pith. sign in

arxiv: 2605.15745 · v1 · pith:MPRZQYZXnew · submitted 2026-05-15 · 💻 cs.DS · cs.CC

The Robotaxi Placement Problem: Minimizing Expected ETA for Stochastic Demand

Ioannis Caragiannis , Kostas Kollias , Mohammad Roghani , Aaron Schild , Ali Kemal Sinop This is my paper

Pith reviewed 2026-05-19 19:30 UTC · model grok-4.3

classification 💻 cs.DS cs.CC
keywords robotaxi placementstochastic optimizationapproximation algorithmsmetric spacestree metricsdynamic programmingride-hailing
0
0 comments X

The pith

Sampling robotaxi locations from the demand distribution provides a randomized 2-approximation for minimizing expected rider wait times.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formalizes the robotaxi placement problem of positioning k robotaxis to minimize the expected total matching distance to k random riders drawn from a demand distribution. A reader would care because efficient placement reduces wait times in autonomous ride-hailing without requiring real-time complex computations. The authors establish that independent sampling from the demand distribution achieves a randomized 2-approximation guarantee. They also give a polynomial-time exact algorithm for tree metrics using dynamic programming and show inapproximability on general metrics via reduction from max coverage. Real-world experiments indicate that a variance-reduced version of this random strategy performs close to optimal.

Core claim

Sampling robotaxi locations independently according to the demand distribution yields a randomized 2-approximation algorithm for the robotaxi placement problem. An explicit inapproximability bound is presented via a gap-preserving reduction from the maximum coverage problem. An exact polynomial-time dynamic programming algorithm exists for k-RP in tree metrics by decoupling the stochastic matching dependencies.

What carries the argument

The independent sampling of robotaxi positions according to the known demand distribution, which provides the 2-approximation by bounding the expected cost of the resulting matching.

If this is right

  • The expected cost of a random placement is at most twice the optimal expected cost.
  • Exact solutions are computable in polynomial time on tree metrics.
  • A variance-reduced random placement strategy is effective on real ride-hailing data.
  • Inapproximability holds for general metrics unless the maximum coverage problem admits better approximations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sampling method could be adapted to settings where demand is estimated from historical data rather than known exactly.
  • Similar randomized placements might apply to other stochastic matching problems such as dynamic server allocation.
  • Empirical success on real data suggests testing the approach on road networks that are nearly trees for practical heuristics.

Load-bearing premise

The demand distribution over rider locations is known in advance and can be sampled from independently for each of the k riders.

What would settle it

Construct a small metric instance with a known optimal placement, compute its expected matching cost exactly, then sample many placements from the demand distribution and measure whether their average cost exceeds twice the optimal by a large margin.

Figures

Figures reproduced from arXiv: 2605.15745 by Aaron Schild, Ali Kemal Sinop, Ioannis Caragiannis, Kostas Kollias, Mohammad Roghani.

Figure 1
Figure 1. Figure 1: Comparison of algorithm UCkM (vertical axis) and the best run of algorithm VRRP (horizontal axis) across 1,000 random rider demand realizations for each of our four distributions A, B, C, and D. In the four plots of [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The 263 points of our metric space corresponding to the centroids of the 263 NYC taxizones. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visual representation of the four probability distributions over the riders’ pickup locations [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

Autonomous ride-hailing platforms must strategically position idle robotaxis to minimize the wait times of prospective riders. We formalize this as the \emph{robotaxi placement problem} ($k$-RP). Given a finite metric space and a demand distribution over its points, the goal is to position $k$ robotaxis to minimize the expected total distance in a perfect matching between the robotaxis and $k$ random riders. We present several theoretical results for this stochastic optimization problem. First, we observe that sampling robotaxi locations independently according to the demand distribution yields a randomized $2$-approximation algorithm. Second, we present an explicit inapproximability bound via a novel gap-preserving reduction from the maximum coverage problem. Furthermore, while it is not even clear whether the exact expected cost of a placement can be computed efficiently on general metrics, we design an exact polynomial-time dynamic programming algorithm for $k$-RP in tree metrics by decoupling the stochastic matching dependencies. Finally, empirical evaluations on real-world ride-hailing data reveal that a variance-reduced random placement strategy is highly effective in practice, yielding expected wait times that are very close to those obtained by computationally heavy exact algorithms for the uniform capacitated $k$-median problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces the robotaxi placement problem (k-RP): given a finite metric space and a demand distribution over its points, position k robotaxis to minimize the expected total distance in a perfect matching between the robotaxis and k random riders sampled from the distribution. It claims a randomized 2-approximation by independently sampling k robotaxi locations from the demand distribution, an inapproximability result via a novel gap-preserving reduction from the maximum coverage problem, an exact polynomial-time dynamic programming algorithm for tree metrics that decouples the stochastic matching dependencies, and empirical evaluations on real-world ride-hailing data showing a variance-reduced random placement strategy yields expected wait times close to those from exact algorithms for the uniform capacitated k-median problem.

