Research  Open  Published:
Optimization problems in correlated networks
Computational Social Networksvolume 3, Article number: 1 (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.
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.
However, often correlations or (inter)dependencies exist among link weights. For example, in overlay [1] or multilayer networks [2], the abstract links in the logical layer are mapped to different links in the physical layer. In this context, two or more abstract links, which use the same physical links, may have correlated latencies [3], bandwidth usage [4], or geographical failures [5, 6]. Or if the path must pass through some specific nodes (e.g., regenerators to boost the signal quality [7]), such important nodes and their links may also introduce dependencies. Correlations also appear in social networks. For instance, a message may be forwarded more rapidly if it came from a close friend rather than from a distant acquaintance. Another example relates to interdependent networks [8], where for instance the electricity network and Internet are coupled and interconnected, and one node or link failure in one network may cause failures of nodes or links in the other network. Similarly, in SharedRisk Link Group (SRLG) networks [9], links in, for example, the same duct will fail simultaneously, if their duct fails. The dependencies in interdependent and SRLG networks can also be seen as correlations, so we use the term correlation throughout this paper and study relevant problems in these socalled correlated networks. Our key contributions are as follows:

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
A network having node and link weights can be transformed into a directed network with only link weights, as done in [10]. Therefore, we assume that nodes are unweighted and only consider correlated link weights. Throughout this paper, we use the term “correlated model” to represent “correlated link weight model.”
Deterministic correlated model
Without loss of generality, we use w(l) to represent the weight of link l. For simplicity, in this paper we call w(l) the cost of l, although it could also reflect other metrics such as delay and energy. In the deterministic correlated model, for any two links \(l_i\) and \(l_j\), their joint total cost is represented by \(w(l_i) \oplus w(l_j)\), where the operator \(\oplus\) indicates the joint total cost of the links, which may differ from the \(+\) operator when they are correlated. When correlated, the use of one link may influence the cost of another in this model. For example, in Fig. 1 where the cost is shown above each link, it is assumed that only links (s, a) and (b, t) are correlated with joint cost of 11, and all the other links have uncorrelated costs. We can see that in path s–b–t, the cost of link (b, t) is 10, since another link (s, b) in this path is not correlated with it. Therefore, this path’s cost is equal to 18. However, in path s–a–b–t, the cost of link (b, t) should be calculated together with link (s, a), leading to a joint cost 11, since they are correlated and both appear in this path. Therefore, this path’s total cost is equal to \(11+4=15\), which is smaller than the sum of the individual link costs (\(6+4+10=20\)).
Equivalently, we could formulate \(w(l_i) \oplus w(l_j)=\rho _{i,j} \cdot (w(l_i)+w(l_j))\), where \(\rho _{i,j}\) stands for the correlation coefficient between links \(l_i\) and \(l_j\), and its value varies in the range of \((0,\infty )\), since we do not consider negative costs. When \(\rho _{i,j}\) is equal to 1, \(l_i\) and \(l_j\) are uncorrelated, when \(\rho _{i,j}\) is greater than 1, \(l_i\) and \(l_j\) have an increasing correlation, and otherwise we say that \(l_i\) and \(l_j\) have a decreasing correlation.
Analogously, for given \(m>1\) links \(l_1,l_2,\ldots ,l_m\) in the deterministic correlated model, their joint total cost can be expressed as follows:
Similarly, if the link l’s weight is multiplicative (e.g., failure probability), then by using \(log(w(l))\) to represent its weight value, Eq. (1) also applies. The decreasing correlation case can also reflect SRLG networks. For instance, in SRLG networks, each link is associated with several SRLG events with their respective failure probabilities. Hence, the total failure probability (represented by \(P_{SRLG}\)) of two correlated links that have at least one SRLG in common will be equal to the product of the failure probabilities of all the distinct SRLG events that belong to these two links. Let us denote by \(P_{l_1}=P_{s_1} \cdot P_{s}\) and \(P_{l_2}=P_{s_2}\cdot P_{s}\) the failure probabilities of these two links, respectively, where \(P_{s}\) denotes the common SRLGs’ failure probability between \(l_1\) and \(l_2\), and \(P_{s_1}\) (\(P_{s_2}\)) is the nonoverlapping SRLGs’ failure probability of \(l_1\) (\(l_2\)). Then \(P_{l_1}\cdot P_{l_2}<P_{SRLG}=P_{s_1} \cdot P_{s_2} \cdot P_{s} <\min (P_{l_1},P_{l_2})\). By taking the \(log\), we have
Or equivalently,
where \(\rho <1\) denotes their correlation coefficient.
In probability theory, given two random variables X and Y with expected values \(\mu _X\) and \(\mu _Y\), and standard deviations \(\sigma _X\) and \(\sigma _Y\), their linear correlation coefficient \(\rho (X,Y)\) is defined as
where Cov[X, Y] represents the covariance of X and Y.
However, the linear correlation coefficient in probability theory is different from and cannot be transformed to the one defined in the deterministic correlated model, because the variances of X and Y in Eq. (3) must be nonzero and finite. However, in the deterministic correlated model, for any link l, when none of its correlated links simultaneously appear on a path, the cost of l is fixed/deterministic with a variance of 0.
Stochastic correlated model
In many reallife networks, the link weights are uncertain because of inaccurate Network State Information (NSI) [11, 12]. For instance, Papagiannaki et al. [13] showed that the queuing delay distribution can be approximated by a Weibull distribution. Since the Cumulative Density Function (CDF) of a Weibull distribution is logconcave and the CDFs of many common distributions (e.g., Exponential distribution, Uniform distribution, etc.) are logconcave [14, 15], we make, as in [11, 16], a mild (general) assumption that the link weights follow a logconcave distribution.
We first define the Correlated Group (CG):
Definition 1
