Lower bound on positive coefficients of the optimum of 0,1-linear programming problem I have an instance linear programming such that the coefficients and the constant terms are 0 or 1.
Formally, the set of variables is denoted as $V$ and $|V| = n$. There are $m$ constraints, formed as
$$ \mathbf{a}_{1}^{\top} \mathbf{v} \leq 1 $$
or
$$ \mathbf{a}_{2}^{\top} \mathbf{v} \geq 1 $$
where $\mathbf{a}_{1}, \mathbf{a}_{2} \in \{0,1\}^{n}$ and $\mathbf{v} = [v_{1}, v_{2}, \ldots, v_{n}]^{\top}$. (Assume that $0 \leq v_{i} \leq 1$).
The objective function is
$$\min \sum_{i}v_{i} = [1,1, \ldots, 1] \cdot\mathbf{v}$$
I am hoping to find a constant $c \geq 0$ (depending only on $n$) such that there always exist an optimum $\mathbf{v}^{*} \in [0,1]^{n}$ satisfying $v^{*}_{i} = 0$ or $v^{*}_{i} \geq c$.
Does it hold for $c = 1/n$? Is there any positive lower bound on $c$?
For example, consider the LP relaxation of the vertex cover problem. The constraints are formed as
$v_{i} + v_{j} \geq 1$, for each edge $v_{i}v_{j} \in E(G)$. We know that there exists a half-integral optimum, i.e., $v_{i} \in \{0,1/2,1\}$.
$\frac 1n$ is not a lower bound.
For example, let $n=10$.
Consider 3 set of equations (you can get equalities by considering both $a^\top v \le 1$ and $a^\top v \ge 1$):

\begin{align}
v_1 + v_2 + v_3 \qquad = 1\\
\qquad v_2 + v_3 + v_4 = 1\\
v_1 \qquad  + v_3 + v_4 = 1\\
v_1 + v_2 \qquad + v_4 = 1\\
\end{align}
It's easy to check that $v_1=v_2=v_3=v_4=\frac 13$ is the unique solution.

Consider $v_5 + v_6 + v_7 + v_8 + 0 \cdot v_9 = 1$ and, similarly to the above case, all cyclic shifts of $(v_5, \ldots, v_9)$. Again, the unique solution is $v_5 = v_6 = v_7 = v_8 = v_9 = \frac 14$.

The last equation is
$$v_1 + v_2 \qquad + v_5 \qquad + v_{10} = 1$$
Then $v_{10} = 1 - 2 \cdot \frac 13 - \frac 14 = \frac 1 {12}$.


Generalizing it, you can always obtain coefficient $\frac 1{k (k+1)}$ using $2k + 4$ variables.
I think that by properly selecting numbers $p_1, \ldots, p_k$, you should be able to get a coefficient as small as $O\left(\frac 1 {\prod_i p_i}\right)$ with $O\left(\sum_i p_i\right)$ variables.