Significance. The randomized 2-approximation is a clean and strong result that relies only on the triangle inequality applied to composed matchings and the identical distributions of the sampled placement Y and demand D, yielding E[cost(match(Y, D))] ≤ 2·OPT without additional parameters or assumptions. The exact DP for trees provides a positive algorithmic result for structured metrics. Empirical findings indicate practical effectiveness of simple sampling heuristics. These elements advance stochastic variants of facility location and matching problems in a manner that is falsifiable and grounded in standard metric properties.

major comments (2)
  1. [Inapproximability] Inapproximability section: the gap-preserving reduction from maximum coverage must explicitly construct the metric and demand distribution such that the expected matching cost preserves the coverage gap; without the precise mapping from coverage sets to rider locations and how the minimum matching cost encodes the objective, it is unclear whether the inapproximability factor carries through for arbitrary k.
  2. [Dynamic programming for tree metrics] Dynamic programming section: the claim that the DP decouples stochastic matching dependencies requires a precise definition of the state (e.g., number of placed robotaxis and expected unmatched demand per subtree) to confirm that the recursion computes the exact minimum expected cost in polynomial time rather than requiring enumeration of all possible matchings.
minor comments (3)
  1. [Abstract] Abstract: the 'variance-reduced random placement strategy' is referenced in the empirical results but should be defined or cross-referenced in the main text for clarity.
  2. [Empirical evaluation] Empirical evaluation: report the concrete parameters of the real-world dataset, including the size of the metric space and the range of k values tested.
  3. [Notation] Notation: ensure the demand distribution and the matching cost function are denoted consistently across the theoretical and experimental sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the results, and recommendation for minor revision. We address the two major comments point by point below, clarifying the technical details and indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Inapproximability] Inapproximability section: the gap-preserving reduction from maximum coverage must explicitly construct the metric and demand distribution such that the expected matching cost preserves the coverage gap; without the precise mapping from coverage sets to rider locations and how the minimum matching cost encodes the objective, it is unclear whether the inapproximability factor carries through for arbitrary k.

    Authors: We thank the referee for this observation. The reduction in the manuscript is gap-preserving, but we agree that the construction can be stated more explicitly. In the revised version we will add a detailed construction: the metric is defined on a ground set consisting of one point per universe element plus auxiliary points for each set, with distances chosen so that the triangle inequality holds while any uncovered element contributes a fixed additive cost to the expected matching distance. The demand distribution places mass on the uncovered elements proportionally to the coverage gap. This mapping ensures that the minimum expected matching cost differs from the optimum by a factor directly tied to the maximum-coverage objective, and the argument holds for any fixed k (the reduction parameterizes the number of robotaxis by the coverage parameter). We will include the full distance matrix definition and the cost calculation to make the preservation of the gap transparent. revision: yes

  2. Referee: [Dynamic programming for tree metrics] Dynamic programming section: the claim that the DP decouples stochastic matching dependencies requires a precise definition of the state (e.g., number of placed robotaxis and expected unmatched demand per subtree) to confirm that the recursion computes the exact minimum expected cost in polynomial time rather than requiring enumeration of all possible matchings.

    Authors: We agree that a formal statement of the DP state will remove any ambiguity. In the revision we will define the state for a subtree T as a triple (v, j, m) where v is the root of T, j is the number of robotaxis allocated to T, and m is the expected number of unmatched riders originating from the demand distribution restricted to T. The recurrence decides how many robotaxis to place at v, how to partition the remaining j-1 robotaxis among the child subtrees, and how many unmatched riders from each child are matched locally versus propagated upward; the expected cost is computed by linearity of expectation over the independent subtrees. Because the global perfect-matching requirement is satisfied by propagating only the expected unmatched count, the DP never enumerates matchings and runs in time polynomial in the tree size and k. We will present the recurrence equations and the polynomial-time bound explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central 2-approximation claim rests on independent sampling from the known demand distribution combined with the triangle inequality in the metric space and identical distributions of the sampled sets Y and D. This yields E[cost(match(Y, D))] ≤ 2·OPT directly from the proof structure without any fitted parameters, self-definitions, or reductions to prior self-citations. The inapproximability reduction from maximum coverage, the tree-metric DP that decouples matching dependencies, and the empirical variance-reduced placement comparisons are all independent of the approximation result and introduce no load-bearing self-referential steps. No equations or claims reduce the stated results to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the input is a finite metric space with a known probability distribution over demand points; no free parameters are fitted inside the approximation or DP statements.