Given is a network \(G({\mathcal {N}},{\mathcal {L}})\) where \(\mathcal {N}\) represents a set of N nodes and \(\mathcal {L}\) denotes a set of L links. A Correlated Group (CG) is a subset of links \(L_{CG} \subseteq \mathcal {L}\), and \(\forall l \in L_{CG}\), \(\exists l' \in L_{CG} \backslash \{l\}\), such that l and \(l'\) are correlated (\(\rho _{l,l'} \ne 1\)).
Accordingly, the Maximum Correlated Group (MCG) is defined as a CG with the maximum number of correlated links. If a link l is uncorrelated/independent with all the other links, then we say \(\{l\}\) is a singleelement MCG. Suppose there are \(\Omega\) Maximum Correlated Groups (MCGs), and there are \(m_i>0\) links (denoted as \(l_1^{i}\), \(l_2^{i},\ldots ,l_{m_i}^{i}\)) in the ith MCG, where \(1 \le i \le \Omega\). In the ith MCG, a multivariate \(m_i\)dimensional logconcave Cumulative Density Function \(CDF_i(x_1,x_2,\ldots ,x_{m_i})\) is given to allocate cost \(x_1\), \(x_2,\ldots , x_{m_i}\) for links \(l_1^{i}\), \(l_2^{i},\ldots , l_{m_i}^{i}\), respectively.
Therefore, if the possible cost of link l ranges from 0 to \(w_{l}^{\max }\) (\(0 <w_{l}^{\max }\)), then the probability of allocating a cost value out of this range is 0. Hence, we have \(CDF_{i}(w_{l}^{\max })=1\) for a singleelement MCG i, and \(CDF_{j}(w_{l^j_1}^{\max },w_{l^j_2}^{\max },\ldots ,w_{l^j_{m_j}}^{\max })=1\) for a multielement MCG j.
For example, Fig. 2 shows a 2dimensional multivariate normal distribution, where both variables are in the range [0, 4] with mean 2 and covariance matrix \(\left[ \begin{array}{cc} 0.9 &{} 0.4 \\ 0.4 &{} 0.3 \\ \end{array} \right].\) Similarly to Eq. (3), the correlation matrix (composed of linear correlation coefficients) can be derived from the covariance matrix and the variables’ standard variances in the multivariate normal distribution. However, we do not explicitly use the linear correlation coefficient in the stochastic correlated model, since we will later prove that via the logconcave property of this model, the shortest path problem can be solved by convex optimization.
Shortest paths in correlated networks
Shortest path under the deterministic correlated model
Definition 2
Given is a directed network \(G({\mathcal {N}},{\mathcal {L}})\), and each link \(l \in \mathcal {L}\) has a cost w(l) following the deterministic correlated model. The Shortest Path under the Deterministic Correlated Model (SPDCM) problem is to find a path from a source s to a destination t with minimum cost.
In conventional deterministic networks, a subpath of a shortest path is also the shortest. We refer to this property as the dominance of the subpath. However, this is not the case in networks with deterministic correlated link weights, which means that a dominated path may also lead to an optimal solution. For instance, in the example of Fig. 1, we can see that although subpath s–b has a smaller cost than subpath s–a–b, path s–a–b–t (instead of path s–b–t) has minimum cost. In the following, we will study the complexity of the SPDCM problem.
Theorem 1
The SPDCM problem is NPhard.
Proof
When the correlation coefficient is equal to 1, the SPDCM problem can be solved in polynomial time by a conventional shortest path algorithm. We therefore prove in the following that the SPDCM problem is NPhard for “increasing correlation” \({>}1\) as well as “decreasing correlation” \({<}1\).
Increasing correlation
When the correlation coefficient is greater than 1, we make a reduction to the forbidden pairs shortest path problem, which is known to be NPhard [17]. In a given network and for a given set of node pairs \(\zeta\), the forbidden pairs shortest path problem looks for the shortest path between s and t such that at most one node from each pair in the set \(\zeta\) lies on this path. Let us consider a network with deterministic correlated link weights. When two nodes i and j form a forbidden pair, their costs are correlated such that \(w(i,.) \oplus w(j,.)=\infty\), where (i, .) and (j, .) represent any link that contains an end node of i and j, respectively. In all the other cases, the link costs are uncorrelated and finite. Since \(w(i,.) \oplus w(j,.)=\infty\), if the two forbidden nodes appear in the same path, then the cost of this path will be \(\infty\), so it will never lead to the shortest path. Now, the SPDCM problem is equivalent to the forbidden pairs shortest path problem.
Decreasing correlation
When the correlation coefficient is less than 1, we make a reduction to the Minimum Color SinglePath (MCSiP) problem, which is NPhard [18]. Given a network \(G({\mathcal {N}},{\mathcal {L}})\), and given the set of colors \(C=\{c_1,c_2,\ldots ,c_g\}\) where g is the total number of colors, and given the color set \({\{c_{l}\}}\) associated to each link \(l\in \mathcal {L}\), the Minimum Color SinglePath (MCSiP) problem is to find one path from source node s to destination node t such that it uses the least amount of colors.
Assume that each color \(c_i\) is associated with cost 1, where \(1 \le i \le g\). We further assume that \(w(l_1)\oplus w(l_2) \oplus \cdots \oplus w(l_m)=q\), where q is the total number of distinct colors belonging to these m links. Therefore, the SPDCM problem is equivalent to the MCSiP problem. \(\square\)
Theorem 2
The SPDCM problem cannot be approximated to arbitrary degree in polynomial time, unless P \(=\) NP.
Proof
We provide a proof by contradiction.
Increasing correlation
Assume that a polynomialtime approximation algorithm exists that can find a path with a cost at most \(\alpha \cdot opt\), where \(\alpha >1\) is an approximation ratio. For a pair of forbidden nodes i and j, we further assume \(w(i,.) \oplus w(j,.)>\alpha * opt\). Therefore, if an approximation algorithm can find a path \(\psi\) with cost at most \(\alpha * opt\) from s to t, then i and j cannot be simultaneously traversed by this path \(\psi\), which means that the forbidden pairs shortest path problem can be solved in polynomial time, which results in a contradiction.
Decreasing correlation
We first introduce the Disjoint Connecting Paths problem [19]. Given a directed network \(G({\mathcal {N}},{\mathcal {L})}\), a collection of disjoint node pairs \((s_{1}, t_{1}), (s_{2}, t_{2}), \ldots , (s_{z}, t_{z}\)), does G contain z mutually linkdisjoint paths, one connecting \(s_{i}\) and \(t_{i}\) for each i, \(1\le i\le z\). This problem is NPhard when \(z \ge 2\). Assume that a polynomialtime approximation algorithm exists that can find a path with a cost at most \(\alpha \cdot opt\), where \(\alpha >1\) is an approximation ratio. Assuming that all the links in the network have weight 1, and link (u, v) and any \(m>0\) links in \({\mathcal {L}} \backslash \{(u,v)\}\) are correlated, with a total cost of \(\frac{1}{\beta } \cdot m\). Moreover, any two or more links in \({\mathcal {L}} \backslash \{ (u,v) \}\) are assumed to be uncorrelated/independent.
