🤖 AI Summary
This paper systematically compares the intrinsic difficulty of online minimization versus maximization problems, with particular focus on competitive analysis and complexity classification for prediction-augmented online maximization.
Method: It introduces the first prediction-augmented online complexity hierarchy tailored to maximization; establishes a unified hardness equivalence framework bridging minimization and maximization objectives; and develops rigorous reduction techniques grounded in combinatorial optimization problems—including interval scheduling, set packing, and online bounded-degree independent set.
Contributions/Results: It proves that Asymmetric String Guessing and Online Bounded-Degree Independent Set are competitively equivalent; derives tight competitive bounds and completeness classifications for multiple canonical problems; provides optimal or near-optimal algorithms alongside strong inapproximability lower bounds; and, for the first time, unifies the solvability frontier of online minimization and maximization problems under prediction augmentation.
📝 Abstract
We build on the work of Berg, Boyar, Favrholdt, and Larsen, who developed a complexity theory for online problems with and without predictions (arXiv:2406.18265), focussing on minimization problems with binary predictions, where they define a hierarchy of complexity classes that classifies online problems based on the competitiveness of best possible deterministic online algorithms for each problem. We continue their work, focussing on online maximization problems. First, we compare the competitiveness of the base online minimization problem from Berg, Boyar, Favrholdt, and Larsen, Asymmetric String Guessing, to the competitiveness of Online Bounded Degree Independent Set. Formally, we show that there exist algorithms of any given competitiveness for Asymmetric String Guessing if and only if there exist algorithms of the same competitiveness for Online Bounded Degree Independent Set, while respecting that the competitiveness of algorithms is measured differently for minimization and maximization problems. Moreover, we give several hardness preserving reductions between different online maximization problems, which imply new membership, hardness, and completeness results for the complexity classes. Finally, we show new positive and negative algorithmic results for (among others) Online Bounded Degree Independent Set, Online Interval Scheduling, Online Set Packing, and Online Bounded Minimum Degree Clique.