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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 147

On the Complexity of and Algorithms for Findingthe Shortest Path With a Disjoint Counterpart

Dahai Xu, Member, IEEE, Yang Chen, Student Member, IEEE, Yizhi Xiong, Chunming Qiao, Member, IEEE, andXin He, Member, IEEE

AbstractFinding a disjoint path pair is an important compo-nent in survivable networks. Since the traffic is carried on the ac-tive (working) path most of the time, it is useful to find a disjointpath pair such that the length of the shorter path (to be used asthe active path) is minimized. In this paper, we first address such aMin-Min problem. We prove that this problem is NP-complete ineither single link cost (e.g., dedicated backup bandwidth) or duallink cost (e.g., shared backup bandwidth) networks. In addition,it is NP-hard to obtain a -approximation to the optimal solu-tion for any 1. Our proof is extended to another open ques-tion regarding the computational complexity of a restricted versionof the Min-Sum problem in an undirected network with ordereddual cost links (called the MSOD problem). To solve the Min-Minproblem efficiently, we introduce a novel concept called conflictinglink set which provides insights into the so-called trap problem, anddevelop a divide-and-conquer strategy. The result is an effectiveheuristic for the Min-Min problem called COnflicting Link Exclu-sion (COLE), which can outperform other approaches in terms ofboth the optimality and running time. We also apply COLE to theMSOD problem to efficiently provide shared path protection andconduct comprehensive performance evaluation as well as compar-ison of various schemes for shared path protection. We show thatCOLE not only processes connection requests much faster than ex-isting integer linear programming (ILP)-based approaches but alsoachieves a good balance among the active path length, bandwidthefficiency, and recovery time.

Index TermsBandwidth sharing, dynamic provisioning, graphtheory, optimization, protection, survivability.

I. INTRODUCTION

SURVIVABILITY is a critical design problem forhigh-speed networks. To protect a mission-critical connec-tion from a single link (node) failure, a common solution is tofind a link (node) disjoint pair of paths from a source (ingress)node to a destination (egress) node. In such path protectionapplication, traffic is routed along one path, called the active(working) path (AP) unless it is affected by a link (node)

Manuscript received December 9, 2003; revised September 13, 2004, March20, 2005, and April 22, 2005; approved by IEEE/ACM TRANSACTIONS ONNETWORKING Editor J. Yates. This work was supported in part by the NationalScience Foundation (NSF) under Contracts ANIR 0208331 and CCR-0309953.A preliminary version of this paper entitled On Finding Disjoint Paths inSingle and Dual Link Cost Networks was presented at the IEEE INFOCOM2004, Hong Kong.

D. Xu, C. Qiao, and X. He are with the Department of Computer Science andEngineering, State University of New York at Buffalo, Buffalo, NY 14260 USA(e-mail: dahaixu@cse.buffalo.edu; qiao@cse.buffalo.edu; xinhe@cse.buf-falo.edu).

Y. Chen is with the College of Computing, Georgia Institute of Technology,Atlanta, GA 30332 USA (e-mail: yangchen@cc.gatech.edu).

Y. Xiong is with Cedars-Sinai Medical Center, Los Angeles, CA 90048 USA(e-mail: xiongy@cshs.org).

Digital Object Identifier 10.1109/TNET.2005.863451

failure, upon which, the traffic is re-routed along the other path,called the backup path (BP).

In this paper, we focus on the problem of finding a pair of link(node) disjoint AP and BP in an on-line setting. More specifi-cally, we assume that not all requests for survivable connectionsarrive at the same time, and a decision as to how to satisfy a re-quest (if possible at all) has to be made without knowing whichrequests will arrive in the future, and for the sake of guaranteedquality of service (QoS) (e.g., in terms of both delay and loss),without being able to rearrange the way existing connectionsare established, including the existing BPs. The latter restric-tion is placed to avoid the overhead and delay involved in co-ordinating the rearrangement of existing connections, and moreimportantly, avoid data loss due to a link (or node) failure onan AP while its corresponding BP is being rearranged, for ex-ample. Techniques for and potential benefits of rearranging theexisting connections (either both APs and BPs or just BPs) as apart of network maintenance have been described in [1][3].