According to this assumption, the minimum value of a shortest path is 1 if link (u, v) is not traversed, i.e., it traverses only one link from s to t. However, the optimal solution which traverses link (u, v) has a total cost of \(opt=\frac{1}{\beta } \cdot c\), where c is the sum of minimum hops from s to u and from v to t. For any given \(\alpha\), let \(\frac{\beta }{c}>\alpha\), then \(1>\alpha \cdot \frac{1}{\beta } \cdot c\), which means \(1>\alpha \cdot opt\). To find a path with cost at most \(\alpha \cdot opt\), the polynomialtime algorithm must find a path which traverses link (u, v). In that case, the algorithm can, in polynomial time, find two linkdisjoint paths from s to u, and from v to t, which results in a contradiction. \(\square\)
Next, we study the performance of a conventional shortest path algorithm running on a graph where each link has an “uncorrelated” weight value.
Lemma 1
When all the correlation coefficients are greater than 1, a conventional shortest path \(\psi\) has a total cost at most \(\frac{\rho _{\max }}{\rho _{\mathrm{{opt}}}} \cdot opt\), where \(\rho _{\max }\) and \(\rho _{\mathrm{{opt}}}\) are the largest correlation coefficient and the optimal solution’s correlation coefficient, respectively, and opt is the cost of the optimal solution.
Proof
Let \(U(\psi )=\sum _{l \in \psi } w(l)\) and let \(C(\psi )=\rho _u \cdot U(\psi ) =\rho _{u} \cdot \sum _{l \in \psi } w(l)\) reflect the total joint cost of path \(\psi\) considering their correlation, where \(\rho _u\) indicates the correlation coefficient of path \(\psi\). On one hand, a conventional shortest path \(\psi\) should satisfy \(U(\psi ) \le \frac{{\mathrm{{opt}}}}{\rho _{\mathrm{{opt}}}}\). On the other hand, \(C(\psi ) \le \rho _{\max } \cdot U(\psi )\) considering that \(\rho _{\max }\) is the largest correlation coefficient. Hence, \(C(\psi )\le \rho _{\max } \cdot U(\psi )\le \frac{\rho _{\max }}{\rho _{\mathrm{{opt}}}} \cdot opt\). \(\square\)
Lemma 2
When all the correlation coefficients are less than 1, a conventional shortest path \(\psi\) has a cost at most \(\frac{1}{\rho _{\min }} \cdot opt\), where \(\rho _{\min }\) is the smallest correlation coefficient among all the correlation coefficients.
Proof
Let \(V(\psi )=\sum _{l \in \psi } w(l)\) and let \(C(\psi )=\rho _{u} \cdot \sum _{l \in \psi } w(l)\) reflect the total joint cost of path \(\psi\) considering their correlation. Since all the correlations are decreasing (\(\rho <1\)), we have \(C(\psi ) \le V(\psi )\). On the other hand, \(\rho _{\min } \cdot V(\psi ) \le opt\) considering that \(\rho _{\min }\) is the smallest correlation coefficient. Hence, \(C(\psi ) \le V(\psi ) \le \frac{1}{\rho _{\min }} \cdot opt\). \(\square\)
Via Lemmas 1 and 2, we obtain Theorem 3.
Theorem 3
In a network with links following the deterministic correlated model, a conventional shortest path can have cost at most \(max(\frac{\rho _{\max }}{\rho _{\mathrm{{opt}}}}, \frac{1}{\rho _{\min }}) \cdot opt\).
Theorem 3 reveals that a conventional shortest path may have arbitrarily bad performance, since either \(\frac{\rho _{\max }}{\rho _{\mathrm{{opt}}}}\) can be infinitely large or \(\rho _{\min }\) can be infinitely small.
An exact algorithm to solve the SPDCM problem
To solve the SPDCM problem exactly, we modify Dijkstra’s algorithm by letting each node store as many subpaths as possible, which is similar to the exact algorithm for solving the multiconstrained routing problem [20]. Since each node can store as many subpaths as possible, its running time is exponential. We start with some notations used in the algorithm:
sus[u][h]: the parent node of the hth subpath from s to u.
dist[u][h]: the cost value of the hth subpath from s to u.
counter[u]: the number of stored subpaths at node u.
u[m]: the mth subpath from s to u.
adj(u): the set of nodes adjacent to node u.
The pseudocode of the exact algorithm is given in Algorithm 1.
The time complexity of Algorithm 1 can be computed as follows. Let \(k_{\max }\) denote the maximum number of subpaths for each node to store, then in Step 2 Q contains at most \(k_{\max }N\) subpaths. According to [21], \(k_{\max } \le \lfloor e(N2)! \rfloor\), where \(e \approx 2.718\). When using a Fibonacci heap to structure the heap, selecting the minimum cost path has a time complexity of \(O(\log (k_{\max }N))\) [22] in Step 3. Steps 7–11 take at most \(O(k_{\max })\) time for each link to be iterated and hence result in \(O(k_{\max }L)\) time; because for a fixed link, the steps within the inner loop (Steps 8–11) all cost O(1) time. Step 12 invokes \(O(k_{\max })\) time for node t to select the minimum cost path. Hence, the overall time complexity of Algorithm 1 is \(O(k_{\max }N \log (k_{\max }N)+k_{\max }L)\).
Shortest path under the nodal deterministic correlated model
In some realworld networks (e.g., SRLG networks), the links that are spatially (geographically) close to each other are usually correlated, whereas the links that are located far from each other are usually uncorrelated. We make an additional assumption, which is that only the links sharing the same node can be correlated. We call this nodal correlation.
Although the SPDCM problem is NPhard, we will show that, by transforming the original graph to an auxiliary graph, the Shortest Path under the Nodal Deterministic Correlated Model (SPNDCM) problem is solvable in polynomial time. For any node a, there are generally two cases of nodal correlation, namely (1) links in the form of (a, b) and (a, c), and (2) links in the form of (a, b) and (b, c) are correlated. When (a, b) and (a, c) are correlated, a simple path cannot traverse both of them, since looping is not allowed. In this sense, any simple path only traverses at most one of them, which means that the links’ correlation will not affect the cost calculation of any simple path. Therefore, we only need to consider the case when (a, b) and (b, c) are correlated. We first define that if (a, b) and (b, c) are correlated, then a and b are called correlated nodes, which is represented by \(C_n\), else they are uncorrelated nodes, which is denoted by \(U_n\). Subsequently, based on the original graph \(G({\mathcal {N}},{\mathcal {L}})\), the auxiliary graph \(G^{A}({\mathcal {N}}^{A},{\mathcal {L}}^{A})\) can be constructed as follows:

1.
For any two links \((u,v) \in \mathcal {L}\) and \((v,y) \in \mathcal {L}\) that are correlated in G, create new nodes \(u_v\), \(v^{uy}\), \(v_y\) , and \(y_y\) in \(G^{A}\) if they do not already exist. For node \(v \in \mathcal {N}\) in \(\mathcal {G}\), in case \(v^{uy}\) and \(v_v\) need to be created, create \(v^{uy}\) only and regard \(v^{uy}\) to be the same as \(v_v\).

2.
For any node \(a \in \mathcal {N}\) and if it is an uncorrelated node (in \(U_n\)), create node \(a_a\) in \(G^{A}\).

3.
For any two correlated links (u, v) and (v, y) in G, create links \((u_v,v^{uy})\), \((v^{uy},v_y)\) , and \((v_y,y_y)\) in \({\mathcal {G}}^{A}\). Assign to the links \((u_v,v^{uy})\) and \((v_y,y_y)\) the weights of w(u, v) and w(v, y), respectively, and the link \((v^{uy},v_y)\) with weight \((\rho _{(u,v)(v,y)}1) \cdot (w(u,v)+w(v,y))\), where \(\rho _{(u,v)(v,y)}\) is the correlation coefficient of links (u, v) and (v, y).

4.
For each link \((a,b) \in \mathcal {L}\) such that both node a and node b are not correlated nodes, create the link \((a_a,b_b)\) also in \(G^{A}\) with the link weight of w(a, b).

5.
For each link \((a,b) \in \mathcal {L}\) such that \(a \in U_n\) and \(b \in C_n\), draw links \((a_a, b_r)\) in \(G^A\), where \(r \in \mathcal {N}\) and \(b_r \in G^A\).

