Combinatorial allocation problem

Many iterative combinatorial auctions, including the DGS auction [36] and the /Bundle and /BEA auctions can be interpreted within the primal-dual design methodology described in Section 2.4. The following three steps are important in extending the primal-dual framework described in Section 2.4 to the combinatorial allocation problem ... 

Problem formulations [ 1-3 ] for designing lead-generation library under different constraints belong to a class of combinatorial resource allocation problems, which have been widely studied. They arise in many different applications such as minimum distortion problems in data compression (11), facility location problems (12), optimal quadrature rules and discretization of partial differential equations (13), locational optimization problems in control theory (9), pattern recognition (14), and neural networks... 

The discrete decisions involved in both design and retrofit situations (for example, equipment number and allocation, task allocation) lead to an inherent combinatorial aspect, which is challenging from the computational complexity point of view (Pekny and Reklaitis, 1998). These problems are identified as NP-hard (Non-deterministic Polynomial-time hard) problems (Garey and Johnson, 1979), so that there is no known algorithm that can solve the problem in a polynomial time. This feature makes stochastic methods particularly attractive to tackle such problems. 

It is important to note that a growing area of multi-unit auction literature that has been left out of the discussion below is the design and use of combinatorial auctions. These auctions, where bidders can submit package or combinational bids, are often desirable when bidders realize synergies across objects in a multi-object auction. While extremely useful in helping to capture synergies, combinatorial auctions can be quite difficult to solve for the allocation that maximizes the seller s revenue (known as the winner determination problem). 

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