Comparing the Hardness of Online Minimization and Maximization Problems with Predictions

We build on the work of Berg, Boyar, Favrholdt, and Larsen, who developed a complexity theory for online problems with and without predictions (IJTCS-FAW, volume 15828 of LNCS, Springer, 2025) 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. Their work focused on online minimization problems and we continue their work by considering online maximization problems. 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 exists algorithms of any given competitiveness for Asymmetric String Guessing if and only if there exists 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. Beyond this, 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 Degree Clique.

💡 Research Summary

This paper extends the recent complexity theory for online problems with predictions—originally developed for minimization problems—to the realm of maximization problems. The authors build on the framework of Berg, Boyar, Favrholdt, and Larsen (IJTCS‑FAW 2025), which classifies online problems into complexity classes based on the competitiveness of the best deterministic online algorithms, measured against a pair of prediction‑error functions (Δ₀, Δ₁). The original theory used (1, Δ)‑Asymmetric String Guessing (ASG_Δ) as the canonical complete problem for each class.

The central contribution is the introduction of Online Δ‑Bounded‑Degree Independent Set with Predictions (IS_Δ) as a maximization counterpart. In the vertex‑arrival model, each arriving vertex comes with a binary prediction indicating whether it belongs to an optimal independent set. The algorithm must output a 0/1 decision; the profit is the number of vertices correctly kept (i.e., 1 − decision). The authors define a “strict online max‑reduction” that maps algorithms and instances between two maximization problems while preserving competitiveness, optimal value up to an additive constant κ, and the error measures. This reduction framework mirrors the strict online reductions used for minimization but is tailored to the maximization setting.

Using this tool, they prove a bidirectional equivalence: for any competitiveness factor ρ, there exists a ρ‑competitive algorithm for ASG_Δ if and only if there exists a ρ‑competitive algorithm for IS_Δ. The proof carefully handles the different definitions of competitiveness (cost ≤ ρ·OPT + c for minimization versus OPT ≤ ρ·profit + c for maximization) and shows that the error terms are unchanged. Consequently, IS_Δ is shown to be 𝒞_{Δ, Δ₀, Δ₁}‑complete, i.e., it lies in the same complexity class as ASG_Δ and is as hard as any problem in that class.

Having established IS_Δ as a complete problem, the paper proceeds to construct hardness‑preserving reductions from IS_Δ to several well‑studied online maximization problems:

• Online Interval Scheduling (Sch_Δ) – jobs are mapped to vertices, overlapping jobs become edges; predictions indicate whether a job belongs to an optimal schedule. The reduction preserves the number of prediction errors and the additive constant, proving Sch_Δ is 𝒞_{Δ, Δ₀, Δ₁}‑hard.
• Online Set Packing (SP_Δ) – each set becomes a vertex, intersecting sets become edges; the same reduction technique shows SP_Δ is 𝒞_{Δ, Δ₀, Δ₁}‑complete.
• Online Bounded‑Degree Clique (Cli_Δ) – via a complementary construction, a clique instance is reduced to an independent‑set instance, establishing hardness for Cli_Δ.
• Additional reductions are given for variants such as Online Bounded‑Degree Vertex Cover, Online Bounded‑Degree Matching, and others, extending the hardness graph originally presented in