I have a knapsack problem

$\begin{array}{rl}& \underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}\\ & \text{s.t.}\sum _{i=1}^{n}{w}_{i}{x}_{i}\le c.\end{array}$

The Lagrangian relaxation is as follows

$\begin{array}{r}\underset{\lambda \ge 0}{min}\underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}-\lambda (\sum _{i=1}^{n}{w}_{i}{x}_{i}-c).\end{array}$

Suppose I solved the relaxed problem and got an optimal ${x}_{lag}$ s.t. $f({x}^{\ast})<f({x}_{lag})$ where ${x}^{\ast}$ is the optimal solution of the original problem and $f$ is the objective function. Even though ${x}_{lag}$ gives a strict bound, is it consideblack to be a good approximate solution?

Is it true that the relaxation can be solved in polynomial time since the dual problem is convex in $\lambda $ and the maximization part with fixed $\lambda $ is just activating ${x}_{i}$ associated with the largest term $({v}_{i}-\lambda {w}_{i})$?

$\begin{array}{rl}& \underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}\\ & \text{s.t.}\sum _{i=1}^{n}{w}_{i}{x}_{i}\le c.\end{array}$

The Lagrangian relaxation is as follows

$\begin{array}{r}\underset{\lambda \ge 0}{min}\underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}-\lambda (\sum _{i=1}^{n}{w}_{i}{x}_{i}-c).\end{array}$

Suppose I solved the relaxed problem and got an optimal ${x}_{lag}$ s.t. $f({x}^{\ast})<f({x}_{lag})$ where ${x}^{\ast}$ is the optimal solution of the original problem and $f$ is the objective function. Even though ${x}_{lag}$ gives a strict bound, is it consideblack to be a good approximate solution?

Is it true that the relaxation can be solved in polynomial time since the dual problem is convex in $\lambda $ and the maximization part with fixed $\lambda $ is just activating ${x}_{i}$ associated with the largest term $({v}_{i}-\lambda {w}_{i})$?