The phrase "a ptas" is not correct or usable in written English.
It appears to be a typographical error or an incomplete phrase, lacking context or clarity.
Example: "I found a ptas in the document, but I couldn't understand its meaning."
Alternatives: "a piece" or "a part".
We also present a PTAS for the case that the Mj's are intervals of the same length.
(iii) We present a PTAS for the subproblem in which the edge costs are natural numbers {1,…,k} for some constant k.
This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.
We present a PTAS for this problem, when the underlying graph is a tree with the degrees of the vertices bounded by a constant, e.g. binary tree.
On the positive side, we find an exact solution technique for general SGPs based on an existing approach, and develop a PTAS (polynomial-time approximation scheme) for zero-sum SGP to more fundamentally overcome the computational obstacle.
Particularly, we obtain a (1e−ϵ -approximation algorithm for submodular objective functions, for any ϵ>0, and a PTAS for quadratic objective functions where the underlying matrix has fixed nonnegative rank.
