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Tight Analysis of the Lazy Algorithm for Open Online Dial-a-Ride

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Algorithms and Data Structures (WADS 2023)

Abstract

In the open online dial-a-ride problem, a single server has to deliver transportation requests appearing over time in some metric space, subject to minimizing the completion time. We improve on the best known upper bounds on the competitive ratio on general metric spaces and on the half-line, for both the preemptive and non-preemptive version of the problem. We achieve this by revisiting the algorithm \(\textsc {Lazy}\) recently suggested in [WAOA, 2022] and giving an improved and tight analysis. More precisely, we show that it has competitive ratio 2.457 on general metric spaces and 2.366 on the half-line. This is the first upper bound that beats known lower bounds of 2.5 for schedule-based algorithms as well as the natural \(\textsc {Replan}\) algorithm.

Supported by DFG grant DI 2041/2.

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Correspondence to Júlia Baligács , Yann Disser , Farehe Soheil or David Weckbecker .

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A Factor-Revealing Approach for the Half-Line

A Factor-Revealing Approach for the Half-Line

We show how to use the factor revealing approach from Sect. 3 for the dial-a-ride problem on the half-line. Consider the following variables (recall that \(k\in \mathbb {N}\) is the number of schedules started by \(\textsc {Lazy}(\alpha )\)).

  • \(t_1 = t^{(k-1)}\), the start time of the second to last schedule

  • \(t_2 = r^{(k)}\), the start time of the last schedule

  • \(s_1 = |S^{(k-1)}|\), the duration of the second to last schedule

  • \(s_2 = |S^{(k)}|\), the duration of the last schedule

  • \(\textsc {Opt}_1 = \textsc {Opt}(t^{(k-1)})\), duration of the optimal tour serving requests released until \(t^{(k-1)}\)

  • \(\textsc {Opt}_2 = \textsc {Opt}(t^{(k)})\), duration of the optimal tour

  • \(p_1 = p^{(k)}\), the position where \(\textsc {Lazy}(\alpha )\) ends the second to last schedule

  • \(p_2 = a^{(k)}_{f,\textsc {Opt}}\), the position of the first request in \(R^{(k)}\) picked up first by the optimal tour

  • \(s_2^a = |S(R^{(k)},a^{(k)}_{f,\textsc {Opt}})|\), duration of the schedule serving \(R^{(k)}\) starting in \(p_2\)

  • \(d = d(p^{(k)},a^{(k)}_{f,\textsc {Opt}})\), the distance between \(p_1\) and \(p_2\)

With these variables

$$ x=\bigl (t_1,t_2,s_1,s_2,\textsc {Opt}_1,\textsc {Opt}_2,p_1,p_2,s_2^a,d\bigr ), $$

we can create the following valid optimization problem.

$$\begin{aligned}{} & {} \max \,\, t_2 + s_2\nonumber \\{} & {} \,\, \text {s.t.}~ \textsc {Opt}_2 = 1 \end{aligned}$$
(21)
$$\begin{aligned} d= & {} |p_1 - p_2| \end{aligned}$$
(22)
$$\begin{aligned} t_2= & {} \max \{t_1+s_1,\alpha \textsc {Opt}_2\} \end{aligned}$$
(23)
$$\begin{aligned} t_1\ge & {} \alpha \textsc {Opt}_1 \end{aligned}$$
(24)
$$\begin{aligned} \textsc {Opt}_1\ge & {} \ p_1 \end{aligned}$$
(25)
$$\begin{aligned} s_2\le & {} \ d + s_2^a \end{aligned}$$
(26)
$$\begin{aligned} \textsc {Opt}_2\ge & {} \ t_1 + s_2^a \end{aligned}$$
(27)
$$\begin{aligned} t_1 + s_1\le & {} \ (1+\alpha )\textsc {Opt}_1 \end{aligned}$$
(28)
$$\begin{aligned} \textsc {Opt}_2\ge & {} \ p_1 + d \quad \quad \quad \quad \quad \quad \text {or} \quad \quad \textsc {Opt}_2 \ge t_1 + d \end{aligned}$$
(29)
$$\begin{aligned} d\ge & {} \ \alpha \textsc {Opt}_2 - \textsc {Opt}_1 \quad \quad \text {or} \quad \quad s_1 - p_1 \le 2(\textsc {Opt}_2 - p_2) \end{aligned}$$
(30)
$$\begin{aligned} x\ge & {} \ 0 \end{aligned}$$
(31)

Note that in (29) and (30), at least one of the two inequalities has to be satisfied in each case. In order to obtain an MILP, one can introduce four binary variables \(b_1,\dots ,b_4\) to model constraints (22), (23), (29), and (30).

With \(M>0\) being a large enough constant, equality (22) can be replaced by the inequalities

$$\begin{aligned} d \ge p_1 - p_2, \\ d \ge p_2 - p_1, \\ d \le p_1 - p_2 + b_1\cdot M, \\ d \le p_2 - p_1 + (1-b_1)\cdot M. \end{aligned}$$

Equality (23) can be replaced by the inequalities

$$\begin{aligned} t_2 \ge t_1 + s_1, \\ t_2 \ge \alpha \textsc {Opt}_2, \\ t_2 \le t_1 + s_1 + b_2\cdot M, \\ t_2 \le \alpha \textsc {Opt}_2 + (1-b_2)\cdot M. \end{aligned}$$

Constraint (29) can be replaced by the inequalities

$$\begin{aligned} \textsc {Opt}_2 \ge p_1 + d - b_3\cdot M, \\ \textsc {Opt}_2 \ge t_1 + d - (1-b_3)\cdot M, \end{aligned}$$

and, likewise, (30) by the inequalities

$$\begin{aligned} d \ge \alpha \textsc {Opt}_2 - \textsc {Opt}_1 - b_4\cdot M, \\ s_1 - p_1 \le 2(\textsc {Opt}_2 - p_2) + (1-b_4)\cdot M. \end{aligned}$$

The resulting MILP has the optimal solution

$$\begin{aligned}&\ \bigl (t_1,t_2,s_1,s_2,\textsc {Opt}_1,\textsc {Opt}_2,p_1,p_2,s_2^a,d,b_1,b_2,b_3,b_4\bigr ) \\ =&\ \Bigl (1,\frac{\alpha +1}{\alpha },\frac{1}{\alpha },2-\alpha ,\frac{1}{\alpha },1,0,2-\alpha ,0,2-\alpha ,0,1,1,1\Bigr ) \end{aligned}$$

and optimal value \(\max \{3+\frac{1}{\alpha }-\alpha ,1+\alpha \}\). For \(\alpha =\frac{1+\sqrt{3}}{2}>1.366\), this expression is minimized.

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Baligács, J., Disser, Y., Soheil, F., Weckbecker, D. (2023). Tight Analysis of the Lazy Algorithm for Open Online Dial-a-Ride. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_4

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