6.
For each link \((a,b) \in \mathcal {L}\) such that \(a \in C_n\) and \(b \in U_n\), draw links \((a^{rz}, b_b)\) in \(G^A\), where \(r,z \in \mathcal {N}\) and \(a^{rz} \in \mathcal {N}^A\).
The idea of the auxiliary graph is that if two links (u, v) and (v, y) are correlated, we create four corresponding nodes \(u_v,\) \(v^{uy}\), \(v_y\), and \(y_y\) and then draw three links \((u_v,v^{uy})\),\((v^{uy},v_y)\) , and \((v_y,y_y)\). We use \((u_v,v^{uy})\) and \((v_y,y_y)\) to indicate their uncorrelated values, respectively, if only one of these two links is traversed, and \((v^{uy},v_y)\) to represent the correlated loss (decreasing correlation) or gain (increasing correlation), respectively, if they are traversed simultaneously. For instance, in Fig. 3 where the link weight is labeled above each link, assuming that links (a, b), (b, c), (c, t), and (b, d) are nodal correlated, then Fig. 4 is its corresponding auxiliary graph with the assigned weight shown on each link. In particular, since (a, b), (b, c) and (a, b), (b, d) may have different correlation coefficients, in Fig. 4 we use \((b^{ac},b_c)\) and \((b^{ad},b_d)\) to represent their correlation value. Meanwhile, when there is a link from an uncorrelated node to a correlated node in the original graph, e.g., (e, b) in Fig. 3, we draw links \((e_e,b_c)\) and \((e_e,b_d )\) (Step 5). When there is a link from a correlated node to an uncorrelated node in the original graph, e.g., (c, d) in Fig. 3, we draw link \((c^{bt},d_d)\) (Step 6). Considering that there are at most \(N(N1)\) nodal links in a graph, the original graph can be transferred to the auxiliary graph in polynomial time.
Consequently, running a shortest path algorithm on the auxiliary graph can return a minimum cost path under the nodal deterministic correlated model. Our auxiliary graph can deal with both decreasing and increasing correlation cases. Considering that \((\rho _{(u,v)(v,y)}1) \cdot (w(u,v)+w(v,y))<0\) in the auxiliary graph under the decreasing correlation case, and Dijkstra’s algorithm cannot handle negative link weights, we could for instance run Bellman–Ford algorithm on the auxiliary graph. No negative loops will exist in the auxiliary graph, since if a path traverses the negative weight link (say \((v^{uy},v_y)),\) it will also traverse the links in the form of \((u_v,v^{uy})\) and \((v_y,y_y)\), whose total cost is always positive.
Shortest path under the stochastic correlated model
Definition 3
The Shortest Path under the Stochastic Correlated Model (SPSCM) problem: In a given directed graph \({\mathcal {G}}({\mathcal {N}},{\mathcal {L}})\) where the link costs follow the stochastic correlated model, it is assumed that there are in total \(\Omega\) Maximum Correlated Groups (MCGs). The SPSCM problem is to find a path from source s to destination t such that its total cost is minimized and the probability to realize this value is no less than \(P_s\).
We present a convex optimization formulation to solve the SPSCM problem. Convex optimization problems can usually be solved quickly and accurately with convex optimization solvers [23]. Let us first introduce how to develop a Linear Programming (LP) formulation to solve the shortest path problem in deterministic networks:
Objective
Constraints
where \(y_{uv}\) indicates whether link (u, v) is part of the shortest path. When \(y_{uv}=1\), it indicates that link (u, v) appears on the path, else \(y_{uv}=0\). The objective is to minimize the total cost value of the path. Constraint Eq. (6) accounts for that except for s and t, the number of incoming and outgoing links that are part of the path must be the same. For the source node s, the number of its outgoing links should be 1, and for the destination node t the number of its incoming links should be 1. The dual of the above Linear Program (LP) can be expressed as follows:
Objective
Constraints
where \(d_u\) is a value between 0 and 1. Similarly, the SPSCM problem can be solved by the following convex formulation:
Objective
Constraints
where the variables \(x(u,v),x(l^i_1),x(l^i_2),\ldots ,x(l^i_{m_i})\) indicate the allocated possible cost of links (u, v), \(l^i_1\), \(l^i_2,\ldots , l^i_{m_i}\), respectively. Constraint (13) ensures that the total probability of realizing the total cost is no less than \(P_s\). In Eq. (13), for each MCG we apply the multidimensional CDF functions to calculate the probability of realizing a cost. Since the multidimensional CDF function is logconcave, \(\log \left( CDF_i(x(l^i_1),x(l^i_2),\ldots ,x(l^i_{m_i}))\right)\) is convex, and by summing all the MCGs’ CDFs together, it remains convex, which indicates that Eq. (13) is convex. The other constraints are also convex, which proves that the above formulation is a convex optimization formulation.
Widest path under the deterministic correlated model
The Widest Path in Deterministic Networks (WPDN) problem is to find a path from s to t such that the minimum link weight among all its traversed links is maximized. This problem appears with bottleneck metrics, such as bandwidth. The WPDN problem is solvable in polynomial time: First, we order all the link weights in the network in increasing order. After that, each round we prune lowest weight links in the graph and run a Depth First Search (DFS) or a Breadth First Search (BFS) algorithm to find a path from s to t. The algorithm will end if there is no path anymore from s to t and return the pruned weight value of the previous round.
