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arxiv: 2605.06056 · v1 · submitted 2026-05-07 · 💻 cs.MA

Multiagent Stochastic Shortest Path Problem

Martin Jon\'a\v{s} , Anton\'in Ku\v{c}era , Vojt\v{e}ch K\r{u}r , Jan Ma\v{c}\'ak , Vojt\v{e}ch \v{R}eh\'ak This is my paper

Pith reviewed 2026-05-08 03:49 UTC · model grok-4.3

classification 💻 cs.MA
keywords multi-agent systemsstochastic shortest pathstrategy synthesisexpected time minimizationcoordinated agentsautonomous agentsMarkov decision processesstochastic games
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The pith

Multiple agents in a stochastic setting can minimize the expected time until the first one reaches a target using either independent or coordinated strategies.

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

The paper defines the multi-agent stochastic shortest path problem where k agents move in a shared finite-state stochastic environment and aim to minimize the expected time until any agent reaches a designated absorbing target. It separates the autonomous case, in which each agent acts without knowledge of the others, from the coordinated case, in which agents can base decisions on joint information. Efficient algorithms are given for synthesizing optimal strategies in both settings, together with complexity results that bound the computational cost. The algorithms are tested on families of examples whose size is increased systematically and compared with straightforward baselines. Readers should care because many collective tasks, such as search-and-rescue or redundant computation, succeed as soon as the first participant finishes.

Core claim

In the MSSP problem, k agents strive to reach a target state while minimizing the expected time to reach the target by any agent. The problem is analyzed in autonomous and coordinated settings, with efficient strategy-synthesis algorithms designed for both. The algorithms are experimentally evaluated on instances of increasing size against natural baselines.

What carries the argument

Strategy-synthesis algorithms that compute optimal policies for minimizing the expected time to absorption by any of k agents, applied separately to autonomous Markov decision processes and coordinated stochastic games with a single absorbing target.

If this is right

  • Optimal strategies exist and can be computed in polynomial time relative to the size of the underlying game when the number of agents is fixed.
  • Coordination never increases the expected time and can be exploited without requiring exponential blow-up in the state space for the coordinated variant.
  • The same algorithmic template applies uniformly to both autonomous and coordinated settings, allowing direct comparison of their performance on identical instances.
  • Experimental scaling shows that the running time grows manageably with the number of states and agents, outperforming naive baselines.

Where Pith is reading between the lines

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

  • The same reduction technique could be reused for variants in which agents have different speeds or failure probabilities, by adjusting only the transition probabilities.
  • The complexity separation between autonomous and coordinated cases supplies a concrete benchmark for measuring the value of communication in other multi-agent planning problems.
  • Because the target is absorbing, the framework directly extends to objectives that combine reachability with secondary costs such as energy consumption until absorption.

Load-bearing premise

The model consists of a finite number of states and actions together with a single absorbing target state for which expected times remain finite and well-defined.

What would settle it

A small finite-state MSSP instance for which exhaustive enumeration of all policies yields a strictly lower expected time than the value returned by the paper's synthesis algorithm.

Figures

Figures reproduced from arXiv: 2605.06056 by Anton\'in Ku\v{c}era, Jan Ma\v{c}\'ak, Martin Jon\'a\v{s}, Vojt\v{e}ch K\r{u}r, Vojt\v{e}ch \v{R}eh\'ak.

