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There is a solution with something like $\tilde{O}(L!)$ complexity. Enumerate all permutations of the $L$ rows. Consider one such permutation, and permute the rows accordingly. Then apply the follow …

Yes, Freivald's algorithm meets all of your requirements.

It is NP-hard, by reduction from the maximum balanced biclique problem. Suppose you have a bipartite graph $G$, and the goal is to test whether there is a balanced biclique of size $d$. Let $M$ denot …

A more efficient approach is to hash each row, insert them into a hash table, and find duplicates in that way. The running time will be $O(n^2)$. You can implement something like this using only line …

This smells like an XY problem. A neural network is not a good tool to use for computing the eigenvalue decomposition. Machine learning is best for problems where you have examples of input-output p …

You're right. It is a sort of 0-1 knapsack problem. It is NP-complete in general. So you will need to settle for approximate solutions or heuristics or algorithms that only work for short strings o …

Your problem is NP-hard. In particular, it is equivalent to maximum edge weight biclique problem (MWBP), which is known to be NP-hard. Maximum edge weight biclique MWBP is the following problem: give …

I believe the problem is NP-hard, even with only set(), and is even hard to approximate within a factor better than 2. See https://mathoverflow.net/q/105164/37212, https://cstheory.stackexchange.com/ …

This looks very hard to me. I wouldn't be surprised if it is NP-hard. If I had to solve it in practice, my first thought would be to try to solve it using a SAT solver or an ILP solver. I outline be …

I expect that solving the problem exactly might be NP-hard. However, here is an approach that I expect will probably be good enough for practical purposes. It uses a subroutine to solve the following …

You can achieve $O(\log n)$ time for all operations, by using a balanced binary search tree for the index that maps from a row index to the row. Such a tree can support $O(\log n)$ time lookup of any …

A random bijective function should satisfy this property with high probability for any specific set of indices. So, I suggest you use a random bijection on $[n^2]$. If $n$ is very large, you can comp …

(Both matrices are in reduced row echelon form, so both are valid inputs.) However this algorithm will output the same thing for both inputs. Hence this algorithm cannot be correct. …

I will assume the original matrix has $n-1$ rows, $n$ columns, and the rows are linearly independent (this is easy to check; and if it is not the case, then the problem is trivial). Adding a new row $ …

The problem is NP-hard; it is at least as hard as the biclique problem. If you can solve the problem for a single shaped box, you can solve for all boxes by just iterating over all the boxes. So your …