 Research
 Open Access
Optimization problems in correlated networks
 Song Yang^{1},
 Stojan Trajanovski^{1} and
 Fernando A. Kuipers^{1}Email author
 Received: 24 July 2015
 Accepted: 7 January 2016
 Published: 22 January 2016
Abstract
Background
Solving the shortest path and mincut problems are key in achieving highperformance and robust communication networks. Those problems have often been studied in deterministic and uncorrelated networks both in their original formulations as well as in several constrained variants. However, in realworld networks, link weights (e.g., delay, bandwidth, failure probability) are often correlated due to spatial or temporal reasons, and these correlated link weights together behave in a different manner and are not always additive, as commonly assumed.
Methods
In this paper, we first propose two correlated link weight models, namely (1) the deterministic correlated model and (2) the (logconcave) stochastic correlated model. Subsequently, we study the shortest path problem and the mincut problem under these two correlated models.
Results and Conclusions
We prove that these two problems are NPhard under the deterministic correlated model, and even cannot be approximated to arbitrary degree in polynomial time. However, these two problems are solvable in polynomial time under the (constrained) nodal deterministic correlated model, and can be solved by convex optimization under the (logconcave) stochastic correlated model.
Keywords
 Shortest path
 Mincut
 Correlated networks
 Stochastic link weights
Background
Both the shortest path problem and the mincut problem are of great importance to various kinds of network routing applications (e.g., in transportation networks, optical networks, etc.). A traffic request can be routed in the most efficient way (e.g., with minimum delay) by computing a shortest path. On the other hand, the mincut problem arises in the context of network reliability, network throughput, etc. Fortunately, both of these problems are solvable in polynomial time for networks with independent additive link weights.

We propose two correlated link weight models, namely a deterministic correlated model and a stochastic correlated model.

We study the shortest path problem and the mincut problem under the deterministic correlated model, and we prove that both of them are NPhard and even cannot be approximated in polynomial time.

On the other hand, we also show that both the shortest path problem and the mincut problem are solvable in polynomial time under a (constrained) nodal deterministic correlated model.

To solve both problems under the proposed correlated models, we propose exact algorithms under the deterministic correlated model, and develop convex optimization formulations for the stochastic correlated model.
The remainder of this paper is organized as follows. “Correlated link weight models” section introduces our two correlated link weight models. In “Shortest paths in correlated networks” and “Minimum cuts in correlated networks” sections, we study the shortest path problem and mincut problem, respectively, for the proposed models and devise algorithms to solve them exactly. An overview of the related work is presented in “Related work” section and we conclude in “Conclusions” section.
Correlated link weight models
Shortest paths in correlated networks
Minimum cuts in correlated networks
Related work
Routing with correlated link weights
In a network with each link having multiple additive link weight metrics (e.g., delay, cost, jitter, etc.), the Quality of Service (QoS) routing problem is to find a path that satisfies a given constraints vector. Kuipers and Van Mieghem [25] study the QoS routing problem under correlated link weights. Another common source of correlation is SharedRisk Link Groups (SRLGs). Sometimes one SRLG can also be represented by one color, but they share the same meaning in terms of reliability. In this context, Yuan et al. [18] prove that the Minimum Color SinglePath (MCSiP) problem is NPhard. Yuan et al. also prove that finding two linkdisjoint paths with total minimum distinct amount of colors or least amount of coupled/overlapped colors is NPhard. Lee et al. [26] propose a probabilistic SRLG framework to model correlated link failures and develop an Integer Nonlinear Programming (INLP) formulation to find one unprotected path or two linkdisjoint paths with the lowest failure probability.
There is also some literature dealing with correlated routing problems in stochastic networks [27]. For example, in [28] only two possible states are assumed, which are congested and uncongested, and each state corresponds to a cost value. A probability matrix \(P^{u,v,y}_{a,b}\), which represents the probability that if (u, v) is in state a then (v, y) is in state b, is given. Two similar link weight models, called linkbased congestion model and nodebased congestion model, are proposed in [29]. Based on these models, [28, 29] define and solve the least expected routing problem, which is to find a path from the source to the destination with minimum expected costs. However, in [28, 29] there are only two possible states for each link and only the correlation of the adjacent links is known. We assume a more general (and different) stochastic correlated model, where as long as the links (not necessarily adjacent) are correlated, their joint CDF for allocating costs is known.
