the proposed INLP and C# to mplement the heurstc algorthms. B. Results We frst evaluate the performance of the algorthms n terms of Acceptance Rato (AR) n generc networks. Acceptance rato (AR) s defned as the percentage of the number of accepted requests over all the requests. We frst analyze the general avalablty requrement case: In USANet, all the algorthms acheved an AR of 1. We therefore omt the fgure of the general avalablty performance for USANet. However, ths s not the case for the GÉANT topology. From Fg. 10(a), we can see that the performance of all algorthms s under 0.95. Snce GÉANT s not as well connected as USANet s, some nodes n GÉANT only have degree one (e.g., nodes 3, 8, etc.), f a one-degree node becomes the source or the destnaton of a certan request, the request can only be served by partal protecton (or a sngle unprotected path). In ths context, a feasble path may not exst n GÉANT, whch wll result n blockng. In terms of performance, the INLP acheves the hghest AR. On the other hand, shows a hgher AR than the other two heurstcs and (Fg. 10(a)). For the hgh avalablty requrement scenaro (shown n Fgs. 10(b) and 10(c)), as expected, the AR of all these algorthms s lower than n the general avalablty requrement case. In ths scenaro, the INLP requres more tme to fnd a soluton, especally when a soluton does not exst. In order to let the INLP return the result n a reasonable tme, we set the tme lmt for t to serve one request to 50 mnutes. Due to ths reason, we can see that INLP has the lowest AR n USANet and often second hghest AR n GÉANT. Meanwhle, stll has the hghest AR n most of the cases. The tme lmt for the INLP s even more constranng n the case of SRLG networks, leadng to a very poor performance for SRLG networks. We have therefore omtted the results of the INLP n SRLG networks. Snce the optmal soluton rarely exsts n the hgh avalablty requrement case, we only provde the smulaton results for the heurstc algorthms n the general avalablty requrement case. Moreover, to have a far comparson, we compare our algorthms wth and a modfed [2], whch s a heurstc routng algorthm proposed for probablstc SRLG networks. Its man dea s that after fndng the frst shortest path, the remanng lnk weghts should be adjusted (We slghtly change ts lnk weght adjustment to be the same wth the Step 12 of for a farer comparson), and then to fnd another lnk-dsjont shortest path. Fg. 11 shows that the proposed heurstc algorthm stll acheves hgher AR than these two algorthms. Smlar to SRLG networks, the exact INLP n mult-layer networks s very tme consumng. We therefore omt the results of the INLP n mult-layer networks. Fg. 12 provdes the

0.95 0.96 0.75 Acceptance rato (AR) 0.9 0.85 0.8 0.75 INLP traffc demand Acceptance rato (AR) 0.94 0.92 0.9 0.88 INLP* traffc demand Acceptance rato (AR) 0.7 0.65 0.6 0.55 0.5 INLP* traffc demand (a) GÉANT (b) USANet (c) GÉANT Fg. 10: AR of four algorthms n generc networks: (a) general avalablty requrement. (USANet has been omtted snce all 4 algorthms always acheved an AR of 1.); (b)-(c) hgh avalablty requrement, * max 50 mns per request. Acceptance rato (AR) 0.4 0.35 0.3 0.25 0.2 traffc demand (a) USANet Acceptance rato (AR) 0.35 0.3 0.25 0.2 traffc demand (b) GÉANT Fg. 11: AR of the heurstc algorthms n SRLG networks for general avalablty requrement. results for all three heurstcs n the two networks for both general and hgh avalablty requrement scenaros. It can be seen that acheves the hghest AR compared to the other two heurstcs. TABLE I: Runnng tmes per request for four algorthms (ms). Networks INLP USA Generc (General δ) 10190 0.187 0.128 0.127 GÉANT Generc (General δ) 29896 0.558 0.143 0.142 USA Generc (Hgh δ) 79764 0.224 0.147 0.146 GÉANT Generc (Hgh δ) 135181 0.679 0.162 0.160 USA SRLG (General δ) > 3.6 10 7 0.461 0.136 0.161 GÉANT SRLG (General δ) > 3.6 10 7 0.663 0.167 0.196 USA Mult-layer (General δ) > 3.6 10 7 0.857 0.135 0.141 GÉANT Mult-layer (General δ) > 3.6 10 7 1.378 0.175 0.181 USA Mult-layer (Hgh δ) > 3.6 10 7 1.136 0.157 0.162 GÉANT Mult-layer (Hgh δ) > 3.6 10 7 1.679 0.262 0.223 Fnally, n Table I, we present the (average) runnng tmes per request for these four algorthms n generc, SRLG and mult-layer networks. It shows that the INLP s sgnfcantly more tme consumng than the three polynomal-tme heurstcs. On the other hand, has only a slghtly hgher runnng tme than and, but t pays off by havng a hgher