Total dual dyadicness and dyadic generating sets

TL;DR

This paper explores the concept of total dual dyadicness in linear systems with integral data, characterizing it through dyadic generating sets, and compares it to total dual integrality, revealing unique properties due to dyadic rationals.

Contribution

It introduces dyadic generating sets as a new tool to characterize total dual dyadicness and provides a co-NP characterization, enhancing understanding of polyhedral integrality.

Findings

Dyadic generating sets are a key to understanding total dual dyadicness.

Total dual dyadicness differs from total dual integrality due to dyadic rationals' density.

Examples include dyadic matrices, T-joins, cycles, and perfect matchings.

Abstract

A vector is \emph{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 $k\geq 0$. A linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ 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 \emph{dyadic generating sets for cones and subspaces}, the former being the dyadic analogue of \emph{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 \emph{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 matrices, $T$-joins, cycles, and perfect matchings of a graph.