LLL on Knapsack-eque problem

If we have a target upper bound $X$ for the $|x_i|$, then we expect the contribution of the $x_i$ to the square of the $\mathcal L_2$ norm of the short vector to be less than $nX^2$. If we therefore introduce a scaling factor $S\approx\sqrt n X$ and instead work with the lattice generated by the columns of $$B'=\begin{pmatrix} 1 & & & & \\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \\ Ss_1 & Ss_2 & \cdots & Ss_n & -St \end{pmatrix},$$ we will generate vectors $(x_1,\ldots,x_n,x_{n+1})^T$ where $$Sx_{n+1}t=x_1s_1+\cdots+x_ns_n.$$ Now the solution that we want with $|x_{n+1}|=1$ will have a squared $\mathcal L_2$ norm less than $S^2+nX^2\approx 2nX^2$ whereas any other solution will have $|x_{n+1}|>1$ and so squared $\mathcal L_2$ norm at least $4S^2\approx 4nX^2$. Thus our causal answer is always shorter than the non-causal answer and our SVP algorithm should be able to detect it.

One can view this as a use of Kannan's embedding to solve a close vector problem and go to greater length optimising $S$.