Exploiting Hidden Structure in Selecting Dimensions that Distinguish Vectors

TL;DR

This paper investigates the computational complexity of the Distinct Vectors problem, revealing a dichotomy based on row distances in binary matrices and connecting it to hitting set problems for broader cases.

Contribution

It establishes a complexity dichotomy for binary matrices based on row distances and links the problem to hitting sets, providing new tractability and intractability results.

Findings

Polynomial-time solvability when H <= 2 ceil(h/2) + 1

NP-completeness otherwise

Connections to hitting set problems for general matrices

Abstract

The NP-hard Distinct Vectors problem asks to delete as many columns as possible from a matrix such that all rows in the resulting matrix are still pairwise distinct. Our main result is that, for binary matrices, there is a complexity dichotomy for Distinct Vectors based on the maximum (H) and the minimum (h) pairwise Hamming distance between matrix rows: Distinct Vectors can be solved in polynomial time if H <= 2 ceil(h/2) + 1, and is NP-complete otherwise. Moreover, we explore connections of Distinct Vectors to hitting sets, thereby providing several fixed-parameter tractability and intractability results also for general matrices.