On the Theory of Function Placement and Chaining for Network Function Virtualization.

Network function virtualization (NFV) can significantly reduce the operation cost and speed up the deployment for network services to markets. Under NFV, a network service is composed by a chain of ordered virtual functions, or we call a "network function chain." A fundamental question is when given a number of network function chains, on which servers should we place these functions and how should we form a chain on these functions? This is challenging due to the intricate dependency relationship of functions and the intrinsic complex nature of the optimization. In this paper, we formulate the function placement and chaining problem as an integer optimization, where each variable is an indicator whether one service chain can be deployed on a configuration (or a possible function placement of a service chain). While this problem is generally NP-hard, our contribution is to show that it can be mapped to an exponential number of min-cost flow problems. Instead of solving all the min-cost problems, one can select a small number of mapped min-cost problems, which are likely to have a low cost. To achieve this, we relax the integer problem into a fractional linear problem, and theoretically prove that the fractional solutions possess some desirable properties, i.e., the number and the utilization of selected configurations can be upper and lower bounded, respectively. Based on such properties, we determine some "good" configurations selected from the fractional solution and determine the mapped min-cost flow problem, and this helps us to develop efficient algorithms for network function placement and chaining. Via extensive simulations, we show that our algorithms significantly outperform state-of-art algorithms and achieve near-optimal performance.