On Finding Maximum Cardinality Subset of Vectors with a Constraint on Normalized Squared Length of Vectors Sum

TL;DR

This paper addresses the NP-hard problem of selecting the largest subset of vectors with a constraint on the normalized squared length of their sum, proposing an exact algorithm and comparing its performance to existing solvers.

Contribution

It introduces a novel exact algorithm for the problem, with pseudo-polynomial complexity in fixed dimensions, and provides computational comparisons with existing solvers.

Findings

Proposed algorithm outperforms COINBONMIN in low-dimensional cases.

The problem is NP-hard even for feasible solutions.

The algorithm is effective for fixed, small dimensions with integer data.

Abstract

In this paper, we consider the problem of finding a maximum cardinality subset of vectors, given a constraint on the normalized squared length of vectors sum. This problem is closely related to Problem 1 from (Eremeev, Kel'manov, Pyatkin, 2016). The main difference consists in swapping the constraint with the optimization criterion. We prove that the problem is NP-hard even in terms of finding a feasible solution. An exact algorithm for solving this problem is proposed. The algorithm has a pseudo-polynomial time complexity in the special case of the problem, where the dimension of the space is bounded from above by a constant and the input data are integer. A computational experiment is carried out, where the proposed algorithm is compared to COINBONMIN solver, applied to a mixed integer quadratic programming formulation of the problem. The results of the experiment indicate superiority of the proposed algorithm when the dimension of Euclidean space is low, while the COINBONMIN has an advantage for larger dimensions.