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In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.:

$$ A x = b $$

and the goal is to find a solution $x$ with as few nonzero variables as possible.

The problem is known to be NP-hard and hard to approximate to within a constant multiplicative factor (see the wikipedia page for details).

My question is: is anything known about additive approximations? In particular: what is the complexity of finding a solution that has at most $\text{OPT}+d$ nonzero variables, where $\text{OPT}$ is the smallest number of nonzero variables in a solution, and $d$ is some constant?

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Unless $P=NP$, there does not exist a polynomial time approximation algorithm that returns solutions of value at most $\text{OPT}+d$.

Suppose otherwise, and consider some fixed $d$ for which there does exist such an approximation algorithm $A$. We construct a new multiplicative polynomial time approximation algorithm from $A$, that always returns solutions of value at most $2\cdot\text{OPT}$. Hence consider an arbitrary instance $I$ of Min-RVLS:

  • For all $k=1,\ldots,d$ check whether $I$ has a solution $x$ with $k$ nonzero variables. This can be done by checking every subset $S$ of $k$ Variables: Set all variables outside $S$ to zero, and solve the linear equation system for the remaining variables. Since there are only $O(n^d)$ subsets to be checked, the overall running time is polynomial. (Here we use that $d$ is a fixed integer.)
  • If the first step does not reveal any feasible solution, we know that $\text{OPT}>d$. We call the additive approximation algorithm $A$ and find in polynomial time a feasible solution of value at most $\text{OPT}+d<2\cdot\text{OPT}$.

As we either detect an optimal solution (first step) or a solutions with value at most twice the optimum (second step), the resulting polynomial time approximation algorithm has a multiplicative approximation guarantee of $2$.

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  • $\begingroup$ In fact it looks like a general scheme for reducing a multiplicative approximation to an additive approximation. Good to keep that in mind. Thanks! $\endgroup$ – Erel Segal-Halevi Jan 8 at 11:16
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Even if $d = d(n) \le n^{1-\epsilon}$ for an $m \times n$ matrix $A$, no additive approximation $\mathrm{OPT} + d$ is possible unless $\mathsf{P} = \mathsf{NP}$. Let $N = n^k$ for some constant $k$, and take the Kronecker product $A' = I_{N} \otimes A$, and $b' = e_{N} \otimes b$, where $I_N$ is the $N\times N$ identity and $e_N$ is the $N\times 1$ all-ones vector. This is equivalent to just repeating the system of equations $N$ times with fresh variables each time. Then clearly the optimal solution to $A' x' = b'$ has $N\cdot\mathrm{OPT}$ nonzero entries. If you have a $d(Nn) \le n^{(1 -\epsilon)(k+1)}$ additive approximation for this problem, then at least one of the $N$ instances of the original system of equations has a solution with at most $\mathrm{OPT} + d(Nn)/N \le \mathrm{OPT} + n^{1-\epsilon(k+1)}$ nonzero entries. But $n^{1-\epsilon(k+1)} < 1$ for a large enough $k$, so one of your $N$ instances is solved optimally.

This is quite common, BTW: most optimization problems allow some kind of "amplification" like this, which usually implies no non-trivial additive approximation is possible.

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