Total dual dyadicness and dyadic generating sets

Abstract

A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form $$\frac{a}{2^k}$$ for some integers a, k with $$k\ge 0$$ . A linear system $$Ax\le b$$ with integral data is totally dual dyadic if whenever $$\min \{b^\top y:A^\top y=w,y\ge \textbf{0}\}$$ for w integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of dyadic generating sets for cones and subspaces, the former being the dyadic analogue of Hilbert bases, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the density of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids, T-joins, cycles, and perfect matchings of a graph.