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* If the total value of the optimal knapsack solution is at most 1, then we say that '''y''' is feasible.
* If the total value of the optimal knapsack solution is larger than 1, then we say that '''y''' is infeasible, and the items in the optimal knapsack solution correspond to a configuration that violates a constraint (since <math>\mathbf{a}\cdot \mathbf{y} > 1</math> for the vector '''a''' that corresponds to this configuration).
The knapsack problem can be solved by [[dynamic programming]] in [[pseudo-polynomial time]]: <math>O(m\cdot V)</math>, where ''m'' is the number of inputs and ''V'' is the number of different possible values. To get a polynomial-time algorithm, we can solve the knapsack problem approximately, using input rounding. Suppose we want a solution with tolerance <math>\delta</math>. We can round each of <math>y_1,\ldots,y_m</math> down to the nearest multiple of ''<math>\delta</math>''/''n''. Then, the number of possible values between 0 and 1 is ''n''/''<math>\delta</math>'', and the run-time is <math>O(m n /\delta)</math>. The solution is at least the optimal solution minus ''<math>\delta</math>''/''n''.
=== Ellipsoid method with an approximate separation oracle ===
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O\left(m^8 \ln{m} \ln^2(\frac{m n}{g h}) + \frac{m^4 n \ln{m}}{h}\ln(\frac{m n}{g h}) \right)</math>,
The expected total run-time of the [[randomized algorithm]] is: <math>O\left(m^7 \log{m} \log^2(\frac{m n}{g h}) + \frac{m^4 n \log{m}}{h}\log(\frac{m n}{g h}) \right)</math>.
== End-to-end algorithms ==
Karmarkar and Karp presented three algorithms, that use the above techniques with different parameters. The run-time of all these algorithms depends on a function <math>T(\cdot,\cdot)</math>, which is a polynomial function describing the time it takes to solve the fractional LP with tolerance ''h''=1, which is, for the deterministic version,<math>T(m,n)\in O(m^8\log{m}\log^2{n} + m^4 n \log{m}\log{n} )</math>.
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