Geodesic Property of Greedy Algorithms for Optimization Problems on Jump Systems and Delta-matroids

TL;DR

This paper investigates the geodesic property of greedy algorithms for optimization on jump systems and delta-matroids, showing that a specific implementation maintains this property, leading to new efficient algorithms.

Contribution

It demonstrates that a special implementation of the greedy algorithm has the geodesic property and introduces a new greedy algorithm for linear optimization on delta-matroids.

Findings

Special greedy algorithm implementation has the geodesic property.

Original greedy algorithm does not have the geodesic property.

New greedy algorithm for delta-matroids with the geodesic property.

Abstract

The concept of jump system, introduced by Bouchet and Cunningham (1995), is a set of integer points satisfying a certain exchange property. We consider the minimization of a separable convex function on a jump system. It is known that the problem can be solved by a greedy algorithm. In this paper, we are interested in whether the greedy algorithm has the geodesic property, which means that the trajectory of solution generated by the algorithm is a geodesic from the initial solution to a nearest optimal solution. We show that a special implementation of the greedy algorithm enjoys the geodesic property, while the original algorithm does not. As a corollary to this, we present a new greedy algorithm for linear optimization on a delta-matroid and show that the algorithm has the geodesic property.