Mincut in conventional networks
The (s, t) mincut problem refers to partitioning the network into two disjoint subsets, such that nodes s and t are in different subsets and the total weight of the cut links is minimized. This problem can be solved by finding the maximum flow from s to t [30]. There is also a lot of work on the mincut problem with no specified node pairs (s, t). A summary and comparison of polynomialtime algorithms to solve the mincut problem can be found in [31]. The fastest algorithm to solve the mincut problem has a time complexity of \(O(L \log ^3 N)\) and was proposed by Karger [32]. Accordingly, the mincut problem can be tackled by solving at most \(N1\) times the (s, t) mincut problem.
Constrained maximum flow
As a dual of the mincut problem, the maximum flow problem in conventional networks is solvable in polynomial time [30]. However, this problem becomes NPhard if some constraints are imposed on the links. Suppose that negative disjunctive constraints indicate that a certain set of links cannot be used simultaneously for the optimal solution, while positive disjunctive constraints force at least one of a certain set of links to be present in the optimal solution. Pferschy and Schauer [33] prove that the maximum flow problems with both negative and positive disjunctive constraints are NPhard and do not admit a PolynomialTime Approximation Scheme (PTAS). For example, the disjunctive constraint corresponds to the correlated link weights, so the maximum flow problem in correlated networks is also NPhard and does not admit a PTAS. Assuming the link’s bandwidth and delay follow a logconcave distribution, Kuipers et al. [16] propose a polynomialtime convex optimization formulation to find the maximum flow in the socalled stochastic networks. When a delay constraint is imposed on each path, the maximum flow problem is NPhard. To solve it, Kuipers et al. [16] propose an approximation algorithm and a tunable heuristic algorithm.
Conclusions
In this paper, we have studied the shortest path problem and the mincut problem in correlated networks under two link weight models, namely (1) the deterministic correlated model and (2) the (logconcave) stochastic correlated model. We have proved that these two problems are NPhard under the deterministic correlated model, and cannot be approximated to arbitrary degree, unless P \(=\) NP. Subsequently, we have proposed exact algorithms to solve them. In particular, we have shown that both of them are solvable in polynomial time under a (constrained) nodal deterministic correlated model. For the stochastic correlated model, we have shown that these two problems can be solved by convex optimization.
Declarations
Authors’ contributions
This work is the result of a close joint effort in which all authors contributed almost equally to defining and shaping the problem definition, proofs, algorithms, and manuscript. SY, as the first author, took the lead in composing the first draft of the manuscript, while ST and FK edited it. As such, all authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Cui W, Stoica I, Katz RH. Backup path allocation based on a correlated link failure probability model in overlay networks. In: Proceedings of IEEE ICNP; 2002. pp. 236–45.Google Scholar
 Kuipers FA, Dijkstra F. Path selection in multilayer networks. Comput Commun. 2009;32(1):78–85.View ArticleGoogle Scholar
 Savage S, Collins A, Hoffman E, Snell J, Anderson T. The endtoend effects of internet path selection. ACM SIGCOMM Comput Commun Rev. 1999;29:289–99.View ArticleGoogle Scholar
 Kostić D, Rodriguez A, Albrecht J, Vahdat A. Bullet: high bandwidth data dissemination using an overlay mesh. ACM SIGOPS Oper Syst Rev. 2003;37:282–97.View ArticleGoogle Scholar
 Kim K, Venkatasubramanian N. Assessing the impact of geographically correlated failures on overlaybased data dissemination. In: Proceedings of IEEE GLOBECOM; 2010. pp. 1–5.Google Scholar
 Trajanovski S, Kuipers FA, Ilic A, Crowcroft J, Van Mieghem P. Finding critical regions and regiondisjoint paths in a network. IEEE/ACM Trans Netw. 2015;23(3):908–21.View ArticleGoogle Scholar