In the above on-line path protection application, one of thechallenges is how to allocate minimal amount of network re-source (e.g., bandwidth) for a connection requesting, say,units of guaranteed bandwidth. In the case of so-called dedi-cated protection, additional units of bandwidth are to be allo-cated along (each link of) both the chosen AP and BP. Hence, theproblem of minimizing the sum of the additional units of band-width to be allocated along the AP and BP, hereafter called theMin-Sum problem, can be easily solved using, e.g., the ShortestPair of Path (SPP) algorithm [4], [5] (by treating as the costof using each link), which has a polynomial time complexity.However, in the case of so-called shared path protection, nosuch polynomial time algorithm exists today which can solve theMin-Sum problem optimally. Intuitively, this is because while

additional units of bandwidth has to be allocated on a chosenAP, the exact number of additional units of bandwidth to be al-located on a link along BP (called backup bandwidth) dependson how the corresponding AP as well as the existing APs andBPs are established. More specifically, denote by the ad-ditional units of backup bandwidth needed on a link, we have

because of the possible sharing of the backupbandwidth among this BP and other existing BPs using the samelink. In fact, in spite of extensive research [6][9], one of theopen questions is whether the Min-Sum problem is NP-com-plete (NPC) in undirected network, where each link has two or-dered costs and the cost incurred by a BP is always a fractionof the cost incurred by an AP. In addition, given the fact thatthe AP is used most of the time while BP is used only when APfails, it is natural to ask the question of how to minimize the AP.

1063-6692/$20.00 2006 IEEE

148 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006

In this paper, as one of our contributions, we answer this openquestion by proving this instance of the Min-Sum problem (tobe called MS problem with ordered dual costs, or MSOD) to beNPC (and hence the general Min-Sum problem is also NPC).We also address the second open problem, which is to find apair of link (node) disjoint paths in (directed or undirected) net-works such that the length of the shorter path (to be used as anAP), in terms of the number of hops or geographical distance ithas, is minimized. We call it a Min-Min problem, as opposedto the Min-Sum problem above. The Min-Min problem is alsodifferent from the so-called Min-Max problem studied in [10]and [11] where the objective was to minimize the length of thelonger one of the two paths (to be used as a BP).

Specifically, as another major contribution, we prove that theMin-Min problem above is also NPC, whether each link in adirected or undirected network has only one cost (as in the caseof dedicated protection) or dual costs (as in the case of sharedpath protection).1 Furthermore, we prove that even the problemof obtaining a -approximation solution to the Min-Minproblem is NPC for any , which means there is andwill be no polynomial time heuristic algorithm that can achieveguaranteed near-optimal performance as long as .

In addition to the above theoretical results, this paper providesa novel and efficient heuristic algorithm to obtain practicallynear-optimal solutions to the above two problems that we proveto be NPC. More specifically, we propose a heuristic calledCOnflicting Link Exclusion (COLE) for the Min-Min problem,and extend it for the Min-Sum problem in shared path protec-tion. Our comprehensive performance evaluation and compar-ison studies reveal that COLE can achieve better results thanother heuristic algorithms for the Min-Min problem. Further-more, a simple application of COLE to the MSOD problemcan achieve near-optimal performance for shared path protec-tion when compared to its counterpart that solves integer linearprogramming (ILP) formulations using the branch and boundtechniques, whose time complexity can be prohibitively highfor large networks.

The rest of this paper is organized as follows. Section IIdescribes related work, and provides the motivations for thiswork, especially the Min-Min problem. In Section III, wefirst prove that the Min-Min problem is NPC, and so is its

-approximation problem, as well as the MSOD problemin undirected networks. Section IV describes the proposedCOLE heuristic for the Min-Min problem. In Section V, wepresent numerical results from the comparison between theproposed heuristics and other existing heuristics as well as ILPbased approaches. Finally, Section VI concludes this paper.Frequently used acronyms are listed at the end of the paper foreasy reference.

II. PRELIMINARY

In this section, we first describe related work on finding a link(node) disjoint path pair and then provide the motivation for ourwork.

1Since having a single cost network is a special case of having a dual-costnetwork, as long as we can prove that the Min-Min problem in a single-costnetwork is NPC, the Min-Min problem in a dual-cost network is also NPC.

A. Related Work

As mentioned earlier, in solving the problem of finding a linkor node disjoint pair of AP and BP, one can model a network aseither a single (link) cost graph or a dual (link) cost graph. Inthe former, the cost of using a link by AP, denoted by , is thesame as that by BP, denoted by . In the latter, could bedifferent from, and often higher than (due to backup band-width sharing as mentioned earlier). In addition, we will con-sider the most general dual-cost network model where each linkis associated with a vector of two costs, ( , ), such thatand can have an arbitrary relationship (e.g., it is possible that

). Accordingly, the single-cost model is a special caseof this dual-cost model. In the following discussion, we will usethe terms network and graph interchangeably.

