Linear Programming complementation

Abstract

In this paper we introduce a new operation for Linear Programming (LP), called LP complementation, which resembles many properties of LP duality. Given a maximisation (resp. minimisation) LP P, we define its complement Q as a specific minimisation (resp. maximisation) LP which has the same objective function as P. Our central result is the LP complementation theorem, that relates the optimal value

of P and the optimal value

of its complement by

. The LP complementation operation can be applied if and only if P has an optimum value greater than 1.

To illustrate this, we first apply LP complementation to hypergraphs. For any hypergraph H, we review the four classical LPs, namely covering $K\left(H\right)$, packing $P\left(H\right)$, matching $M\left(H\right)$, and transversal $T\left(H\right)$. For every hypergraph $H=(V,E)$, we call

the complement of H. For each of the above four LPs, we relate the optimal values of the LP for the dual hypergraph

to that of the complement hypergraph

(e.g.

).

We then apply LP complementation to fractional graph theory. We prove that the LP for the fractional in-dominating number of a digraph D is the complement of the LP for the fractional total out-dominating number of the digraph complement

of D. Furthermore we apply the hypergraph complementation theorem to matroids. We establish that the fractional matching number of a matroid coincide with its edge toughness.

As our last application of LP complementation, we introduce the natural problem Vertex Cover with Budget (VCB): for a graph $G=(V,E)$ and a positive integer b, what is the maximum number $t_b$ of vertex covers $S_1,\dots ,S_{t_b}$ of G, such that every vertex $v\in V$ appears in at most b vertex covers? The integer b can be viewed as a “budget” that we can spend on each vertex and, given this budget, we aim to cover all edges for as long as possible. We relate VCB with the LP $Q_G$ for the fractional chromatic number ${\chi }_f$ of a graph G. More specifically, we prove that, as $b\to \infty$, the optimum for VCB satisfies $t_b\sim t_f\cdot b$, where $t_f$ is the optimal solution to the complement LP of $Q_G$. Finally, our results imply that, for any finite budget b, it is NP-hard to decide whether $t_b\ge b+c$ for any $1\le c\le b-1$.