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The Weighted Sparsity Problem: Complexity and Algorithms

Many optimization algorithms involve repeated processing of a fixed set of linear constraints. If the constraint matrix A is preprocessed to make it sparser, algebraic operations should become faster. In many applications there is a priori information about the likelihood that each column will appear in a basis, which can be expressed as weights on the columns. This leads to considering the weighted sparsity problem (WSP): Find a row-equivalent constraint matrix with as small a weight of nonzeros as possible. The WSP is shown to be NP-hard even with a nondegeneracy assumption, and even if restricted to instances with at most three nonzeros per both row and column. WSP is shown to have a polynomial algorithm when the number of nonzeros per either row or column is limited to at most two. This contrasts with previous results that, assuming only nondegeneracy, the unweighted version of WSP does have a polynomial algorithm (this has proven to be practically useful in tests on real data). The polynomial algorit...