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Complexity Classes for Online Problems with and without Predictions

Magnus Berg, Joan Boyar, Lene M. Favrholdt, Kim S. Larsen

TL;DR

The paper develops a formal complexity-theoretic framework for online minimization problems with binary predictions, anchored by Online ASG_t as the canonical hard problem. It defines hierarchies of complexity classes C_{η_0,η_1}^t based on reductions that preserve competitiveness with respect to prediction errors, and proves that ASG_{t+1} is strictly harder than ASG_t, yielding a strict, transitive hierarchy. Using a general Reduction Template, the authors establish C_{η_0,η_1}^t-hardness and completeness for a variety of problems (e.g., VC_t, Interval Rejection, k-Spill, 2-SatD, Dominating Set), while also relating the framework to paging lower bounds to derive strong universal lower bounds. Purely online variants are covered, showing the framework applies beyond predictions and that results extend to classical online problems. The work provides a unifying lens to reason about the limits of predictive online algorithms and informs when richer prediction schemes are necessary for tractable performance.

Abstract

With the developments in machine learning, there has been a surge in interest and results focused on algorithms utilizing predictions, not least in online algorithms where most new results incorporate the prediction aspect for concrete online problems. While the structural computational hardness of problems with regards to time and space is quite well developed, not much is known about online problems where time and space resources are typically not in focus. Some information-theoretical insights were gained when researchers considered online algorithms with oracle advice, but predictions of uncertain quality is a very different matter. We initiate the development of a complexity theory for online problems with predictions, focusing on binary predictions for minimization problems. Based on the most generic hard online problem type, string guessing, we define a family of hierarchies of complexity classes (indexed by pairs of error measures) and develop notions of reductions, class membership, hardness, and completeness. Our framework contains all the tools one expects to find when working with complexity, and we illustrate our tools by analyzing problems with different characteristics. In addition, we show that known lower bounds for paging with predictions apply directly to all hard problems for each class in the hierarchy based on the canonical pair of error measures. Our work also implies corresponding complexity classes for classic online problems without predictions, with the corresponding complete problems.

Complexity Classes for Online Problems with and without Predictions

TL;DR

The paper develops a formal complexity-theoretic framework for online minimization problems with binary predictions, anchored by Online ASG_t as the canonical hard problem. It defines hierarchies of complexity classes C_{η_0,η_1}^t based on reductions that preserve competitiveness with respect to prediction errors, and proves that ASG_{t+1} is strictly harder than ASG_t, yielding a strict, transitive hierarchy. Using a general Reduction Template, the authors establish C_{η_0,η_1}^t-hardness and completeness for a variety of problems (e.g., VC_t, Interval Rejection, k-Spill, 2-SatD, Dominating Set), while also relating the framework to paging lower bounds to derive strong universal lower bounds. Purely online variants are covered, showing the framework applies beyond predictions and that results extend to classical online problems. The work provides a unifying lens to reason about the limits of predictive online algorithms and informs when richer prediction schemes are necessary for tractable performance.

Abstract

With the developments in machine learning, there has been a surge in interest and results focused on algorithms utilizing predictions, not least in online algorithms where most new results incorporate the prediction aspect for concrete online problems. While the structural computational hardness of problems with regards to time and space is quite well developed, not much is known about online problems where time and space resources are typically not in focus. Some information-theoretical insights were gained when researchers considered online algorithms with oracle advice, but predictions of uncertain quality is a very different matter. We initiate the development of a complexity theory for online problems with predictions, focusing on binary predictions for minimization problems. Based on the most generic hard online problem type, string guessing, we define a family of hierarchies of complexity classes (indexed by pairs of error measures) and develop notions of reductions, class membership, hardness, and completeness. Our framework contains all the tools one expects to find when working with complexity, and we illustrate our tools by analyzing problems with different characteristics. In addition, we show that known lower bounds for paging with predictions apply directly to all hard problems for each class in the hierarchy based on the canonical pair of error measures. Our work also implies corresponding complexity classes for classic online problems without predictions, with the corresponding complete problems.
Paper Structure (23 sections, 36 theorems, 18 equations, 4 figures, 1 algorithm)

This paper contains 23 sections, 36 theorems, 18 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymmetric String Guessing: A Collection of Hard Problems
  4. A Hierarchy of Complexity Classes
  5. Relative Hardness and Reductions
  6. Defining the Complexity Classes
  7. Establishing the Hierarchy
  8. Purely Online Algorithms
  9. Remark
  10. Remark
  11. A Template for Online Reductions via Simulation
  12. The Reduction Template
  13. A List of $\mathcal{C}_{\eta_0,\eta_1}^{t}\xspace$-Hard Problems
  14. Vertex Cover
  15. $\boldsymbol{k}$-Minimum-Spill
  16. ...and 8 more sections

Key Result

Theorem 5

Let $t \in \mathbb{Z}\xspace^+$ and let $0 < \varepsilon < 1$. Then, for $\textsc{ASG}_{t}\xspace$, the following hold.

Figures (4)

  • Figure 1: A hardness graph based on our complexity hierarchy. The problems shown are defined in Definitions \ref{['def:asg_t']}, \ref{['def:bdvc_t']}, \ref{['def:dom']}, \ref{['def:k-spill']}, \ref{['def:ir_t']}, \ref{['def:2-SAT']}, and \ref{['def:paging']}. Given two problems $P$ and $Q$, we write $P \rightarrow Q$ to indicate that $Q$ is as hard as $P$ (see Definition \ref{['def:as_hard_as']}). If the arrowhead is only outlined, $P$ is not as hard as $Q$. If the arrow is dashed, $P$ is as hard as $Q$ in the weak sense (see Definition \ref{['def:weaklyhard']}). We leave out most arrows that can be derived by transitivity. The gray arrows hold with respect to the pair of error measures $(\mu_0,\mu_1)\xspace$ (see Definition \ref{['def:error_measures_for_ASG_analysis']}), and the remaining arrows hold with respect to all pairs of insertion monotone error measures (see Definition \ref{['def:insertion_monotone_error_measures']}).
  • Figure 5: Example reduction for $\textsc{IR}_{t}\xspace$, with $t = 3$, $x = 01001$, and $y' = 00011(\_\_\_)(\_)$. The first five bits of $y'$ are the $\textsc{IR}_{t}\xspace$ algorithm's responses to the five challenge request, and the bits in parenthesis correspond to its responses to the two blocks. Bits that do not influence the definition of the graph are simply shown as '$\_$'.
  • Figure 8: Connectified graph.
  • Figure 9: Example reduction for dominating set. The vertices with thicker boundary constitute an optimal vertex cover in the left graph, and the corresponding optimal dominating set in the right graph.

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 9
  • Theorem 10
  • Definition 11
  • ...and 43 more