Approximation

Approximation ratio

【Definition】 An algorithm has an approximation ratio of ρ\rhoρ(n) if, for any input of size n, the cost C of the solution produced by the algorithm is within a factor of ρ\rhoρ (n) of the cost C* of an optimal solution: max(CC∗,C∗C)≤ρ(n)\frac{C}{C^*},\frac{C^*}{C}) \leq \rho(n)C∗C,CC∗)≤ρ(n)
If an algorithm achieves an approximation ratio of ρ\rhoρ(n), we call it a ρ\rhoρ(n)-approximation algorithm.

【Definition】 An approximation scheme for an optimization problem is an approximation algorithm that takes as input not only an instance of the problem, but also a value ϵ\epsilonϵ > 0 such that for any fixed ϵ\epsilonϵ, the scheme is a (1+ ϵ\epsilonϵ)-approximation algorithm.
We say that an approximation scheme is a polynomial-time approximation scheme (PTAS) if for any fixed ϵ\epsilonϵ > 0, the scheme runs in time polynomial in the size n of its input instance.
eg. O((1/ϵ)2n3)O((1/ \epsilon)^2 n^3)O((1/ϵ)2n3) --> FPTAS (fully …) , O(nϵ/2)O(n^{\epsilon/2})O(nϵ/2), …

Approximate Bin Packing

Online Algorithms

next fit

void NextFit ( )
{   read item1;while ( read item2 ) {if ( item2 can be packed in the same bin as item1 )place item2 in the bin;elsecreate a new bin for item2;item1 = item2;} /* end-while */
}

Let M be the optimal number of bins required to pack a list I of items. Then next fit never uses more than 2M – 1 bins. There exist sequences such that next fit uses 2M – 1 bins.

first fit

void FirstFit ( )
{   while ( read item ) {scan for the first bin that is large enough for item;if ( found )place item in that bin;elsecreate a new bin for item;} /* end-while */
}

Let M be the optimal number of bins required to pack a list I of items. Then first fit never uses more than 17M / 10 bins. There exist sequences such that first fit uses 17(M – 1) / 10 bins.
best fit

Off-line Algorithm

first fit decreasing
Let M be the optimal number of bins required to pack a list I of items. Then first fit decreasing never uses more than 11M / 9 + 6/9 bins. There exist sequences such that first fit decreasing uses 11M / 9 + 6/9 bins.
best fit decreasing

Exercises

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