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Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and an integer $m$ is there a polynomial in $n$ algorithm to find a $\{-1,0,+1\}$ matrix $T$ such that $$\mathsf{det}(L+T\odot M)=m$$ where $\odot$ refers to $ij$th entry of $T\odot M$ being $T_{ij}M_{ij}$?

  1. Is there any canonical approach to this problem that can beat $2^{O(n)}$ time?

  2. An evidence $2^{O(n)}$ could not be beaten would come if we show that 'Given two $\{0,1\}^{n\times n}$ matrices $L$ and $M$ and integers $m,t$ is there a $\{-1,0,+1\}$ matrix $T$ with $\|T\|_F\leq t$ such that $\mathsf{det}(L+T\odot M)=m$ holds?' is $NP$-complete (it is in $NP$ however there seems no canonical reduction from any $NP$ complete problem and I find every theoretical support for an $NP$ complete reduction seems to turn to absurdity here after the presented answer I think it should be $NP$ complete)?

It seems in all likelihood this problem is in $P$.

If $T$ is a planar bipartite signed biadjacency then the case could be special and might be handlable within $P$. This possibly yields difficulties in reductions.

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    $\begingroup$ This does not look like a matrix completion problem to me. $\endgroup$ – Rodrigo de Azevedo Jun 22 '19 at 22:43
  • $\begingroup$ Might as well write just "polynomial in $n$", because $\log |m| \ll n \log n$ for any achievable $m$: each entry of $L + T \odot M$ is between $-1$ and $2$, so $\left|\det(L + T \odot M)\right| \leq 2^n n!$. (Yes, Hadamard's inequality gives an even better bound, but this gives only a factor of $2$ improvement in $\log|m|$.) $\endgroup$ – Noam D. Elkies Jun 23 '19 at 4:14
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I will prove it is NP-complete if $T$ is restricted to $\pm 1$.

Let $k_1,\ldots,k_n$ be an arbitrary list of integers.

Suppose the cofactors of $L$ along the top row are $c_1,\ldots,c_n$ and all not zero. Define $M$ to be $k_1/c_1,\ldots,k_n/c_n$ along the top row and 0 everywhere else. Define $m=0$.

Now the problem is to divide $k_1,\ldots,k_n$ into two parts of equal sum, which is the NP-complete problem PARTITION.

I doubt if allowing zeros in $T$ will suddenly make it polynomial, but I don't see the details. Maybe someone else does.

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  • $\begingroup$ So you think there is no sub exp algorithm? $\endgroup$ – T.... Jun 23 '19 at 9:08
  • $\begingroup$ If the conjecture that NP-complete problems need exponential time is correct, yes. $\endgroup$ – Brendan McKay Jun 23 '19 at 9:10
  • $\begingroup$ No not all NP complete problems need fully exponential time and that is known but many canonical reductions do need. $\endgroup$ – T.... Jun 23 '19 at 9:11
  • $\begingroup$ The sum of absolute values of entries in your reduction is $\Omega(n)$ and so it seems highly likely some variant should handle $0$ case as well. $\endgroup$ – T.... Jun 23 '19 at 9:14
  • $\begingroup$ I was taking "exponential" to mean $\Omega(e^{n^{\epsilon}})$ for some $\epsilon\gt 0$, which is one of the two common definitions. $\endgroup$ – Brendan McKay Jun 23 '19 at 9:16

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