In this paper we study a problem faced by network service providers in which a set of Virtual Network Functions (VNFs) has to be installed in a telecommunication network at minimum cost. For each given origin-destination pair of nodes (commodities), a latency-constrained routing path has to be found that visits the required VNFs in a pre-defined order. A limited number of functions can be installed at each node. We first prove that the problem is NP-hard in a strong sense, even for a single commodity and without node-capacity, latency and precedence constraints. We then provide a compact Mixed Integer Linear Programming (MILP) formulation, along with several families of valid inequalities. To tackle the problem from a computational perspective, we provide theoretical results that allow us to derive Benders reformulation of the problem. We also exploit an alternative path-based MILP formulation to derive heuristic solutions. All these ingredients are combined in a Branch-and-Benders-Cut framework and computationally tested on a wide range of realistic instances. Our results are also compared with the Automatic Benders decomposition provided by Cplex. Computational results indicate that our decomposition approach is more efficient compared to the two methods provided by the off-the-shelf solver, both in terms of the CPU time and the overall solution quality. The results also indicate that our MILP-heuristic provides high-quality solutions.

在本文中，我们研究了网络服务提供商面临的一个问题，即必须以最小的成本将一组虚拟网络功能（VNF）安装在电信网络中。对于节点（商品）的每个给定的起点-终点对，必须找到受延迟限制的路由路径，该路径以预定义的顺序访问所需的VNF。每个节点上只能安装数量有限的功能。我们首先证明，即使对于单个商品，并且没有节点容量，等待时间和优先级约束，该问题也具有很强的NP难度。然后，我们提供了紧凑的混合整数线性规划（MILP）公式，以及多个有效不等式族。为了从计算角度解决问题，我们提供了理论结果，使我们能够导出问题的Benders重新表述。我们还利用基于路径的替代MILP公式来得出启发式解决方案。将所有这些成分组合在“分支和弯曲弯曲”框架中，并在各种实际情况下进行了计算测试。我们的结果也与Cplex提供的自动Benders分解进行了比较。计算结果表明，与现成求解器提供的两种方法相比，我们的分解方法在CPU时间和整体解决方案质量方面都更加有效。结果还表明，我们的MILP启发式方法提供了高质量的解决方案。我们的结果也与Cplex提供的自动Benders分解进行了比较。计算结果表明，与现成求解器提供的两种方法相比，我们的分解方法在CPU时间和整体解决方案质量方面都更加有效。结果还表明，我们的MILP启发式方法提供了高质量的解决方案。我们的结果也与Cplex提供的自动Benders分解进行了比较。计算结果表明，与现成求解器提供的两种方法相比，我们的分解方法在CPU时间和整体解决方案质量方面都更加有效。结果还表明，我们的MILP启发式方法提供了高质量的解决方案。