AR as shown n Fgs. 10-12. Another observaton s that, for the same algorthm n the same network, the runnng tme s hgher for the hgh avalablty requrement case than n the general avalablty requrement case. IX. NETWORK VULNERABILITY ASSESSMENT A. Problem Defnton and Complexty Analyss Fndng the most vulnerable part of a network as well as the prevously consdered problem of avalablty-based path selecton are both mportant elements for network robustness. In ths secton, we study the (s,t) Network Vulnerablty Assessment (NVA) problem. That s, fnd one or a set of (equal-weght) network cuts whose falure probablty of the lnks n the cut s hghest. A network cut refers to a set of lnks, whose removal wll result n the dsconnecton of the network. Formally, the NVA problem can be defned as follows: Defnton 3: The (s,t) Network Vulnerablty Assessment (NVA) problem: Gven s a network G(N, L), and each lnk l L s assocated wth a falure probablty f l = 1 A l. Gven a source s and a target t, fnd an s t cut C for whch l C f l s maxmzed. In case there are multple cuts of hghest weght all of them should be returned. When the node par (s,t) s not specfed, we denote ths problem as the NVA problem, whch can be solved by solvng the (s,t) NVA problem at most N 1 tmes. Therefore, these two problems share the same hardness. We use log(f l ) for the weght of lnk l n the network. In generc networks, the NVA problem can be solved by fndng all the mn-cuts n O(L 2 N + N 2 L) tme accordng to [21]. On the other hand, we wll prove that the NVA problem n SRLG networks s NP-hard. Recall that n SRLG networks, ntroduced n Secton V, two types of falures/avalabltes should be ncorporated, namely Shared-Rsk Lnk Group (SRLG) falures and sngle lnk falures/avalabltes, therefore the probablty that all the lnks belongng to path p fal smultaneously (denoted by F (p)) can be calculated as: 1 (1 π ) + f l 1 (1 π ) l p ;srlg p ;srlg p l p f l (15)

Acceptance rato (AR) 1 0.99 0.98 0.97 0.96 traffc demand (a) USANet Acceptance rato (AR) 0.78 0.76 0.74 0.72 0.7 0.68 0.66 traffc demand (b) GÉANT Acceptance rato (AR) 0.65 0.6 0.55 0.5 0.45 traffc demand (c) USANet Acceptance rato (AR) 0.5 0.4 0.3 0.2 0.1 0 traffc demand (d) GÉANT Fg. 12: AR of heurstcs n mult-layer networks: (a)-(b) general avalablty requrement; (c)-(d) hgh avalablty requrement. Please note that Eq. (15) s dfferent from the falure probablty of a path n SRLG networks: 1 Eq. (11). Before we prove the NP-hardness of the NVA problem, let us frst study the Maxmum Color Path Selecton (MCPS) problem. Contrary to the MCSP problem, the MCPS problem s to fnd a sngle path such that t uses the largest amount of colors. Theorem 7: The Maxmum Color Path Selecton (MCPS) problem s NP-hard. Proof: We dstngush two cases, namely, (1) all the lnks have exclusve colors,.e., there does not exst any color that s shared/overlapped by two or more lnks, and (2) two or more lnks may contan the same color(s). Case (1): The MCPS problem s equvalent to the NP-hard Longest Path problem [18], whch s to fnd a path from s to t such that ts weght s maxmzed. Fg. 14: Example network. x 1 x 2 x 3 x n s y 1 y 2 y 3 y n-1 t z 1 z 2 z 3 z n Fg. 13: Reducton of the MCPS problem to the Dsjont Connectng Paths problem. Case (2): We frst ntroduce the Dsjont Connectng Paths problem [18]. Gven a network G(N, L), and a collecton of dsjont node pars (s 1, t 1 ), (s 2, t 2 ),..., (s z, t z ), does G contan z mutually lnk-dsjont paths, one connectng s and t for each, 1 z? Ths problem s proved NP-hard when z 3, and we reduce the MCPS problem to t. In Fg. 13, assume that there are n total g colors and lnks (a, b) and (c, d) share 0 < x < g common colors, but they together contan g dstnct colors. Except for these two lnks, the other lnks are assgned 0 < y < g colors, but there does not exst two or more lnks contanng g dstnct colors. In ths context, fndng a path from s to t wth the largest amount of colors s equvalent to fndng three mutually lnk-dsjont paths between three node pars (s, a), (b, c) and (d, t). Theorem 8: The Network Vulnerablty Assessment (NVA) problem n SRLG Networks s NP-hard. Fg. 15: Reducton of the (s,t) NVA problem n SRLG networks to the MCPS problem. Proof: It s equvalent to prove that the (s,t) NVA problem s NP-hard. In Fg. 14, assume all lnk avalabltes are 1 and all lnks have non-zero SRLG falure probabltes, except for lnks (x, x +1 ) and (z, z +1 ) whch have 0 SRLGs, where 1 n 1. Assume there are g SRLG events n total, and each SRLG event occurs wth a probablty of 1 g. In ths context, for a path p wthout lnks (x, x +1 ) and (z, z +1 ), accordng to Eq. (15), F (p) can be calculated as: 1 (1 1 g )m (16) where m s the total number of dstnct SRLGs traversed by p. Therefore, to maxmze Eq. (16) one needs to maxmze m,.e., to fnd a path wth the greatest probablty that all ts traversed lnks fal smultaneously. Ths s equvalent to fndng a path havng the largest number of dstnct SRLGs. We want to solve the (x 1,z 1 ) NVA problem. Based on Fg. 14, we frst

