Online Dial-a-Ride Problem (OLDARP)

• OLDARP is an online routing problem where transportation requests arrive over time with specific pickup, dropoff, and release times in a metric space.
• The problem features open and closed variants that critically affect route geometry and algorithmic strategies, with competitive ratios ranging from approximately 2.0585 to 2.457.
• Advanced algorithmic approaches like the Lazy family, Smartstart, and learning-augmented methods leverage waiting, batching, and interruption techniques to optimize completion time.

Searching arXiv for recent and foundational papers on the Online Dial-a-Ride Problem (OLDARP). The Online Dial-a-Ride Problem (OLDARP) is an online routing problem in which transportation requests arrive over time and must be served by a server moving in a metric space. In the standard single-server formulation, the server starts at a distinguished origin, moves at unit speed, and each request specifies a pickup location, a dropoff location, and a release time. In the non-preemptive setting, once a request is picked up it may only be unloaded at its destination. The canonical objective in the modern OLDARP literature is to minimize the completion time, that is, the time when all requests have been served. A major structural distinction is between the open variant, where the server may stop anywhere after the last delivery, and the closed variant, where the route must return to the origin. This distinction is central in competitive analysis, because it changes both the geometry of optimal routes and the online algorithm’s waiting and interruption decisions (Baligacs et al., 2022).

1. Formal model and core variants

The metric-space formulation used in recent work fixes a metric space $(M,d)$ with a distinguished origin $O\in M$. A single server starts at $O$, moves at unit speed, and may have capacity $c\in\mathbb N\cup\{\infty\}$. Requests arrive online over time; in the open non-preemptive model studied by Baligács, Böhmová, Disser, and Hackfeld, each request is

$r=(a,b;t),$

where $a\in M$ is the pickup point, $b\in M$ is the dropoff point, and $t\ge 0$ is the release time (Baligacs et al., 2022). The algorithm learns $r$ only at time $t$, and the total number of requests is not known in advance. The completion time of an online algorithm on request sequence $O\in M$0 is denoted $O\in M$1, while $O\in M$2 denotes the offline optimum under the same release times.

A closely related formulation appears in learning-augmented work, where the route is required to return to the origin; there the objective route $O\in M$3 must both serve all requests and end at the origin (Liang et al., 17 Jul 2025). This is the closed variant. The same paper makes explicit the difference between OLDARP and the online traveling salesman problem (OLTSP): OLTSP requests specify a single service location, whereas OLDARP requests specify a pickup-delivery pair (Liang et al., 17 Jul 2025). In Appendix formulations devoted specifically to OLDARP with predictions, the problem is further specialized to the uncapacitated case $O\in M$4, again with a single server, non-preemption, and completion-time objective $O\in M$5 (Hu et al., 2022).

The competitive-analysis benchmark is standard. For open OLDARP, the strict competitive ratio is

$O\in M$6

This formulation is used explicitly in the 2022 open-problem analysis (Baligacs et al., 2022). In the learning-augmented closed setting, an online algorithm is $O\in M$7-competitive if $O\in M$8 for every input (Liang et al., 17 Jul 2025). This shared benchmark has made OLDARP a canonical testbed for online routing techniques based on waiting, batching, interruption, and prediction.

2. Open versus closed OLDARP and the role of schedule structure

The open and closed variants differ in a way that is algorithmically decisive. In the open problem, after serving the last request, the server need not return to the origin. This allows the online algorithm to benefit from ending “far away,” but it also makes online timing subtler because interruption and endpoint selection interact (Baligacs et al., 2022). In the closed problem, by contrast, any schedule must absorb a final return-to-origin leg, which often makes waiting-at-origin strategies more natural and supports sharper guarantees for schedule-based algorithms (Liang et al., 17 Jul 2025).

