# Generic Cospark of a Matrix Can Be Computed in Polynomial Time

## TL;DR

This paper proves that the generic cospark of a matrix, based on its sparsity pattern, can be computed efficiently in polynomial time, despite the NP-hardness of the general problem.

## Contribution

The paper introduces a polynomial-time algorithm to compute the generic cospark of a matrix from its sparsity pattern, linking probabilistic properties to computational efficiency.

## Key findings

- The generic cospark equals the maximum cospark over matrices with the same sparsity pattern.
- With probability one, the cospark matches the generic cospark for matrices with entries drawn from continuous distributions.
- The proposed algorithm computes the generic cospark in polynomial time.

## Abstract

The cospark of a matrix is the cardinality of the sparsest vector in the column space of the matrix. Computing the cospark of a matrix is well known to be an NP hard problem. Given the sparsity pattern (i.e., the locations of the non-zero entries) of a matrix, if the non-zero entries are drawn from independently distributed continuous probability distributions, we prove that the cospark of the matrix equals, with probability one, to a particular number termed the generic cospark of the matrix. The generic cospark also equals to the maximum cospark of matrices consistent with the given sparsity pattern. We prove that the generic cospark of a matrix can be computed in polynomial time, and offer an algorithm that achieves this.