Relationship of Two Formulations for Shortest Bibranchings

TL;DR

This paper explores the relationship between two mathematical formulations of the shortest bibranching problem, revealing how they are interconnected through Benders decomposition and discrete convex analysis.

Contribution

It demonstrates how the valuated matroid intersection formulation can be derived from the linear programming formulation via Benders decomposition, linking polyhedral combinatorics and discrete convex analysis.

Findings

Valuated matroid intersection can be derived from LP formulation.

Integrality is preserved through Benders decomposition.

Connections between different formulations are established.

Abstract

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis.