|
Within computer science and operations research, many combinatorial optimization problems are computationally intractable to solve exactly (to optimality). Many such problems do admit fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input. Randomized rounding is a widely used approach for designing and analyzing such approximation algorithms.〔 〕〔 〕 The basic idea is to use the probabilistic method to convert an optimal solution of a relaxation of the problem into an approximately optimal solution to the original problem. == Overview == The basic approach has three steps: # Formulate the problem to be solved as an integer linear program (ILP). # Compute an optimal fractional solution to the linear programming relaxation (LP) of the ILP. # Round the fractional solution of the LP to an integer solution of the ILP. (Although the approach is most commonly applied with linear programs, other kinds of relaxations are sometimes used. For example, see Goeman's and Williamson's semi-definite programming-based Max-Cut approximation algorithm.) The challenge in the first step is to choose a suitable integer linear program. Familiarity with linear programming is required, in particular, familiarity with how to model problems using linear programs and integer linear programs. But, for many problems, there is a natural integer linear program that works well, such as in the Set Cover example below. (The integer linear program should have a small integrality gap; indeed randomized rounding is often used to prove bounds on integrality gaps.) In the second step, the optimal fractional solution can typically be computed in polynomial time using any standard linear programming algorithm. In the third step, the fractional solution must be converted into an integer solution (and thus a solution to the original problem). This is called ''rounding'' the fractional solution. The resulting integer solution should (provably) have cost not much larger than the cost of the fractional solution. This will ensure that the cost of the integer solution is not much larger than the cost of the optimal integer solution. The main technique used to do the third step (rounding) is to use randomization, and then to use probabilistic arguments to bound the increase in cost due to the rounding (following the probabilistic method from combinatorics). There, probabilistic arguments are used to show the existence of discrete structures with desired properties. In this context, one uses such arguments to show the following: : ''Given any fractional solution of the LP, with positive probability the randomized rounding process produces an integer solution that approximates '' according to some desired criterion. Finally, to make the third step computationally efficient, one either shows that approximates with high probability (so that the step can remain randomized) or one derandomizes the rounding step, typically using the method of conditional probabilities. The latter method converts the randomized rounding process into an efficient deterministic process that is guaranteed to reach a good outcome. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Randomized rounding」の詳細全文を読む スポンサード リンク
|