# General Bounds for Incremental Maximization

## TL;DR

This paper develops a theoretical framework for incremental solutions to cardinality constrained maximization problems, providing bounds on their competitive ratios and analyzing the performance of greedy algorithms under relaxed submodularity conditions.

## Contribution

It introduces a general framework for incremental maximization, establishes competitive ratio bounds, and analyzes greedy algorithms with relaxed submodularity assumptions.

## Key findings

- A 2.618-competitive incremental algorithm for a broad class of problems.
- No incremental algorithm can have a competitive ratio below 2.18.
- The greedy algorithm achieves a 2.313 competitive ratio under relaxed submodularity.

## Abstract

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $k\in\mathbb{N}$ that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all $k$ between the incremental solution after $k$ steps and an optimum solution of cardinality $k$. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general $2.618$-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below $2.18$ in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be $1.58$-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) ($b$-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) $2.313$ for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.