11 can be reduced to 10

TL;DR

This paper improves bounds on the integer completion property of 2x2 doubly nonnegative matrices, reducing the maximum value from 11 to 10, and for some small matrices, down to 9.15.

Contribution

It proves that the upper bound for the integer completion property can be lowered from 11 to 10, and further to 9.15 for specific small matrices.

Findings

The maximum icpr(A) for 2x2 doubly nonnegative matrices is at most 10.

For certain small matrices, the icpr(A) can be as low as 9.15.

The previous bound of 11 is not tight for all cases.

Abstract

Laffey and Smigoc proved that for every 2x2 doubly nonnegative integer matrix A, icpr(A) is less than or equal to 11. We prove that 11 can be replaced by 10, and show that for many small matrices, even by 9.15