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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., $v_{i} \in \{0,1/2,1\}$.

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$\frac 1n$ 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 $a^\top v \ge 1$):

  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.

  2. 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 $v_5 = v_6 = v_7 = v_8 = v_9 = \frac 14$.

  3. 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.

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  • $\begingroup$ +1. By the way, do you know anything about 0,1-quadratic programming? This question of mine didn't get a lot of attention: cs.stackexchange.com/q/109039/61663, it's about a quadratic optimization problem on variables that can be in {-1,1}, which is equivalent to {0,1} if you just add 1 and divide by 2 on all variables. $\endgroup$ – Nike Dattani Apr 25 at 2:57
  • $\begingroup$ @NikeDattani, sorry, don't know anything about it. About the question you've mentioned: I'm confused what kind of solution you want (I have the same confusion as Juho in comments). Do you want a poly-time approximation (in this case check ".878-Approximation Algorithms for MAX CUT") or do you want something that works reasonably well in practice (possibly without any guarantees)? $\endgroup$ – user114966 Apr 25 at 3:12
  • $\begingroup$ It doesn't need to be poly-time. It just needs to work reasonably well in practice, and possibly without any guarantees. I can check out the .878-approximation algorithms, and if you were to suggest that in an answer, I wouldn't mind that either! $\endgroup$ – Nike Dattani Apr 25 at 4:52
  • $\begingroup$ @NikeDattani, you may try approaches I mention here, except for spectral partitioning. They are for MIN-CUT, and it should be straightforward to adjust them for MAX-CUT. $\endgroup$ – user114966 Apr 26 at 1:00
  • $\begingroup$ Thanks! That question was from 2 years ago and it will be a while before I can try these, would you consider writing a short answer on the question with some of these suggestions so that when I get around to doing the tests, I'll have the suggestions in one place? I may not remember to look at the comments of this question! $\endgroup$ – Nike Dattani Apr 26 at 1:06

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