derve Fg. 15 wth the same nodes except that we add one more node s. We regard s = y 0, and t = y n. The lnk weght n Fg. 15 s set as follows: (y 1, x ) and (y 1, z ) have 0 SRLGs, whle (x, z ) and (z, y ) have the same SRLGs as n Fg. 14, where 1 n. In Fg. 15, we are asked to solve the MCPS problem from the source s to the destnaton t. Snce we want to fnd a cut that separates x 1 and z 1, any cuts n the form of (x, y ) and (y, z ), where 1 n are not part of the optmal soluton. Moreover, consderng the lnk n the form of (x j, x j+1 ) or (y j, y j+1 ) has 0 SRLGs and sngle lnk avalablty 1, whch means ts avalablty s 1 or falure probablty s 0, t cannot lead to the optmal soluton as well. Based on the above observatons, any feasble cut C should contan one lnk ether (x, y ) or (y, z ), for all 1 n. We prove n the followng that the (s,t) NVA problem n Fg. 14 can be reduced to the MCPS problem n Fg. 15 n polynomal tme, where, for smplcty, we assume only one optmal cut exsts n the (s,t) NVA problem. (s,t) NVA to MCPS: An optmal soluton of the (s,t) NVA problem should be composed of ether (x, y ) or (y, z ), where 1 n. Let C NV A reflect the set of lnks n the optmal soluton. Because C NV A has the largest amount of dstnct SRLGs, C NV A together wth (s, x 1 ) or (s, z 1 ) forms a path from s to t wth the maxmum number of SRLGs. MCPS to (s,t) NVA: Let R MCP S denote the set of lnks n the optmal soluton of the MCPS problem. Because R MCP S has the largest amount of SRLGs, let C NV A = R MCP S \{(y, x +1 ), (y, z +1 )}. Consderng that the lnks (y, x +1 ) and (y, z +1 ) have 0 SRLGs, C NV A also has the largest amount of SRLGs. Therefore solvng the MCPS problem yelds a soluton to the (s,t) NVA problem. Theorem 9: The NVA problem n mult-layer networks s NP-hard. Proof: For any two lnks l and m n a mult-layer network, we have that A l A m = (1 f l ) (1 f m ) = 1 f m f l + f l f m. Smlar to Theorem 6, we can also prove that t s NP-hard to fnd one path from the source to the destnaton n mult-layer networks such that the probablty that all ts lnks fal s maxmzed. Subsequently, we can prove that the NVA problem n mult-layer networks s NP-hard, whch follows analogously wth Theorem 8. We therefore omt the detals. X. CONCLUSION The avalablty of a connecton represents how relably a connecton can carry data from a source to a destnaton. In ths paper, we have frst studed the Avalablty-Based Path Selecton (ABPS) problem, whch s to establsh a connecton over at most k (fully or partally) lnk-dsjont paths, such that the total avalablty s no less than δ (0 < δ 1). We have proved that, n general, the ABPS problem s NP-hard and cannot be approxmated n polynomal tme for k 2, unless P=NP. We have further proved that n SRLG networks and mult-layer networks, even the sngle-path (k = 1) varant of the ABPS problem s NP-hard. We have proposed a polynomal-tme heurstc algorthm and an exact INLP to solve the ABPS problem n generc networks, SRLG networks and mult-layer networks. Va smulatons, we have found that our heurstc algorthm outperforms two exstng algorthms n terms of acceptance rato wth only slghtly hgher runnng tme. On the other hand, the runnng tme of the exact INLP s sgnfcantly larger (by several orders of magntude) than the runnng tme of the heurstc algorthms. Fnally, we have proved that the Network Vulnerablty Assessment (NVA) problem, whch s to fnd a cut of the network for whch the falure probablty of all ts lnks s hghest, s solvable n polynomal tme n generc networks, but NP-hard to solve n SRLG networks and mult-layer networks. REFERENCES [1] Q. She, X. Huang, and J. Jue, How relable can two-path protecton be? 