Because of the (mathcal {NP} -hardness of Problem (8), new algorithms are needed that can solve the TARP optimally but more efficiently than regular ILP solvers.
The proof of the NP-hardness of problems, or, NP-completeness of its decision version, involves a polynomial-time reduction from one of the well-known NP-complete problems.
No adversary can forge a valid signature due to the hardness of DL problem.
They proved NP hardness of the problem and linearized it in three different ways.
We develop a series of relaxed and equivalent models to reduce the hardness of the problem and provide theoretical results to show the equivalences.
Owing to the hardness of the problem, we propose an heuristic approach based on the combined utilization of evolutionary algorithms and other existing algorithms.
The hardness of this problem rests on the non-linear characteristics of the multidimensional well-production and pressure-drop functions, as well as the discrete routing decisions.
The method we describe in this section is based on the idea that notwithstanding this inherent computational hardness of the problem, it is reasonable to expect that if good n 1-sized and n 2-sized solutions are known, a (n 1+n 2 -sized solution better than one which would on average be obtained by a random draw can be hypothesized.
We first study the hardness of this problem and prove the following result: a succinct (or a compact) index cannot answer PRSS queries efficiently in the pointer machine model, and also not in the RAM model unless bounds on the well-researched orthogonal range query problem improve.
In particular, we study the problem of adding a minimum size set of points to a given unit disk graph in such a way that in the resulting graph any two original points have hop-distance at most a given bound D. After having proved the hardness of the problem, we propose two different bi-criteria algorithms that, conjunctively, provide logarithmic approximation ratio on both criteria.
Despite the hardness of the problem, homology-based methods appeared to be quite successful due to improved alignment methods and more and more available template structures.
