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 a 2 k for some integers a, k with k ≥ 0. A linear system Ax ≤ b with integral data is totally dual dyadic if whenever min{b ⊺ y ∶ A ⊺ y = w, y ≥ 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 matrices, T -joins, circuits, and perfect matchings of a graph.

Total Dual Dyadicness and Dyadic Generating Sets

“…Ax = 0, and for each i ∈ [n], we have q i | xi , so if xi = 0 then xi has a prime factor greater than or equal to p k+1 . Thus the 1-by-n matrix A satisfies the conclusion of Lemma 5.4 (1).…”

“…In Section 4, we focus on the size of the denominators of a solution to a feasible dyadic linear program. In particular, we show that if Ax ≤ b, x dyadic is feasible, where A ∈ Z m×n and b ∈ Z m , then there exists a 1 2 k -integral solution, where k ≤ log 2 n + (2n + 1) log 2 ( A ∞ √ n + 1) . Here A ∞ denotes the largest absolute value of an entry in A.…”

“…Proof. Denote by L 1 the set L as defined by (1) and by L 2 the set L as defined by (2). We need to show L 1 = L 2 .…”

“…A rational number is dyadic if it is an integer multiple of 1 2 k for some nonnegative integer k. Dyadic numbers are important for numerical computations because they have a finite binary representation, and therefore they can be represented exactly on a computer in floating-point arithmetic. When real or rational numbers are approximated by dyadic numbers on a computer, approximation errors may propagate and accumulate throughout the computations.…”

“…Take a mathematical point of view: Given a prime integer p ≥ 2, we say that a rational number is finitely p-adic if it is of the form r p k for some integer r and nonnegative integer k. This concept is closely related to the notion of the p-adic numbers introduced by Hensel, formally defined as the set of "finite-tailed" infinite series +∞ i=N a i p i where N ∈ Z, and a i ∈ Z and 0 ≤ a i < p for each i ≥ N . 1 The study of p-adic numbers gives rise to beautiful and powerful mathematics; see the excellent book by Gouvêa for more [17]. It can be readily checked that the set of finitely p-adic numbers is the set of finite series of the form M i=N a i p i , where M, N ∈ Z, M ≥ N , and 0 ≤ a i < p, a i ∈ Z for all N ≤ i ≤ M , justifying our terminology.…”