Computational Complexity of Various Problems: Manyproblems related to finding a pair of link or node disjoint pathsin single cost networks have been studied [4], [5], [10][12].For instance, the Min-Sum problem mentioned earlier, in whichthe objective is to minimize the sum of the costs of the twopaths can be solved using a polynomial time algorithm calledSPP [4], [5]. Unlike Min-Sum, the Min-Max problem, whoseobjective is to minimize the length of the longer one of thetwo paths was proved to be NPC [10], [11]. However, as faras we know, no existing work has addressed what we call theMin-Min problem, in which the objective is to minimize thelength of the shorter one of the two paths. In later subsections,we will provide the motivations for the Min-Min problem aswell as its NP completeness proof.

While it is clear that the Min-Max problem in a more generaldual-cost network is NPC, it is not straightforward to show thatthe Min-Sum problem in a dual-cost network is also NPC (giventhat a polynomial time algorithm exists in a single-cost net-work). In [9], it was shown that in a network where the relation-ship between and on each link is arbitrary, the Min-Sumproblem is NPC. A more restricted version of the problem wasalso studied, where for each and every link, . Thisversion of the Min-Sum problem, referred to as the MS withordered dual cost (or MSOD) problem, is also NPC [7]. Re-cently, an even more restricted version of the MSOD problemwas studied, where for each and every link, , and

is a constant for all the links. It has been shown thatsuch a MS with uniformly ordered dual link costs (or MSOD-U)is NPC but only for a directed network [8]. It was left as anopen question as to whether the MSOD-U problem is NPC foran undirected network.2

Later, we will prove the MSOD-U problem is NPC for anundirected network. Our proof can be applied to a directed net-work, yielding a different proof from that in [8]. Note that,the MSOD-U problem is more restricted than the previouslymentioned two versions, namely, ordered but only nonuni-formly, and arbitrary (or unordered). Hence, our proof canalso be used to show that these versions are NPC. In partic-ular, we note that the proof of NP completeness of the Min-Sumproblem in [7] only applies to the nonuniformly ordered dual-

2Because the proof in [8] used a known NPC problem that is only applicableto a directed network in its reduction process, the proof cannot be extended toan undirected network.

XU et al.: ON THE COMPLEXITY OF AND ALGORITHMS FOR FINDING THE SHORTEST PATH WITH A DISJOINT COUNTERPART 149

TABLE ICOMPUTATIONAL COMPLEXITY OF VARIOUS PROBLEMS OF FINDING

TWO DISJOINT PATHS

cost case, where of a link to be used by a BP depends onhow its corresponding AP and other existing APs and BPs areestablished. However, our proof will also lead to the proof ofthe NP-completeness of a more general version of the Min-Sumproblem in the nonuniformly ordered case, where of a linkis independent of the other factors.

Table I summarizes the main results on the computationalcomplexity of various problems related to finding a link or nodedisjoint pair of paths, where our major contributions are high-lighted.

Algorithms: In this work, we focus on the algorithms usefulfor path protection. Note that, although SPP, which is an al-gorithm developed for single-cost networks, can be applied toshared path protection, the paths found by SPP are not optimalfor shared path protection in that the sum of the costs (band-width) of AP and BP is not minimum. Accordingly, several al-gorithms have been developed for shared path protection, whichessentially are solutions to the MSOD problem.

One typical class of algorithms develop ILP formulationswhose objective is to jointly optimize the selection of bothAP and BP for the MSOD problem, and then solve the ILPformulations using the branch-and-bound search techniques toachieve optimal results. Two representative schemes belongingto this class, which have previously undergone extensive quan-titative performance evaluation, are Sharing with CompleteInformation (SCI) [13] and Distributed Partial InformationManagement (DPIM) [14].

In addition to the above mentioned ILP formulation basedalgorithms, whose time complexity could be prohibitively highfor a large network, another class of algorithms for the MSODproblem use the active-path-first (or APF) heuristic [2], [15],[16]. In these APF-based heuristics, an AP is found first by usingthe Dijkstra algorithm (or any other shortest path algorithms)without considering the need to find a corresponding BP forthe time being, and the BP is found (again using the Dijkst...