An important fundamental problem in e-commerce is to design a mechanism that computes optimal system-wide solutions despite the self-interest of individual users (buyers, sellers, brokers, etc.) and computational agents (buying agents, selling agents, etc.). This class of problems, formally called computational mechanism design problems, are similar in nature to the economic mechanism design problems arising in economics and game theory. Classic game-theoretic solutions for these problems are often prohibitively expensive computationally. For example, the Generalized Vickrey Auction (GVA) is a classical mechanism to solve the Combinatorial Allocation Problem (CAP). Despite its rich game theoretic properties, the GVA mechanism is impractical from a computational point of view as it presents a number of computational challenges starting from the NP-hard winner determination problem through hard valuation problem of the agents, to prohibitively high communication costs. A plethora of proposals has emerged in recent times to address these problems. One of the popular approaches proposed is based on using the primal-dual algorithm for solving the mechanism design problem. The objective of this paper is two fold. First, we bring out the challenges and difficulties in adopting the classical game-theoretic solutions to computational mechanism design problems. Next we show that LP duality can play a useful role in computational mechanism design. We discuss a representative algorithm based on duality, ”COMBAUCTION” for solving the combinatorial allocation problem.