Online Algorithms for Geometric Independent Set

TL;DR

This paper studies online algorithms for the maximum independent set problem in geometric graphs, providing bounds and algorithms that leverage geometric properties and randomization to improve performance.

Contribution

It introduces new bounds for deterministic and randomized online algorithms based on the independent kissing number and geometric representations, extending previous results to broader graph classes.

Findings

Greedy algorithm achieves a competitive ratio of 6 on graphs with bounded independent kissing number.

Randomized algorithms with geometric knowledge outperform deterministic bounds in specific geometric graph classes.

Expected competitive ratios depend polylogarithmically on object diameter ratios, with substantial performance guarantees.

Abstract

In the classical online model, the maximum independent set problem admits an $\Omega(n)$ lower bound on the competitive ratio even for interval graphs, motivating the study of the problem under additional assumptions. We first study the problem on graphs with a bounded independent kissing number $\zeta$, defined as the size of the largest induced star in the graph minus one. We show that a simple greedy algorithm, requiring no geometric representation, achieves a competitive ratio of $\zeta$. Moreover, this bound is optimal for deterministic online algorithms and asymptotically optimal for randomized ones. This extends previous results from specific geometric graph families to more general graph classes. Since this bound rules out further improvements through randomization alone, we investigate the power of randomization with access to geometric representation. When the geometric representation of the objects is known, we present randomized online algorithms with improved guarantees. For unit ball graphs in $\mathbb{R}^3$, we present an algorithm whose expected competitive ratio is strictly smaller than the deterministic lower bound implied by the independent kissing number. For $\alpha$-fat objects and for axis-aligned hyper-rectangles in $\mathbb{R}^d$ with bounded diameters, we obtain algorithms with expected competitive ratios that depend polylogarithmically on the ratio between the maximum and minimum object diameters. In both cases, the randomized lower bound implied by the independent kissing number grows polynomially with the ratio between the maximum and minimum object diameters, implying substantial performance guarantees for our algorithms.