A second major distinction in the literature is between schedule-based and non-schedule-based algorithms. Schedule-based algorithms repeatedly compute a route for the currently known unserved requests, execute that route, and ignore any requests arriving during its execution. The recent learning-augmented paper "Waiting is worth it and can be improved with predictions" models OLDARP exactly this way: when the server is at the origin and the known unserved set $O\in M$9 is nonempty, it computes a schedule $O$0, starts it at a threshold time, and ignores all new requests until the schedule completes and the server returns to the origin (Liang et al., 17 Jul 2025). The overall route is thus a concatenation of origin-to-origin batch schedules.

By contrast, the open-problem algorithmic line culminating in the Lazy family allows interruption under controlled conditions and is therefore not schedule-based in the classical sense (Baligacs et al., 2022). This distinction matters because previous lower bounds had shown a $O$1 barrier for schedule-based algorithms and for the natural Replan-style approach, whereas improved analyses of Lazy beat that barrier (Baligacs et al., 2022). This suggests that in open OLDARP, progress beyond classical waiting-and-batching methods requires more adaptive reset mechanisms.

3. Classical waiting algorithms on the line and in general metrics

The modern theory of OLDARP has been shaped by waiting-based algorithms. On the line, the influential Smartstart algorithm was subjected to tight analysis by Böhmová, Disser, Královič, Královič, and Schewior. In the open, non-preemptive line setting, Smartstart waits until the earliest time $O$2 satisfying

$O$3

where $O$4 is the shortest schedule length from current position $O$5 at time $O$6 serving released requests $O$7, and $O$8 is the waiting parameter (Birx et al., 2019). Once it starts a batch, it executes an optimal offline schedule for the currently known requests and ignores later arrivals until that batch finishes.

The tight line analysis yields an exact competitive ratio for Smartstart in the open non-preemptive setting: $O$9 attained at an optimal parameter $c\in\mathbb N\cup\{\infty\}$0 (Birx et al., 2019). The same paper proves that for the closed non-preemptive line case, Smartstart has exact competitive ratio $c\in\mathbb N\cup\{\infty\}$1, uniquely optimized at $c\in\mathbb N\cup\{\infty\}$2 (Birx et al., 2019). These results established the importance of the “wait, batch, ignore arrivals” paradigm, while also showing its limitations in the open problem.

A later improvement for the open non-preemptive line problem came from SmarterStart, which changed Smartstart’s waiting trigger. Instead of basing the wait condition on the server’s current position, SmarterStart bases it on the offline optimum from the origin for all requests released so far: $c\in\mathbb N\cup\{\infty\}$3 This prevents adversarial “luring” that exploits position-dependent waiting (Birx et al., 2019). The resulting competitive ratio on the line is approximately $c\in\mathbb N\cup\{\infty\}$4, improving the previous upper bound $c\in\mathbb N\cup\{\infty\}$5, and the paper proves this analysis is tight for the SmarterStart family (Birx et al., 2019). The same work also gives a lower bound $c\in\mathbb N\cup\{\infty\}$6, the first to separate open Dial-a-Ride on the line from online TSP on the line (Birx et al., 2019).

These line results were subsequently overtaken, for the open problem, by new analyses of the Lazy family in general metrics and on restricted line-like spaces (Baligacs et al., 2022).

4. The Lazy family and the current best deterministic upper bounds for open OLDARP

The decisive advance in open OLDARP came from the parameterized algorithm Lazy($c\in\mathbb N\cup\{\infty\}$7), introduced and refined in 2022. Its high-level logic is to wait until enough information has accumulated, start an optimal schedule for the currently known unserved requests, and interrupt only if a newly arriving request is “cheap enough” to justify resetting: namely, the server must be able to finish the currently loaded requests and return to the origin by time $c\in\mathbb N\cup\{\infty\}$8 (Baligacs et al., 2022). When the server becomes idle, it either waits until time $c\in\mathbb N\cup\{\infty\}$9 or starts a new shortest schedule $r=(a,b;t),$0 for the current unserved set from its current position (Baligacs et al., 2022).

