Learning-Augmented Online TRP on a Line

TL;DR

The paper addresses online traveling repairman on a line under a learning-augmented framework that provides predicted request locations. It introduces a $3$-competitive lower bound, a $2+\sqrt{3}$-competitive half-line algorithm that extends to lines, and a line-algorithm that retains $2+\sqrt{3}$-competitiveness under perfect predictions and achieves $\min\{2+\sqrt{3}+4\delta,4\}$ with imperfect predictions characterized by $\delta$. The approach combines an optimal half-line strategy with the offline AFRATI_ALG tour to form a predicted tour (PRED_TOUR) and then analyzes the cost relative to $OPT$, including robust adjustments for prediction error $\delta$. These results constitute the first explicit online TRP guarantees in the learning-augmented setting, advancing understanding of prediction-assisted routing on a line and motivating further exploration to broader metrics and practical robustness.

Abstract

We study the online traveling repairperson problem on a line within the recently proposed learning-augmented framework, which provides predictions on the requests to be served via machine learning. In the original model (with no predictions), there is a stream of requests released over time along the line. The goal is to minimize the sum (or average) of the completion times of the requests. In the original model, the state-of-the-art competitive ratio lower bound is $1+\sqrt{2} > 2.414$ for any deterministic algorithm and the state-of-the-art competitive ratio upper bound is 4 for a deterministic algorithm. Our prediction model involves predicted positions, possibly error-prone, of each request in the stream known a priori but the arrival times of requests are not known until their arrival. We first establish a 3-competitive lower bound which extends to the original model. We then design a deterministic algorithm that is $(2+\sqrt{3})\approx 3.732$-competitive when predictions are perfect. With imperfect predictions (maximum error $δ> 0$), we show that our deterministic algorithm becomes $\min\{3.732+4δ,4\}$-competitive, knowing $δ$. To the best of our knowledge, these are the first results for online traveling repairperson problem in the learning-augmented framework.

Figures (5)

• Figure 1: The lower bound construction on a half-line $\mathcal{L}$ with $|\mathcal{L}|=10$. The request locations are shown. The 8 requests at 1, 4--10 arrive at time $t=0$. The 3 remaining requests arrive over time.
• Figure 2: An illustration of how Algorithm \ref{['algorithm:half-line']} visits half-line $\mathcal{L}$ in round-trips $RT_j, j\geq 1,$ starting from and ending at origin $o$. The segment lengths of $\mathcal{L}$ visited in each $RT_j$ is also shown at bottom.
• Figure 3: An illustration of $OFF\_TOUR$ for offline TRP or $PRED\_TOUR$ of predicted locations for online TRP, computed using $AFRATI\_ALG$. The points on $\mathcal{L}$ where the tour changes direction are denoted by $x_i$ on $\mathcal{L}_{left}$ and $y_i$ on $\mathcal{L}_{right}$.
• Figure 4: An alternative illustration of $OFF\_TOUR$ or $PRED\_TOUR$ of Fig. \ref{['fig:tour']} as a half-line starting from $o$ at the right end (despite it visits $o$ multiple times). Consider any request $r_i$ (denoted as black circle) that arrives at a location on the bold intervals. We have that $OPT(r_i)\geq \{|OFF\_TOUR_i|,t_i\}$, where $|OFF\_TOUR_i|$ is the length of $OFF\_TOUR$ starting from $o$ (right end) to the position $p_i$ of $r_i$ (shown as a arrow line).
• Figure 5: An illustration of how round-trips in Fig. \ref{['fig:halfline']} are adjusted to accommodate error $\Delta$ in predicted locations.

Theorems & Definitions (14)

• Theorem 1: lower bound
• proof
• Corollary 1
• Theorem 2
• proof
• Corollary 2
• proof
• Theorem 3: afrati1986complexity
• Theorem 4
• proof