In the Widest Path under the Deterministic Correlated Model (WPDCM) problem, if \(m>1\) correlated links in a path have a joint weight value W, then for each link the maximum average/amortized weight is \(\frac{W}{m}\). For instance, if a path traverses three correlated links with a joint weight value of 15 and passes another uncorrelated link with a weight of 6, then this path has a “width” value of 5. The reason is that the maximum (average/amortized) weight for each of these three correlated links is \(15/3=5\), and this value is less than for another uncorrelated link (6).
However, the WPDCM problem is still NPhard and cannot be approximated to arbitrary degree. The proof follows analogously from Theorems 1 and 2.
Minimum cuts in correlated networks
Mincut under the deterministic correlated model
Definition 4
The MinCut under the Deterministic Correlated Model (MCDCM) problem: Given is a network \(G({\mathcal {N}},{\mathcal {L}})\), and each link \(l \in \mathcal {L}\) is associated with a cost w(l). It is assumed that two or more link costs are correlated under the deterministic correlated model. Given a source s and a target t, find a cut \(\mathcal {C}\) that partitions G into two disjoint subsets X (\(X \in \mathcal {N}\)) and \({\mathcal {N}}X\) such that s and t are in different subsets and the cost of the cut C is minimized.
The mincut value is not equal to the maximum flow value under the deterministic correlated model. For example, in Fig. 5 assume that links (s, a) and (s, t) are correlated with a joint cost of 8. In this example, the maximum flow from s to t is \(sat\) with value 10, while the mincut is composed of links (s, a) and (s, t) and has a cost of 8.
Theorem 4
The MCDCM problem is NPhard.
Proof
In Fig. 6, we assume that the links in the form of \((x_i,x_{i+1})\) and \((z_{i+1},z_{i})\) have infinite uncorrelated cost and the link costs of \((x_i,y_i)\) and \((y_i,z_i)\) follow the deterministic correlated model, where \(1 \le i \le n\). We want to find a mincut to separate \(x_1\) and \(z_1\). Based on Fig. 6, we first derive Fig. 7 with the same nodes except that we add one more node s. We set \(s=y_0\), and \(t=y_{n}\). The link weight in Fig. 7 is set as follows: \((y_{i1}, x_{i})\) and \((y_{i1}, z_{i})\) have 0 uncorrelated cost, while \((x_{i}, y_{i})\) and \((z_{i}, y_{i})\) have the same (correlated) costs with \((x_i, y_i)\) and \((y_i,z_i)\) in Fig. 6, respectively, where \(1 \le i \le n\). In Fig. 7, we want to solve the SPDCM problem from the source s to the destination t.
Since we want to find a mincut that separates \(x_1\) and \(z_1\), any cut in the form of \((x_i,y_i)\) and \((y_i,z_i)\), where \(1 \le i \le n\), is not the optimal solution. The reason is that this kind of cut only separates \(y_i\) and other nodes, but not \(x_1\) and \(z_1\). Moreover, considering that the links in the form of \((x_j,x_{j+1})\) or \((y_j,y_{j+1})\) have infinite costs, they cannot be in the optimal solution. Based on above analysis, any feasible cut C should contain one link of either \((x_i,y_i)\) or \((y_i,z_i)\), for all \(1 \le i \le n\). We prove in the following that the MCDCM problem in Fig. 6 can be reduced to the SPDCM problem in Fig. 7 in polynomial time.
The SPDCM problem to the MCDCM problem: Considering an optimal solution of the SPDCM problem, denote by \(R_{SPDCM}\) the set of links in the optimal solution of the SPDCM problem. Because \(R_{SPDCM}\) has minimum cost, let \(C_{MCDCM}=R_{SPDCM} \backslash \{(y_i,x_{i+1}), (y_i,z_{i+1})\}\) and then \(\forall (z_i,y_i) \in C_{MCDCM}\), change it to \((y_i,z_i)\) in \(C_{MCDCM}\). Since the links \((y_i,x_{i+1})\) and \((y_i,z_{i+1})\) have 0 cost, \(C_{MCDCM}\) also has minimum cost (the same with \(R_{SPDCM}\)). Therefore, solving the SPDCM problem yields a solution to the MCDCM problem.
The MCDCM problem to the SPDCM problem: An optimal solution of the MCDCM problem should be composed of either \((x_i, y_i)\) or \((y_i,z_i)\), where \(1 \le i \le n\). Denote by \(C_{MCDCM}\) the set of links in the optimal solution of the MCDCM problem. Let \(R_{SPDCM}=C_{MCDCM}\) and then \(\forall (y_i,z_i) \in R_{SPDCM}\), change it to \((z_i,y_i)\) in \(R_{SPDCM}\). Because \(C_{MCDCM}\) has minimum cost value and the links in the form of \((y_{i1},x_{i})\) or \((y_{i1},z_{i})\) have 0 cost, \(R_{SPDCM}\) together with \((y_{i1},x_{i}),\) if \((x_i,y_i) \in R_{SPDCM}\) or \((y_{i1},z_{i})\) if \((z_i,y_i) \in R_{SPDCM}\), can form a path from s to t with minimum cost. Hence, a solution to the MCDCM problem can also solve the SPDCM problem. \(\square\)
Theorem 5
The MCDCM problem cannot be approximated to arbitrary degree in polynomial time, unless P \(=\) NP.
Proof
The proof follows from the fact that the SPDCM problem cannot be approximated to arbitrary degree in polynomial time according to Theorem 2. \(\square\)
Theorem 6