A first analysis established that for $r=(a,b;t),$1, where $r=(a,b;t),$2, Lazy is $r=(a,b;t),$3-competitive, yielding the bound

$r=(a,b;t),$4

for open online dial-a-ride on general metric spaces and any capacity $r=(a,b;t),$5 (Baligacs et al., 2022). This improved the best previously known upper bounds from $r=(a,b;t),$6 in general metrics and $r=(a,b;t),$7 on the line (Baligacs et al., 2022). The proof is phase-based, showing by induction that each started schedule $r=(a,b;t),$8 satisfies

$r=(a,b;t),$9

The golden ratio enters when closing the inductive step on completion time (Baligacs et al., 2022).

A tighter analysis of the same Lazy family soon improved the deterministic upper bound further. Baligács, Böhmová, Disser, and Hackfeld showed that for

$a\in M$0

Lazy has competitive ratio

$a\in M$1

on general metric spaces (Baligacs et al., 2022). On the half-line, the stronger geometry permits

$a\in M$2

(Baligacs et al., 2022). The same bounds also carry over to the preemptive setting by replacing shortest non-preemptive schedules with preemptive shortest schedules (Baligacs et al., 2022).

This tight analysis is especially significant because it beats the known $a\in M$3 lower bounds for schedule-based algorithms and for the natural Replan family (Baligacs et al., 2022). The paper explicitly identifies this as the first upper bound below that barrier. It also proves that the new constants are tight for Lazy itself: the general-metric bound $a\in M$4 and the half-line bound $a\in M$5 are lower bounds for the family over all parameter choices (Baligacs et al., 2022). This does not settle the true optimal competitive ratio of open OLDARP, but it clarifies both the power and the limits of this interruption-based paradigm.

5. Lower bounds and structural hardness

OLDARP is known to admit nontrivial deterministic lower bounds even on highly structured metrics. On the real line, the paper "Improved Bounds for Open Online Dial-a-Ride on the Line" proves a lower bound of approximately $a\in M$6, defined as the second largest root of

$a\in M$7

This is the first lower bound that strictly separates online Dial-a-Ride on the line from online TSP on the line in competitive terms (Birx et al., 2019). The proof uses a batch of transportation requests that force committed motion under non-preemption and finite capacity, exploiting exactly the transport aspect absent from TSP (Birx et al., 2019).

For open OLDARP in general, the 2022 general-metric Lazy paper recalls lower bounds around $a\in M$8 for the line, emphasizing that a substantial gap remains between the best known upper and lower bounds for the problem itself (Baligacs et al., 2022). By contrast, its lower bound

$a\in M$9

applies only to the Lazy($b\in M$0) family, not to all algorithms (Baligacs et al., 2022). Optimizing this family lower bound yields the universal family-wide obstruction

$b\in M$1

showing that no parameter choice within Lazy can beat the improved upper bound (Baligacs et al., 2022). The tighter analysis in (Baligacs et al., 2022) matches this value from above.

Learning-augmented work reveals a parallel structural limitation for the closed, schedule-based setting. The 2025 paper on online predictions proves that any schedule-based algorithm with online predictions has competitive ratio at least $b\in M$2, even for OLTSP on the line, and therefore also for OLDARP because OLDARP generalizes OLTSP (Liang et al., 17 Jul 2025). Thus perfect online predictions do not break the classical barrier of $b\in M$3 within the schedule-based model (Liang et al., 17 Jul 2025). This does not contradict the open-problem upper bounds above $b\in M$4; rather, it identifies a framework-specific limit in the closed, origin-returning setting.

The literature also includes a distinct spatial locality model in which each newly arriving request’s pickup lies within distance $b\in M$5 of the current position of some server. In that model, deterministic lower bounds of $b\in M$6 persist regardless of $b\in M$7 for general non-sequential arrivals, including OLDARP (Guragain et al., 20 Jun 2025). This suggests that even limited geometric side information does not eliminate the fundamental adversarial structure of online routing.

