Comparing the Hardness of Online Minimization and Maximization Problems with Predictions

TL;DR

This work extends the binary-predictions online complexity framework to online maximization and proves that, for every $t$, the base problem $\textsc{ASG}_{t}$ is exactly as hard as Online $t$-Bounded Degree Independent Set ($\textsc{IS}_{t}$) with respect to the canonical error pair $(\mu_0,\mu_1)$, via Pareto-optimal reductions. It develops a set of strict online max-reductions and derives membership and completeness results for a family of problems—$\textsc{SP}_{t}$, $\textsc{Sch}_{t}$, $\textsc{Cli}_{t}$, $\textsc{MkCS}_{kt}$, and $\textsc{MM}_{t}$—in the complexity class $\mathcal{C}_{\mu_0,\mu_1}^{t}$, including their unbounded variants. By connecting these problems through reductions, the paper translates hardness into concrete algorithmic consequences, establishing tight relationships between maximization problems and the dual minimization problem $\textsc{VC}_{t}$. The results unify several online problems under predictions, revealing that (under binary predictions and the chosen error measures) their competitive landscapes are tightly coupled and largely equivalent in hardness. This framework lays groundwork for future exploration of randomized predictions and alternative prediction schemes in online optimization.

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.

Figures (1)

• Figure 1: An (incomplete) representation of the hardness graph for $t \in \mathbb{Z}^+$ with respect to $(\mu_0,\mu_1)\xspace$. The vertices are online problems, and there is an arc $P \rightarrow Q$ when $Q$ is as hard as $P$ (see Definition \ref{['def:as_hard_as']}). If the arrowhead of an arc $P \rightarrow Q$ is only outlined (e.g. $\textsc{IS}_{t}\xspace \rightarrow \textsc{IS}\xspace$) then $Q \rightarrow P$ cannot exist. All problems and arcs in gray are from BBFL24. The remaining problems are defined in Definitions \ref{['def:bdis_t']}, \ref{['def:sp_t']}, \ref{['def:sch_t']}, \ref{['def:cli_t']}, \ref{['def:mcs_kt']}, and \ref{['def:mat_t']}, and the existence of the remaining the arcs are proven in Sections \ref{['sec:independent_set_vs_asg_t']} and \ref{['sec:a_collection_of_problems_from_CCWM_max']}. All arcs in the transitive closure of the hardness graph exists but are omitted for simplicity. Hence, if there is a $(P,Q)$-path in the hardness graph, then $P \rightarrow Q$ also exists.

Theorems & Definitions (58)

• Definition 1
• Definition 2: BBFL24
• Lemma 3: BBFL24
• Remark 1
• Definition 4
• Theorem 5: BBFL24
• Remark 2
• Lemma 6
• proof
• Lemma 6