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Setting Targets is All You Need:Improved Order Competitive Ratio for Online Selection

Liyan Chen, Nuozhou Sun, Zhihao Gavin Tang

TL;DR

This paper studies the online single-selection problem under the order-competitive ratio benchmark, where algorithms compete against the online optimum (order-aware benchmark) without knowing arrival order. It introduces targeted value algorithms (TVA) that set and update a guessed target payoff $g_0$ and show that a deterministic choice yields $1/\varphi$-order-competitiveness while a randomized choice achieves $0.732$, improving upon the prior deterministic bound of $1/\varphi$. A robustness-enhanced variant, Targeted Value algorithms with Detection (TVD), switches to a conservative mode when overestimation is detected, achieving the same $0.732$-level with an upper bound of $0.7582$ for TVD and a general upper bound of $0.8293$ for randomized order-unaware strategies. The results illuminate the value of calibrating an online guess of the optimal online benchmark and show that randomization helps surpass deterministic limits under the online benchmark. Together, the techniques bridge online optimal benchmark theory with robust, practically implementable strategies for online selection.

Abstract

There is a rising interest for studying the online benchmark as an alternative of the classical offline benchmark in online stochastic settings. Ezra, Feldman, Gravin, and Tang (SODA 2023) introduced the notion of order-competitive ratio, defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm, to quantify the loss incurred by the lack of knowledge of the arrival order. They showed in the online single selection setting (a.k.a. the prophet problem), the optimal order-competitive ratio achieved by deterministic algorithms is $1/\varphi \approx 0.618$, and left with an open question whether randomized algorithms can do better. We answer the open question firmly by introducing a novel family of algorithms called \emph{targeted value algorithms}. We show that the task of online selection is as easy as guessing the optimal online benchmark. Specifically, we provide 1) an alternative optimal $1/\varphi$ order-competitive algorithm by setting the targeted value deterministically, and 2) a $0.732$ order-competitive algorithm by setting the targeted value randomly. We further provide a $0.758$ upper bound on the order-competitive ratio of our algorithm, showing that our analysis is close to the best possible, and establish an upper bound of $0.829$ on the order-competitive ratio for general randomized order-unaware algorithms.

Setting Targets is All You Need:Improved Order Competitive Ratio for Online Selection

TL;DR

This paper studies the online single-selection problem under the order-competitive ratio benchmark, where algorithms compete against the online optimum (order-aware benchmark) without knowing arrival order. It introduces targeted value algorithms (TVA) that set and update a guessed target payoff and show that a deterministic choice yields -order-competitiveness while a randomized choice achieves , improving upon the prior deterministic bound of . A robustness-enhanced variant, Targeted Value algorithms with Detection (TVD), switches to a conservative mode when overestimation is detected, achieving the same -level with an upper bound of for TVD and a general upper bound of for randomized order-unaware strategies. The results illuminate the value of calibrating an online guess of the optimal online benchmark and show that randomization helps surpass deterministic limits under the online benchmark. Together, the techniques bridge online optimal benchmark theory with robust, practically implementable strategies for online selection.

Abstract

There is a rising interest for studying the online benchmark as an alternative of the classical offline benchmark in online stochastic settings. Ezra, Feldman, Gravin, and Tang (SODA 2023) introduced the notion of order-competitive ratio, defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm, to quantify the loss incurred by the lack of knowledge of the arrival order. They showed in the online single selection setting (a.k.a. the prophet problem), the optimal order-competitive ratio achieved by deterministic algorithms is , and left with an open question whether randomized algorithms can do better. We answer the open question firmly by introducing a novel family of algorithms called \emph{targeted value algorithms}. We show that the task of online selection is as easy as guessing the optimal online benchmark. Specifically, we provide 1) an alternative optimal order-competitive algorithm by setting the targeted value deterministically, and 2) a order-competitive algorithm by setting the targeted value randomly. We further provide a upper bound on the order-competitive ratio of our algorithm, showing that our analysis is close to the best possible, and establish an upper bound of on the order-competitive ratio for general randomized order-unaware algorithms.
Paper Structure (21 sections, 11 theorems, 34 equations, 1 figure)

This paper contains 21 sections, 11 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Our Techniques.
  4. Hardness Results.
  5. Related Works
  6. Preliminaries
  7. Online Benchmark / Order-aware algorithms.
  8. Order-Competitive Ratio / Order-unaware algorithms.
  9. Single-threshold algorithms.
  10. Targeted value algorithms: A Warm-up
  11. Targeted value algorithms (TVA)
  12. Analysis.
  13. Proof of Lemma \ref{['lem:targeted_value']}
  14. Underestimation ($g_0 \le \mathsf{OPT}$).
  15. Overestimation ($g_0 \ge \mathsf{OPT}$).
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a (randomized) $~0.732$ order-competitive algorithm for the online single selection problem.

Figures (1)

  • Figure 1: Illustration of arrival order $\pi_x$

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Claim 3.1
  • proof
  • ...and 18 more