Search

A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.

A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that, given any clutter $\mathcal{C}$ over $2n$ elements via a filter oracle, decides whether or not $\mathcal{C}$ is ideal. We prove that in the worst case, $\mathfrak{A}_{2n}$ must make at least $2^n$ calls to the filter oracle. Our proof uses the theory of cuboids.

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., Comment: 22 pages