No Price Tags? No Problem: Query Strategies for Unpriced Information

TL;DR

This work introduces an online variant of the priced query model in which per-variable costs are unknown and revealed only as cumulative investment crosses thresholds. It establishes a separation between the offline (cost-aware) and online settings, proving that online strategies must incur overhead in general, exemplified by Tribes and AND-based lower bounds. The authors design online algorithms anchored in Boolean-function influence, notably the Influence-Proportional Round Robin (IPRR) framework and its warmup variant, achieving competitive guarantees against strong benchmarks like $\mathrm{opt}^{\mathrm{avg}}_\varepsilon$ up to polylogarithmic factors, with dependence on the total influence $\mathbf{I}[f]$. They further develop specialized strategies for symmetric functions and shallow decision trees, including following or pruning decision trees and learning of trees when not given, to obtain tighter or instance-specific bounds. The results connect priced-information query strategies with online decision-making and Boolean-function analysis, yielding practical insights for adaptive information acquisition under uncertainty and suggesting broad directions for partial-information and dynamic-cost extensions.

Abstract

The classic *priced query model*, introduced by Charikar et al. (STOC 2000), captures the task of computing a known function on an unknown input when each input variable can only be revealed by paying an associated cost. The goal is to design a query strategy that determines the function's value while minimizing the total cost incurred. However, all prior work in this model assumes complete advance knowledge of the query costs -- an assumption that fails in many realistic settings. We introduce a variant of the priced query model that explicitly handles *unknown* variable costs. We prove a separation from the traditional priced query model, showing that uncertainty in variable costs imposes an unavoidable overhead for every query strategy. Despite this, we design strategies that essentially match our lower bound and are competitive with the best cost-aware strategies for arbitrary Boolean functions. Our results build on a recent connection between priced query strategies and the analysis of Boolean functions, and draw techniques from online algorithms.

Figures (3)

• Figure 1: The function $f: \{0,1\}^{\log \sqrt{n}} \times \{0,1\}^{n} \to \{0,1\}$ from \ref{['def:address-func']}.
• Figure 2: The function $h: \{0,1\}^{\ell}\times \{0,1\}^{\log \sqrt{n}} \times \{0,1\}^{n} \to \{0,1\}$ where $f$ is as in \ref{['def:address-func']}.
• Figure 3: An illustration of the behaviors of the optimal offline algorithm and $\textsc{Warmup}\text{-}\textsc{IPRR}$ on a symmetric function. Here, we assume that the costs are ordered as $c_1 \leq c_2 \leq \cdots \leq c_n$, and $i^\ast$ denotes the index s.t. $\mathrm{bias}(f) \leq\varepsilon$ after fixing $x_1, \dots, x_{i^\ast}$. The cross-hatched region denotes the additional cost incurred by $\textsc{Warmup}\text{-}\textsc{IPRR}$ in contrast to the optimal offline algorithm.

Theorems & Definitions (62)

• Theorem 1: Informal version of \ref{['thm:tribes-lower-bound']}
• Definition 2
• Theorem 3: Theorem 1 of blanc2021query
• Theorem 4: Informal version of \ref{['thm:iprr-main']} and \ref{['prop:thresholded-IPRR-on-correct-cost']}
• Theorem 5: Informal version of \ref{['thm:symmetric']}
• Theorem 6: Informal version of \ref{['thm:follow-the-tree-general']} and \ref{['remark:unknown-tree']}
• Theorem 7: Informal version of \ref{['thm:warmup-iprr']}
• Proposition 8: Informal version of \ref{['prop:warmup-iprr-hard-instance']}
• Proposition 9: Informal version of \ref{['prop:and-lower-bound']}
• Definition 9