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[19] S. Yuan, S. Varma, and J. Jue, Mnmum-color path problems for relablty n mesh networks, INFOCOM 2005, pp. 2658 2669, 2005. [20] J. W. Suurballe and R. E. Tarjan, A quck method for fndng shortest pars of dsjont paths, Networks, vol. 14, no. 2, pp. 325 336, 1984. [21] H. Nagamoch, K. Nshmura, and T. Ibarak, Computng all small cuts n an undrected network, SIAM Journal on Dscrete Mathematcs, vol. 10, no. 3, pp. 469 481, 1997. APPENDIX A SINGLE-LINK FAILURE In ths secton, we assume that only 1 lnk n the network can fal at a tme. Apart from that, we also assume that two successve falures occur wth a tme dfference longer than the traversal tme of any path n the network. For ths scenaro, we present a polynomal-tme algorthm to solve the ABPS problem. More specfcally, n Algorthm 2, when k = 1, n Steps 1-2, we elmnate the lnks wth avalablty less than δ, such that we obtan a new graph G where each lnk has avalablty at least δ. Subsequently, by runnng Djkstra s shortest path algorthm on G from s to t, we can solve the ABPS problem for k = 1. When k 2, f the optmal soluton conssts of k fully lnk-dsjont paths, then 2 fully lnk-dsjont paths also exst and under the sngle-lnk falure scenaro have avalablty s 1, whch s optmal. Hence, by applyng Suurballe s algorthm [20] n Step 3, the soluton can be found, f t exsts. When the optmal soluton conssts of k partally lnkdsjont paths, then 2 partally lnk-dsjont paths are also enough. The reason s that the avalablty of partally lnkdsjont paths s decded by one unprotected lnk (say l). Hence, t suffces to fnd k = 2 partally lnk-dsjont paths. In Steps 4-5, for each lnk (u, v) whose avalablty s no less than δ, we create another (parallel) lnk between u and v wth the same avalablty. After that, we call Suurballe s algorthm [20] from s to t. Snce n the optmal soluton the unprotected lnk has avalablty at least δ, by creatng the parallel lnks whose avalablty s at least δ, the paths p 1 and p 2 returned by Suurballe s algorthm [20] are two fully lnk-dsjont paths. After that, f p 1 and p 2 traverse the parallel lnks, we then merge these two lnks nto one lnk. Ths knd of lnk reflects the unprotected lnk n the optmal soluton. On the other hand, the lnks whose avalablty s less than δ are protected n the returned soluton because of the correctness of Suurballe s algorthm. Therefore, an optmal soluton can be found by algorthm 2. APPENDIX B HARDNESS OF THE ABBPS PROBLEM FOR K=2 For some varants, lke the fully lnk dsjont case, the ABBPS problem s polynomally solvable 4, however t s NPhard n ts general settng. Theorem 10: The partally lnk-dsjont ABBPS problem for k = 2 s NP-hard. Proof: We provde a proof for when k = 2. As t s shown n Fg. 16, assume we are gven a path (denoted by GP) s-a-bc-d-t wth the avalablty labeled on each lnk and all the other Algorthm 2 ABPSSngleLnkFalure(G, s, t, δ, k) 1: Elmnate the lnks wth avalablty less than δ on G, thereby obtanng a new graph G. 2: Run Djkstra s algorthm on G from s to t. If the soluton s found then return the result; Else f k > 1, contnue; Otherwse output there s no soluton. 3: Run Suurballe s algorthm [20] on G from s to t. Return the result f the soluton s found, otherwse contnue. 4: Create another (parallel) lnk between u and v wth the same avalablty A (u,v), for each (u, v) L f A (u,v) δ. The graph s denoted as G. 5: Run Suurballe s algorthm [20] on G from s to t. Return the result f the soluton s found, otherwse output there s no soluton. Fg. 16: Reducton of the ABBPS problem wth partally lnk dsjont paths to the Dsjont Connectng Paths problem. lnks have an avalablty of 1. We now want to fnd a partally lnk-dsjont path wth GP such that ther combned avalablty s no less than 1. Snce the requested avalablty s 1, only lnk (a, b) and lnk (c, d) can be unprotected n an optmal soluton. Suppose that when lnk (a, b) s elmnated, there do not exst paths from node s to nodes b, c, d and t, and when lnk (c, d) s elmnated, there are no paths from node b to nodes d and t. In ths context, to solve the partally lnk-dsjont ABBPS problem for k = 2, both (a, b) and (c, d) should be unprotected n the optmal soluton. Ths s equvalent to fndng three pars of lnk-dsjont paths between node pars (s, a), (b, c) and (d, t) (.e., the Dsjont Connectng Paths problem). Hence, solvng the partally lnk-dsjont ABBPS problem for k = 2 yelds a soluton to the NP-hard Dsjont Connectng Paths problem. 4 Trvally, by lookng for an unprotected path wth maxmum avalablty n the network where all the lnks from the prmary path are excluded.