axioms (2)
  • domain assumption The underlying space is a metric (satisfies triangle inequality and symmetry).
    Invoked to define distances in the perfect matching cost.
  • domain assumption Demand consists of k independent draws from a known distribution.
    Required for the expectation in the objective and for the independent sampling algorithm.

pith-pipeline@v0.9.0 · 5760 in / 1350 out tokens · 35773 ms · 2026-05-19T19:30:33.624106+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Optimal matchings of random points.Combi- natorica, 4(4):259–264, 1984

    Mikl´os Ajtai, J´anos Koml´os, and G´abor Tusn´ady. Optimal matchings of random points.Combi- natorica, 4(4):259–264, 1984

    work page 1984

  2. [2]

    On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment.Proceedings of the National Academy of Sciences, 114(3):462–467, 2017

    Javier Alonso-Mora, Samitha Samaranayake, Alex Wallar, Emilio Frazzoli, and Daniela Rus. On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment.Proceedings of the National Academy of Sciences, 114(3):462–467, 2017

    work page 2017

  3. [3]

    Bertsimas.Probabilistic combinatorial optimization problems

    Dimitris J. Bertsimas.Probabilistic combinatorial optimization problems. PhD thesis, Mas- sachusetts Institute of Technology, 1988

    work page 1988

  4. [4]

    Bertsimas, Patrick Jaillet, and Amedeo R

    Dimitris J. Bertsimas, Patrick Jaillet, and Amedeo R. Odoni. A priori optimization.Operations Research, 38(6):1019–1033, 1990

    work page 1990

  5. [5]

    Empty-car routing in ridesharing systems.Operations Research, 67(5):1437–1452, 2019

    Anton Braverman, Jian Gang Dai, Xin Liu, and Lei Ying. Empty-car routing in ridesharing systems.Operations Research, 67(5):1437–1452, 2019

    work page 2019

  6. [6]

    Bi-factor approxi- mation algorithms for hard capacitatedk-median problems

    Jaroslaw Byrka, Krzysztof Fleszar, Bartosz Rybicki, and Joachim Spoerhase. Bi-factor approxi- mation algorithms for hard capacitatedk-median problems. InProceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 722–736, 2015

    work page 2015

  7. [7]

    Moses Charikar, Sudipto Guha, ´Eva Tardos, and David B. Shmoys. A constant-factor ap- proximation algorithm for the k-median problem.Journal of Computer and System Sciences, 65(1):129–149, 2002

    work page 2002

  8. [8]

    Approximating k-median with non-uniform capacities

    Julia Chuzhoy and Yuval Rabani. Approximating k-median with non-uniform capacities. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 952–958, 2005

    work page 2005

  9. [9]

    A (2+ϵ)-approximation algorithm for metric k-median

    Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee, Chris Schwiegelshohn, and Ola Svensson. A (2+ϵ)-approximation algorithm for metric k-median. InProceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC), pages 615–624, 2025

    work page 2025

  10. [10]

    On the fixed-parameter tractability of capacitated clus- tering

    Vincent Cohen-Addad and Jason Li. On the fixed-parameter tractability of capacitated clus- tering. InProceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP), pages 41:1–41:14, 2019

    work page 2019

  11. [11]

    A threshold oflnn for approximating set cover.Journal of the ACM, 45(4):634–652, 1998

    Uriel Feige. A threshold oflnn for approximating set cover.Journal of the ACM, 45(4):634–652, 1998

    work page 1998

  12. [12]

    Alaya, Aur´elie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, L ´eo Gautheron, Nathalie T.H

    R´emi Flamary, Nicolas Courty, Alexandre Gramfort, Mokhtar Z. Alaya, Aur´elie Boisbunon, Stanislas Chambon, Laetitia Chapel, Adrien Corenflos, Kilian Fatras, Nemo Fournier, L ´eo Gautheron, Nathalie T.H. Gayraud, Hicham Janati, Alain Rakotomamonjy, Ievgen Redko, Antoine Rolet, Antony Schutz, Vivien Seguy, Danica J. Sutherland, Romain Tavenard, Alexander T...

    work page 2021

  13. [13]

    COIN-OR Branch and Cut (Cbc), 2020

    John Forrest, Ted Ralphs, Stefan Vigerske, Lou Hafer, Bjarni Kristjansson, Matthew Saltzman, Miles Lubin, and Haroldo Gambini Santos. COIN-OR Branch and Cut (Cbc), 2020

    work page 2020

  14. [14]

    Openstreetmap: User-generated street maps.IEEE Pervasive Computing, 7(4):12–18, 2008

    Mordechai Haklay and Patrick Weber. Openstreetmap: User-generated street maps.IEEE Pervasive Computing, 7(4):12–18, 2008

    work page 2008

  15. [15]

