# Improved Bounds for the Greedy Strategy in Optimization Problems with Curvature

## TL;DR

This paper introduces a new concept called partial curvature to improve greedy strategy bounds in optimization problems constrained by matroids, especially when the objective function is only defined on the matroid.

## Contribution

It provides necessary and sufficient conditions for extending polymatroid functions and introduces partial curvature to tighten performance bounds for greedy algorithms.

## Key findings

- Partial curvature bounds are generally tighter than total curvature bounds.
- Conditions for extending polymatroid functions to larger uniform matroids are characterized.
- An algorithm for constructing incremental extensions of polymatroid functions is provided.

## Abstract

Consider the problem of choosing a set of actions to optimize an objective function that is a real-valued polymatroid function subject to matroid constraints. The greedy strategy provides an approximate solution to the optimization problem, and it is known to satisfy some performance bounds in terms of the total curvature. The total curvature depends on the value of objective function on sets outside the constraint matroid. If we are given a function defined only on the matroid, the problem still makes sense, but the existing bounds involving the total curvature do not apply. This is puzzling: If the optimization problem is perfectly well defined, why should the bounds no longer apply? This motivates an alternative formulation of such bounding techniques. The first question that comes to mind is whether it is possible to extend a polymatroid function defined on a matroid to one on the entire power set. This was recently shown to be negative in general. Here, we provide necessary and sufficient conditions for the existence of an \emph{incremental} extension of a polymatroid function defined on the uniform matroid of rank $k$ to one defined on the uniform matroid of rank $k+1$, together with an algorithm for constructing the extension. Whenever a polymatroid objective function defined on a matroid can be extended to the entire power set, the greedy approximation bounds involving the total curvature of the extension apply. However, these bounds still depend on sets outside the constraint matroid. Motivated by this, we define a new notion of curvature called \emph{partial curvature}, involving only sets in the matroid. We derive necessary and sufficient conditions for an extension of the function to have a total curvature that is equal to the partial curvature. Moreover, we prove that the bounds in terms of the partial curvature are in general improved over the previous ones......