6. Learning augmentation, prediction models, and locality-aware extensions

Recent work has expanded OLDARP beyond classical worst-case online analysis into learning-augmented and information-augmented settings. The paper "Online TSP with Predictions" extends three prediction models to OLDARP in an appendix devoted specifically to the learning-augmented dial-a-ride problem (Hu et al., 2022). In that formulation, requests are $b\in M$8, the problem is uncapacitated and non-preemptive, and the server starts and ends at the origin (Hu et al., 2022). The three prediction models are: full sequence prediction without identity, full sequence prediction with identity, and prediction of the last arrival time (Hu et al., 2022).

Corresponding algorithms include LADAR-Trust, LADAR-NID, LADAR-ID, and LADAR-Last. Their guarantees are expressed in the language of consistency, robustness, and smoothness. For example, LADAR-ID satisfies

$b\in M$9

hence is $t\ge 0$0-consistent, $t\ge 0$1-robust, and $t\ge 0$2-smooth (Hu et al., 2022). LADAR-Last satisfies

$t\ge 0$3

hence is $t\ge 0$4-consistent and $t\ge 0$5-robust (Hu et al., 2022). These are not best-known competitive ratios for OLDARP itself; rather, they show how learning-augmented methodologies can transfer from OLTSP to dial-a-ride.

The 2025 schedule-based prediction paper develops a more lightweight prediction model. Instead of predicting all future requests, it predicts a binary decision per schedule: whether the next batch should start earlier or later than the SmartStart threshold (Liang et al., 17 Jul 2025). The resulting algorithm SSOP adds a predicting state to SmartStart and achieves, with Christofides-based approximate schedules, a tradeoff of

$t\ge 0$6

and

$t\ge 0$7

for $t\ge 0$8, where $t\ge 0$9 and $r$0 comes from Christofides’ heuristic (Liang et al., 17 Jul 2025). As $r$1, the consistency approaches $r$2, close to the exact-schedule schedule-based benchmark, but robustness deteriorates sharply (Liang et al., 17 Jul 2025).

Another information-augmented direction is spatial locality. For uncapacitated, nomadic OLDARP, locality-based algorithms improve the best known $r$3 bound when the locality radius $r$4 is small relative to metric diameter $r$5. On arbitrary metrics, the bound becomes

$r$6

for a single server (Guragain et al., 20 Jun 2025). On the line, the bound becomes

$r$7

where $r$8 measures how centrally located the origin is (Guragain et al., 20 Jun 2025). For multiple servers, a parallelized version yields

$r$9

on any metric (Guragain et al., 20 Jun 2025). The paper explicitly notes that locality is imposed only on the pickup point, not the destination (Guragain et al., 20 Jun 2025). This suggests a promising but still highly specialized route toward improved competitive ratios under side information.

7. Related variants: revenue maximization and dynamic ridesharing extensions

OLDARP has also been studied under objectives other than completion time. A prominent alternative is revenue maximization under a global time horizon. In "Maximizing Revenues for Online-Dial-a-Ride," each request is $t$0, where $t$1 is the revenue earned if the request is served before a deadline $t$2. In the general weighted-edge setting, the offline problem is NP-complete and the online problem is not competitive at all; in the unit-weight setting, however, the deterministic Greatest Revenue First (grf) algorithm is $t$3-competitive with an additive term equal to the value of the last request served by the offline optimum (Christman et al., 2013). This variant departs fundamentally from completion-time OLDARP because requests may be rejected and the objective is profit, not makespan.

A later refinement studies the Segmented Best Path (SBP) algorithm for the revenue-maximization version on weighted graphs. Under bounded-edge assumptions, SBP has exact competitive ratio $t$4 in the nonuniform-revenue case and a $t$5-competitive guarantee up to an additive $t$6 term in the uniform-revenue case (Christman et al., 2019). These results belong to a different objective regime, but they show how the OLDARP framework can be adapted to acceptance-control settings with hard horizons.