By assigning each link l with the cost w(l), running a conventional mincut algorithm will return a cut with total cost at most \(\max (\frac{\rho _{\max }}{\rho _{\mathrm{{opt}}}}, \frac{1}{\rho _{\min }}) \cdot opt\).
Proof
The proof follows analogously from Theorem 3. \(\square\)
Since the MCDCM problem is NPhard and even does not admit a polynomialtime approximation algorithm, we suggest a bruteforce approach to solve it. The idea is that we start with two sets A and B, with s in A and t in B. Then we have \(N2\) nodes left, and there are \({N2 \atopwithdelims ()0}+{N2 \atopwithdelims ()1}+ \cdots +{N2 \atopwithdelims ()N2}=O(2^{N})\) combinations to assign these \(N2\) nodes to sets A and B. Each combination assignment corresponds to a cut to separate A and B, and the one with minimum cost is returned as the optimal solution.
Mincut under the SRLGlike correlated model
In “Correlated link weight models” section, we introduced and formulated the joint failure calculation in SRLG networks, which follows the decreasing correlated model. We define the SRLGlike correlated model as follows:
Definition 5
The SRLGlike correlated model: Suppose \(l_1\), \(l_2,\ldots ,l_m\) (\(1<m \le L\)) form a Correlated Group (CG), then \(w(l_1) \oplus w(l_2) \cdots \oplus w(l_j)\) is greater than the sum of at most \(j1\) link costs, but smaller than \(w(l_1)+ w(l_2) \cdots + c(l_j)\), where \(1< j \le m\).
The Shortest Path under the SRLGlike model (SPSRLG) problem is NPhard, since it is a general case of the MCSiP problem introduced in “Shortest paths in correlated networks” section, which is NPhard [18]. Also the mincut under the SRLGlike correlated model (MCSRLG) problem is NPhard. Similar to the proof that the MCDCM problem is NPhard in “Minimum cuts in correlated networks” section, the MCSRLG problem in the form of Fig. 6 can be reduced to the NPhard SPSRLG problem.
In the Nodal SRLGlike correlated model, we assumed that only the links that share the same node follow the SRLGlike correlated model. As “Shortest path under the nodal deterministic correlated model” section shows that the Shortest Path under the Nodal Deterministic Correlation Model problem is solvable in polynomial time, we address the mincut under the Nodal SRLGlike correlated model (MCNSRLG) problem in the following. In general, the MCNSRLG problem is still NPhard. The reason is that for the MCNSRLG problem in Fig. 8, where the links in the form of \((x_1,y_i)\) are assumed to be correlated, we could derive a graph like in Fig. 7 by duplicating \(n1\) more x nodes, z nodes, and one y node. We set \(s=y_0\) and \(t=y_n\). The link weights in Fig. 7 are set as follows: For any two \((x,y_i)\) and \((x,y_j)\) [or \((y_i,z_1)\) and \((y_j,z_1)\)] in Fig. 8, \((x_i,y_i)\) and \((x_j,y_j)\) [or \((y_i,z_i)\) and \((y_j,z_j)\)] in Fig. 7 follow the same correlation. This link weight setting also applies to more than two links. The link weights in the form of \((y_{i1},x_i)\) and \((y_{i1},z_i)\) have 0 uncorrelated cost, for \(1 \le i <n\). Consequently, based on these link weight assignments, the MCNSRLG problem in Fig. 8 can be reduced to the SPSRLG problem in Fig. 7, similar to the proof of the MCDCM problem in “Minimum cuts in correlated networks” section.
However, we found that the MCNSRLG problem is solvable in polynomial time when (1) only the nodal links in the form of (u, v) and (v, y) follow the SRLGlike correlated model and/or (2) for any node \(u \in \mathcal {N}\), at most two nodal links (u, v) and (u, x) follow the SRLGlike correlated model. To prove case (1), let us first study the following theorem:
Theorem 7
Any two links in the form of ( u, v) and ( v, y) will never both appear in the optimal solution of the MCNSRLG problem in case (1).
Proof
Suppose s and t are separated by a mincut C such that s is in the subset A and t is in the subset B. A proof by contradiction: we assume that (u, v) and (v, y) are both in the mincut C. Since C is the mincut that separates s and t, then node u should be in subset A, otherwise if node u is in subset B, there is no need to use (u, v) and (v, y) as the cut links, since their existence does not affect the connectedness between A and B. Similarly, node v is in subset B, otherwise if node v is also in A, there is no need to cut link (u, v). Based on this analysis, if y is in A, then (v, y) is not necessarily the link in the cut C, since link (v, y) does not affect the connectedness from A to B. However, if node y is in B, link (v, y) is also not necessarily the link in the mincut, since nodes v and y are in the same subsets, which results in a contradiction. \(\square\)
Based on Theorem 7, we can use conventional Linear Programming (LP) for solving the MCNSRLG problem under case (1). Following [24], the LP is as follows:
Objective
Constraints
where c(u, v) stands for the link weight (i.e., capacity) of link (u, v) in the deterministic network and \(h_{u,v}\) is an indicator denoting whether (u, v) belongs to the cut.
Under case (2), the MCNSRLG problem can be solved in polynomial time by running the above LP (Eqs. (14)–(17)) on an auxiliary graph \(G^{U}({\mathcal {N}}^U,{\mathcal {L}}^U)\). The auxiliary graph can be derived from the original graph G as follows:

1.
For each pair of two links \((u,v) \in \mathcal {L}\) and \((u,x) \in \mathcal {L}\) that are correlated in G, create a new node \(u'\), and draw link \((u',u)\) with weight \(\rho _{vx} (w(u,v)+w(u,x))\) to represent the joint cost of links (u, v) and (u, x), where \(\rho _{vx}\) represents the correlation coefficient between (u, v) and (u, x).

2.
For any link \((a,u) \in \mathcal {L}\) and \((u,b) \in {\mathcal {L}} \backslash \{v,x\}\) such that (u, v) and (u, x) are correlated in G, draw link \((a,u')\) and \((u',b)\) in \(G^{U}\) with weights w(a, u) and w(u, b), respectively.

3.
For each pair of two links \((v,u) \in \mathcal {L}\) and \((x,u) \in \mathcal {L}\) that are correlated in G, create a new node \(u'\), and draw link \((u,u')\) with weight \(\rho _{vx} (w(v,u)+w(x,u))\) to represent the total cost of links (v, u) and (x, u), where \(\rho _{vx}\) represents the correlation coefficient between (v, u) and (x, u).

4.
For any link \((a,u) \in {\mathcal {L}} \backslash \{v,x\}\) and \((u,b)\in \mathcal {L}\) such that (v, u) and (x, u) are correlated in G, draw link \((a,u')\) and \((u',b)\) in \(G^{U}\) with weights w(a, u) and w(u, b), respectively.

5.
For the other links \((c,d) \in \mathcal {L}\), create link (c, d) also in \(G^U\) with the same weight.
The proposed auxiliary graph shares similarities with the auxiliary graph in Fig. 4. For example, Fig. 10 is an auxiliary graph of the original graph shown in Fig. 9. Moreover, we mention that the proposed auxiliary graph also applies when there are \(m>2\) links starting from the same node that follow the SRLGlike correlated model, but the failure of one correlated link will trigger the other \(m1\) links to simultaneously “fail” (e.g., SRLG networks, interdependent networks).
Mincut under the stochastic correlated model
Definition 6
The MinCut under Stochastic Correlated Model (MCSCM) problem: In a given directed graph \({\mathcal {G}}({\mathcal {N}},{\mathcal {L}})\) with link costs following the stochastic correlated model, the MCSCM problem is to find a cut \(\mathcal {C}\) which partitions G into two disjoint subsets X (\(X \in \mathcal {N}\)) and \(\mathcal {N}X\) such that

s and t are in different subsets;

the allocated cost of the cut C is minimum;

the total probability of realizing the cost value is no less than \(P_c\).
We propose a corresponding convex optimization formulation based on Eqs. (14)–(17):
Objective
Constraints
where \(x(u,v),x(l^i_1),x(l^i_2),\ldots ,x(l^i_{m_i})\) indicate the allocated possible cost by links (u, v), \(l^i_1\), \(l^i_2,\ldots , l^i_{m_i}\), respectively. In particular, Eq. (19) ensures that the probability of realizing that the mincut cost is no less than \(P_c\). More specifically, for each MCG we apply the multidimensional CDF functions to calculate the probability of realizing a cost. Since the CDF function is logconcave, \(\log \left( CDF_i(x(l^i_1),x(l^i_2),\ldots ,x(l^i_{m_i}))\right)\) is convex, and by summing all the MCGs together, it remains convex, which indicates that Eq. (19) is convex.
It remains to show that Eq. (18) is convex. In general, the product of two convex functions is not always convex; however, according to [23, pp. 119], one special case is as follows: “If functions f and g are convex, both nondecreasing (or nonincreasing) and positive (nonnegative) functions on an interval, then \(f \cdot g\) is convex.” Therefore, for each \((u,v)\in \mathcal {L}\), \(x(u,v) \cdot h_{u,v}\) is convex.
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.
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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.
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The authors declare that they have no competing interests.
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Keywords
 Shortest path
 Mincut
 Correlated networks
 Stochastic link weights