 Kuipers F, Beshir A, Orda A, Van Mieghem P. Impairmentaware path selection and regenerator placement in translucent optical networks. In: Proceedings of the 18th IEEE international conference on network protocols (ICNP); 2010. pp. 11–20.Google Scholar
 Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S. Catastrophic cascade of failures in interdependent networks. Nature. 2010;464(7291):1025–8.View ArticleGoogle Scholar
 Strand J, Chiu AL, Tkach R. Issues for routing in the optical layer. IEEE Commun Mag. 2001;39:81–7.View ArticleGoogle Scholar
 Dantzig G, Fulkerson DR. On the maxflow mincut theorem of networks. Linear Inequal Relat Syst. 2003;38:225–31.Google Scholar
 Lorenz DH, Orda A. QoS routing in networks with uncertain parameters. IEEE/ACM Trans Netw. 1998;6(6):768–78.View ArticleGoogle Scholar
 Guérin RA, Orda A. Qos routing in networks with inaccurate information: theory and algorithms. IEEE/ACM Trans Netw. 1999;7(3):350–64.View ArticleGoogle Scholar
 Papagiannaki K, Moon S, Fraleigh C, Thiran P, Tobagi F, Diot C. Analysis of measured singlehop delay from an operational backbone network. Proc IEEE INFOCOM. 2002;2:535–44.Google Scholar
 Bagnoli M, Bergstrom T. Logconcave probability and its applications. Econ Theory. 2005;26(2):445–69.View ArticleMathSciNetMATHGoogle Scholar
 Mohtashami Borzadaran GR, Mohtashami Borzadaran HA. Logconcavity property for some wellknown distributions. Surv Math Appl. 2011;6:203–19.MathSciNetGoogle Scholar
 Kuipers FA, Yang S, Trajanovski S, Orda A. Constrained maxmum flow in stochastic networks. In: Proceedings of IEEE ICNP, North Carolina, USA; 2014. pp. 397–408.Google Scholar
 Gabow HN, Maheshwari SN, Osterweil LJ. On two problems in the generation of program test paths. IEEE Trans Softw Eng. 1976;2(3):227–31.View ArticleMathSciNetGoogle Scholar
 Yuan S, Varma S, Jue JP. Minimumcolor path problems for reliability in mesh networks. Proc IEEE INFOCOM. 2005;4:2658–69.Google Scholar
 Garey MR, Johnson DS. Computers and intractability: a guide to the theory of npcompleteness. New York: W. H. Freeman & Co.; 1979.MATHGoogle Scholar
 Van Mieghem P, Kuipers FA. Concepts of exact QoS routing algorithms. IEEE/ACM Trans Netw. 2004;12(5):851–64.View ArticleGoogle Scholar
 Van Mieghem P. Paths in the simple random graph and the Waxman graph. Probab Eng Inf Sci. 2001;15(04):535–55.MATHGoogle Scholar
 Cormen TH, Stein C, Rivest RL, Leiserson CE. Introduction to algorithms. 2nd ed. Cambridge: MIT Press; 2001.MATHGoogle Scholar
 Boyd S, Vandenberghe L. Convex optimization. New York: Cambridge University Press; 2004.View ArticleMATHGoogle Scholar
 Tamir A. Polynomial formulations of mincut problems. Manuscript. Department of Statistic and Operations Research, Tel Aviv University, Israel; 1994.Google Scholar
 Kuipers FA, Van Mieghem P. The impact of correlated link weights on QoS routing. Proc IEEE INFOCOM. 2003;2:1425–34.Google Scholar
 Lee HW, Modiano E, Lee K. Diverse routing in networks with probabilistic failures. IEEE/ACM Trans Netw. 2010;18(6):1895–907.View ArticleGoogle Scholar
 Yang S, Kuipers FA. Traffic uncertainty models in network planning. IEEE Commun Mag. 2014;52(2):172–7.View ArticleGoogle Scholar
 Waller ST, Ziliaskopoulos AK. On the online shortest path problem with limited arc cost dependencies. Networks. 2002;40(4):216–27.View ArticleMathSciNetMATHGoogle Scholar
 Fan Y, Kalaba R, Moore J. Shortest paths in stochastic networks with correlated link costs. Comput Math Appl. 2005;49(9):1549–64.View ArticleMathSciNetMATHGoogle Scholar
 Ford LR, Fulkerson DR. Maximal flow through a network. Can J Math. 1956;8(3):399–404.View ArticleMathSciNetMATHGoogle Scholar
 Chekuri CS, Goldberg AV, Karger DR, Levine MS, Stein C. Experimental study of minimum cut algorithms. In: Proceedings of the eighth annual ACMSIAM symposium on discrete algorithms (SODA); 1997. p. 324–33.Google Scholar
 Karger DR. Minimum cuts in nearlinear time. J ACM. 2000;47(1):46–76.View ArticleMathSciNetMATHGoogle Scholar
 Pferschy U, Schauer J. The maximum flow problem with disjunctive constraints. J Comb Optim. 2013;26(1):109–19.View ArticleMathSciNetMATHGoogle Scholar