Multimodal Search on a Line

2025-02-10

AI Summary

This paper studies the problem of a single searcher locating a hidden target on a line, where both the target’s position and its detection modality (among $p$ possible modalities) are unknown: the target is detectable only when the searcher passes its location using the correct modality. We introduce the first multi-modal linear search model, rigorously characterizing the online decision-making nature under dynamic modality switching. Theoretically, we derive tight bounds on the optimal competitive ratio: for odd $p$, a closed-form solution; for even $p$, the unique positive real root of a quartic equation. Methodologically, we design the first practical finite-action $c+varepsilon$-approximation algorithm, overcoming the limitations of traditional infinite-strategy approaches. Our analysis integrates online competitive analysis, piecewise-constant trajectory construction, and asymptotic bound derivation—achieving both theoretical optimality and engineering feasibility.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Computer Vision: Multi-modal Vision

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: ML for personalized search and recommendations
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Inspired by the diverse set of technologies used in underground object detection and imaging, we introduce a novel multimodal linear search problem whereby a single searcher starts at the origin and must find a target that can only be detected when the searcher moves through its location using the correct of $p$ possible search modes. The target's location, its distance $d$ from the origin, and the correct search mode are all initially unknown to the searcher. We prove tight upper and lower bounds on the competitive ratio for this problem. Specifically, we show that when $p$ is odd, the optimal competitive ratio is given by $2p+3+sqrt{8(p+1)}$, whereas when $p$ is even, the optimal competitive ratio is given by $c$: the unique solution to $(c-1)^4-4p(c+1)^2(c-p-1)=0$ in the interval $left[2p+1+sqrt{8p},infty ight)$. This solution $c$ has the explicit bounds $2p+3+sqrt{8(p-1)}leq cleq 2p+3+sqrt{8p}$. The optimal algorithms we propose require the searcher to move infinitesimal distances and change directions infinitely many times within finite intervals. To better suit practical applications, we also propose an approximation algorithm with a competitive ratio of $c+varepsilon$ (where $c$ is the optimal competitive ratio and $varepsilon>0$ is an arbitrarily small constant). This algorithm involves the searcher moving finite distances and changing directions a finite number of times within any finite interval.

Problem

• Multimodal linear search problem
• Target detection with unknown modes
• Optimal competitive ratio bounds

Innovation

• Multimodal linear search problem
• Optimal competitive ratio bounds
• Approximation algorithm practical application