Introduction

Discrete optimization is a problem in which the decision variables assume discrete values from a specified set. Combinatorial optimization problems, on the other hand, are problems of choosing the best combination out of all possible combinations. Most combinatorial problems can be formulated as integer programs. In wireless networking and resource allocation, integer/combinatorial optimization problems are investigated with the efficient allocation of limited resources to meet desired objectives when the values of some or all of the variables are restricted to be integral. Constraints on basic resources, such as modulation, channel allocation, and coding rate restrict the possible alternatives that are considered feasible. For example, in 3G cellular networks, discrete processing gains for different codes give users different bandwidths for transmission. In a WLAN, the available time slots are occupied by different users. Consequently the allocation of time is restricted to a discrete nature. InWiMAX or Flash-OFDM, the distinct time-frequency slot is also allocated to the admitted users. Moreover, for practical implementation, the coding rate and adaptive modulation can have only discrete values. Even for the power control, the minimal step for the current cellular system is 1 dB. To design future wireless networks, it is of importance to study these integer optimizations, especially from an industrial implementation point of view.

The versatility of the integer/combinatorial optimization model stems from the fact that, in many practical problems, activities and resources, such as channel, user, and time slot, are indivisible.