Beyond competitive analysis, there are dynamic shared-mobility problems explicitly described as related to dial-a-ride. The dynamic tree algorithm for peer-to-peer ride-sharing matching treats drivers as servers with private origin-destination trips and incremental pickup/dropoff insertions. Its lower-level route representation is a dynamic tree of feasible schedules that is updated as new requests are inserted, while an upper-level assignment problem selects feasible driver-request combinations (Yao et al., 2021). This is not classical OLDARP—drivers are not centrally controlled fleet vehicles—but the paper’s use of precedence, capacity, waiting-time, and excess-travel constraints makes it an OLDARP-adjacent extension.

Similarly, work on dynamic electric dial-a-ride with charging management focuses on a subset of OLDARP decisions, namely recharge planning and real-time charger assignment under stochastic demand. The two-stage approach of Ma and Xie plans daily charging schedules and then solves an online vehicle–charger assignment problem using Lagrangian relaxation, with empirical savings in charging waiting time, charging duration, and energy cost (Ma, 2020). This does not solve full OLDARP jointly with routing, but it shows how energy constraints and queueing at chargers create a new online control layer within dynamic dial-a-ride systems.

A plausible implication of these variants is that OLDARP has evolved into a family of online pickup-and-delivery problems distinguished less by request structure than by objective and side constraints: completion-time minimization, profit maximization, prediction augmentation, locality promises, shared-mobility detour constraints, and battery-aware fleet control.

8. Current state of the field and open directions

The current deterministic state of the art for open OLDARP is dominated by the Lazy family. For general metric spaces, the best known upper bound is

$t$7

(Baligacs et al., 2022). On the half-line, it is

$t$8

(Baligacs et al., 2022). Earlier milestones on the line include Smartstart’s exact $t$9 ratio for the open case (Birx et al., 2019) and SmarterStart’s improved $O\in M$00 ratio with tight analysis (Birx et al., 2019). The best known lower bound for the open problem itself remains around $O\in M$01 on the line and therefore for general metrics as well (Birx et al., 2019).

For the closed schedule-based setting, exact schedules inside SmartStart attain the classical ratio $O\in M$02, while Christofides-based polynomial-time schedules yield the benchmark approximately $O\in M$03 (Liang et al., 17 Jul 2025). Learning-augmented schedule-level predictions can improve polynomial-time consistency toward $O\in M$04 without violating the schedule-based lower-bound barrier (Liang et al., 17 Jul 2025). Appendix results on more explicit prediction models provide additional consistency/robustness tradeoffs, but not improved worst-case competitive ratios for the base problem (Hu et al., 2022).

Several misconceptions recur in the literature. One is to conflate line-specific results with general-metric progress; many strong constants, including SmarterStart’s $O\in M$05, depend essentially on line geometry (Birx et al., 2019). Another is to interpret lower bounds for Lazy or for schedule-based algorithms as lower bounds for OLDARP itself; the 2022 Lazy lower bounds are explicitly family-specific (Baligacs et al., 2022), and the schedule-based prediction lower bound of $O\in M$06 is framework-specific (Liang et al., 17 Jul 2025). A third is to treat OLDARP as merely OLTSP with paired nodes. The line lower bound $O\in M$07, which separates dial-a-ride from TSP, shows that the transport constraint creates genuinely harder online structure (Birx et al., 2019).

The main open questions remain those already highlighted by the best recent open-problem papers. Can one beat $O\in M$08 for open OLDARP with a deterministic algorithm outside the Lazy paradigm (Baligacs et al., 2022)? Can problem-level lower bounds be pushed substantially above $O\in M$09 (Birx et al., 2019)? Can information-augmented models such as spatial locality or online schedule predictions be generalized into broader algorithmic principles (Guragain et al., 20 Jun 2025, Liang et al., 17 Jul 2025)? This suggests that the next advances may require either stronger adaptive interruption schemes than Lazy or a shift away from the prevailing schedule-centric abstractions.