Performance Bounds for the $k$-Batch Greedy Strategy in Optimization Problems with Curvature

TL;DR

This paper analyzes the performance bounds of the $k$-batch greedy algorithm for optimization problems with curvature, providing harmonic and exponential bounds under submodularity and matroid constraints, and compares different batch sizes.

Contribution

It introduces performance bounds for the $k$-batch greedy strategy using total curvature, extending previous results to general and uniform matroid constraints.

Findings

The $k$-batch greedy strategy has a harmonic bound of 1/(1+α_k) for general matroids.

For uniform matroids, the strategy satisfies an exponential bound involving α_k.

The $k$-batch greedy strategy outperforms the $k_1$-batch strategy in bounds when k_1 divides k.

Abstract

The $k$-batch greedy strategy is an approximate algorithm to solve optimization problems where the optimal solution is hard to obtain. Starting with the empty set, the $k$-batch greedy strategy adds a batch of $k$ elements to the current solution set with the largest gain in the objective function while satisfying the constraints. In this paper, we bound the performance of the $k$-batch greedy strategy with respect to the optimal strategy by defining the total curvature $\alpha_k$. We show that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy satisfies a harmonic bound $1/(1+\alpha_k)$ for a general matroid constraint and an exponential bound $\left(1-(1-{\alpha}_k/{t})^t\right)/{\alpha}_k$ for a uniform matroid constraint, where $k$ divides the cardinality of the maximal set in the general matroid, $t=K/k$ is an integer, and $K$ is the rank of the uniform matroid. We also compare the performance of the $k$-batch greedy strategy with that of the $k_1$-batch greedy strategy when $k_1$ divides $k$. Specifically, we prove that when the objective function is nondecreasing and submodular, the $k$-batch greedy strategy has better harmonic and exponential bounds in terms of the total curvature. Finally, we illustrate our results by considering a task-assignment problem.