    Data-driven model predictive control of autonomous mobility-on-demand systems

    Ram´on Iglesias, Federico Rossi, Kevin Wang, David Hallac, Jure Leskovec, and Marco Pavone. Data-driven model predictive control of autonomous mobility-on-demand systems. InProceed- ings of the 2018 IEEE International Conference on Robotics and Automation (ICRA), pages 1–7, 2018

    work page 2018

  16. [16]

    Korupolu, C.Greg Plaxton, and Rajmohan Rajaraman

    Madhukar R. Korupolu, C.Greg Plaxton, and Rajmohan Rajaraman. Analysis of a local search heuristic for facility location problems.Journal of Algorithms, 37(1):146–188, 2000

    work page 2000

  17. [17]

    Tight bounds for minimax grid matching with applications to the average case analysis of algorithms.Combinatorica, 9(2):161–187, 1989

    Tom Leighton and Peter Shor. Tight bounds for minimax grid matching with applications to the average case analysis of algorithms.Combinatorica, 9(2):161–187, 1989. 10

    work page 1989

  18. [18]

    On uniform capacitatedk-median beyond the natural LP relaxation.ACM Transactions on Algorithms, 13(2):22:1–22:18, 2017

    Shi Li. On uniform capacitatedk-median beyond the natural LP relaxation.ACM Transactions on Algorithms, 13(2):22:1–22:18, 2017

    work page 2017

  19. [19]

    Efficient large-scale fleet management via multi-agent deep reinforcement learning

    Kaixiang Lin, Renyu Zhao, Zhe Xu, and Jiaping Zhou. Efficient large-scale fleet management via multi-agent deep reinforcement learning. InProceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (KDD), pages 1774–1783, 2018

    work page 2018

  20. [20]

    Real-time routing with OpenStreetMap data

    Dennis Luxen and Christian Vetter. Real-time routing with OpenStreetMap data. InProceedings of the 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (SIGSPATIAL), pages 513–516, 2011

    work page 2011

  21. [21]

    Fei Miao, Shuo Lin, Sirajum Munoz, Cheng-Zhong Lin, Jianping Huang, and George J. Pappas. Taxi dispatch with real-time sensing data in metropolitan areas: A receding horizon control approach.IEEE Transactions on Automation Science and Engineering, 13(2):463–478, 2016

    work page 2016

  22. [22]

    PuLP: A linear programming toolkit for Python

    Stuart Mitchell, Michael O’Sullivan, and Iain Dunning. PuLP: A linear programming toolkit for Python. Technical report, The University of Auckland, Auckland, New Zealand, 2011

    work page 2011

  23. [23]

    Smith, Emilio Frazzoli, and Daniela Rus

    Marco Pavone, Stephen L. Smith, Emilio Frazzoli, and Daniela Rus. Robotic load balancing for mobility-on-demand systems.International Journal of Robotics Research, 31(7):839–854, 2012

    work page 2012

  24. [24]

    Deep reinforcement learning for ride-sharing dispatching and repositioning

    Zhiwei (Tony) Qin, Xiaocheng Tang, Yan Jiao, Fan Zhang, Chenxi Wang, and Qun (Tracy) Li. Deep reinforcement learning for ride-sharing dispatching and repositioning. InProceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pages 6566–6568, 2019

    work page 2019

  25. [25]

    Multicommodity facility location

    Ramamoorthi Ravi and Amitabh Sinha. Multicommodity facility location. InProceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 350–359, 2004

    work page 2004

  26. [26]

    Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover’s distance as a metric for image retrieval.International Journal of Computer Vision, 40(2):99–121, 2000

    work page 2000

  27. [27]

    X S∈K 1− |US| |U1| |L∩U1|# ≥E  k· 1− 1 k X S∈K |US| |U1| !|L∩U1|  =E

    David B. Shmoys and Chaitanya Swamy. An approximation scheme for two-stage stochastic optimization with recourse. In45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 228–237, 2004. 11 A Tightness of Corollary 3 We show that the result in Corollary 3 is tight with the next lemma. Lemma 10.There exists an instance of k-RP with n po...

    work page 2004

  28. [28]

    is higher than the slope of the line connecting points (z1, zr

    work page

  29. [29]

    The inequality here follows usingz 1 = 1 e −ε,z 2 = 1 e , andr= E[|L∩U1|] k

    and (z2, zr 2), i.e., zr 1 z1 ≥ zr 2 −zr 1 z2−z1 and, therefore, zr 1 ≥z r 2 · z1 z2 . The inequality here follows usingz 1 = 1 e −ε,z 2 = 1 e , andr= E[|L∩U1|] k . Now, let T be the number of points in L that contribute 2−ε to the cost of matching M. Clearly, all points in L∩U 0 have this property. Furthermore, since Y of the set points in K are at dista...

    work page 2023