  We consider the chance-constrained binary knapsack problem (CKP), where the
item weights are independent and normally distributed. We introduce a
continuous relaxation for the CKP, represented as a non-convex optimization
problem, which we call the non-convex relaxation. A comparative study shows
that the non-convex relaxation provides an upper bound for the CKP, at least as
tight as those obtained from other continuous relaxations for the CKP.
Furthermore, the quality of the obtained upper bound is guaranteed to be at
most twice the optimal objective value of the CKP. Despite its non-convex
nature, we show that the non-convex relaxation can be solved in polynomial
time. Subsequently, we proposed a polynomial-time 1/2-approximation algorithm
for the CKP based on this relaxation, providing a lower bound for the CKP.
Computational test results demonstrate that the non-convex relaxation and the
proposed approximation algorithm yields tight lower and upper bounds for the
CKP within a short computation time, ensuring the quality of the obtained
bounds.
