How can you find the best approximation ratio?

It measures how well an algorithm's answer approximates the ideal solution for a certain issue. The ratio is defined as the greatest value of the algorithm's solution divided by the smallest (optimal) solution. A lower approximation ratio suggests a more accurate method whereas a higher ratio indicates a greater departure from the ideal answer. Finding algorithms with low approximation ratios is critical in cases when obtaining the precise best solution is computationally difficult or impracticable.

Approximation is one to the vital tasks for engineering because it always has to happen, its just about cost trade offs. First thing is to establish the sensitivity of the measure, shockingly often for the inexperienced, getting in the ballpark is good enough, in which case just having easy to remember standards could solve the problem. If there are significant costs associated with an error then start linear (principle component analysis, first derivative, continued fraction, or similar) and if the cost structure says there is still low hanging fruit then up the algorithm order by one. Repeat until you have the simplest thing that's too expensive to improve.

It is worth noting the famous 0.878 approximation ratio for MAXCUT and MAX2SAT by Goemans and Williamson. The maximum cut problem (MAXCUT) involves finding a set of vertices S that maximizes the weight of the edges in the cut (S,¬S); that is, the weight of the edges with one endpoint in S and the other in ¬S. The problem is NP-hard. The approximation involves relaxing the constraints of vertices belonging to S or ¬S (e.g., 1 or -1) to them being on a unit sphere. The Goemans-Williamson algorithm then chooses a random hyperplane through the origin and partitions the vertices according to the side of the hyperplane they fall on. The value of the approximation guarantee α is given by the expression α=min [0<θ≤π] 2/π θ/(1−cosθ)

This is a difficult question to answer for a combinatorics problem but easier when it's a simple computational problem. For instance. the fast square root algorithm is able to use the IEEE data storage system to approximate the square root from its logarithmic representation and thus it's able to get within 1% of the final value in a single Newton step. Without this initial optimization, an average of 5 iterations is required. So to get best approximation ratio, a great understanding of the problem , the programming language as well as data type and number system is required. since all these skills played a role in that extreme optimization of inverse square root algorithm.