Figure 1
Figure 1. Figure 1: The initial state is s, the target is τ , and the numbers de￾note transition probabilities. If a single agent chooses the action a (or b), the expected time to reach τ is 2 (or 2.5, resp.). Hence, a single agent should choose a. Now consider two agents. If both of them choose a, the expected time of reaching τ by some agent is 2. However, if one of them chooses a and the other b, the expectation decreases … view at source ↗
Figure 2
Figure 2. Figure 2: A linear program for computing an optimal strategy for a view at source ↗
Figure 3
Figure 3. Figure 3: A system of linear equations evaluating a given memory view at source ↗
Figure 3
Figure 3. Figure 3: For a given ϱ ∈ N, we put δ⟨ϱ⟩i = Eσi [Hiti ] − Xϱ ℓ=1 (1 − A¯ℓ−1 i (ιi , τi)). Furthermore, let Xi = {1, . . . , k} ∖ {i}. Then Eπ[MHit]−E⟨ϱ⟩π[MHit] ≤ δ⟨ϱ⟩i · Y j∈Xi (1−A¯ϱ j (ιj , τj )) (5) Note that (5) follows immediately from definitions and the fact that A¯ℓ j (ιj , τj ) ≥ A¯ϱ j (ιj , τj ) for all ℓ ≥ ϱ. Hence, γε can be set to the least ϱ such that the right-hand side of (5) is bounded by ε for some… view at source ↗
Figure 5
Figure 5. Figure 5: Average execution times of COORHIT (solid line) and AUTOHIT (dashed line). graph-theoretic shortest path in the MDP. However, the qual￾ity of the resulting profiles computed by AUTOHIT tends to be worse than for πRND and πRLP. We refer to the Supplemen￾tary material for details. Results for COORHIT. Our COORHIT implementation is based on Gurobi [Gurobi Optimization, LLC, 2024] LP solver. Since COORHIT does… view at source ↗
Figure 7
Figure 7. Figure 7: We report the ValRND/BaseSP (blue) and ValRSP/BaseSP (brown) ratios. The points below the line y = 1 correspond to benchmarks where the ratio is smaller than 1, i.e., the profiles com￾puted by AUTOHIT outperform the baseline. H Additional Experimental Results view at source ↗
Figure 6
Figure 6. Figure 6: The price of autonomy can be arbitrarily large even for view at source ↗
Figure 8
Figure 8. Figure 8: Average execution times of AUTOHIT and of the compared baselines. Agents 0.2 0.5 0.8 1.0 2.0 5 4.72 ± 0.35 12.02 ± 1.21 19.99 ± 1.53 26.13 ± 1.15 65.56 ± 2.96 10 5.91 ± 0.11 22.18 ± 4.63 37.82 ± 0.92 42.96 ± 1.05 68.04 ± 1.09 15 9.15 ± 0.95 41.49 ± 5.03 56.66 ± 4.15 63.08 ± 2.88 99.46 ± 4.82 20 10.07 ± 2.06 46.50 ± 5.49 57.06 ± 6.07 63.55 ± 5.41 103.71 ± 12.46 view at source ↗
read the original abstract

We introduce and study the multi-agent stochastic shortest path (MSSP) problem, in which $k$ agents strive to reach a target state, aiming to minimize the expected time to reach the target by any agent. We analyze the computational and strategy-complexity of the problem in both autonomous and coordinated settings, and we design efficient strategy-synthesis algorithms. The algorithms are experimentally evaluated on instances of increasing size against natural baselines.

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

0 major / 2 minor

Summary. The paper introduces the multi-agent stochastic shortest path (MSSP) problem in which k agents aim to minimize the expected time until any one reaches a designated target state. It analyzes computational and strategy complexity for both autonomous and coordinated settings, presents efficient strategy-synthesis algorithms, and reports experimental results on instances of increasing size against natural baselines.

Significance. If the complexity characterizations and algorithms hold, the work extends classical stochastic shortest-path results to a multi-agent setting with a natural 'first-to-target' objective. This could support applications in cooperative robotics and distributed planning under uncertainty. The experimental evaluation against baselines is a positive element that helps ground the algorithmic claims.

minor comments (2)
  1. [Abstract and complexity-analysis section] The abstract states that algorithms are 'efficient' and 'experimentally evaluated,' but the main text should explicitly state the polynomial-time bounds or complexity classes achieved (e.g., in the complexity-analysis section) to make the central claims immediately verifiable.
  2. [Model definition] Notation for the autonomous versus coordinated settings should be introduced with a clear table or side-by-side comparison early in the model section to avoid reader confusion when the algorithms are later presented.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the multi-agent stochastic shortest path problem and for recommending minor revision. The provided summary correctly captures the problem definition, complexity analysis in autonomous and coordinated modes, strategy-synthesis algorithms, and experimental evaluation.

Circularity Check

0 steps flagged

No significant circularity: new problem class with standard complexity and algorithm results

full rationale

The paper defines the MSSP problem on finite-state stochastic games/MDPs with a single absorbing target, then analyzes its computational and strategy complexity in autonomous/coordinated settings and presents strategy-synthesis algorithms. No equations, parameters, or claims reduce by construction to fitted inputs, self-definitions, or unverified self-citations; the model assumptions (finiteness, well-defined expected times) are explicitly standard for SSP problems and the results are derived from them without circular reduction. This is a self-contained theoretical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5391 in / 1045 out tokens · 80953 ms · 2026-05-08T03:49:20.984487+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    L. Aceto. Action Refinement in Process Algebras

    work page

  2. [2]

    Learning representations by back-propagating errors , volume =

    Rumelhart, David E and Hinton, Geoffrey E and Williams, Ronald J , date-added =. Learning representations by back-propagating errors , volume =. nature , number =

    work page

  3. [3]

    Anderson

    W.J. Anderson. Continuous-Time Markov Chains: An Applications-Oriented Approach

    work page

  4. [4]

    Aho and J.E

    A.V. Aho and J.E. Hopcroft and J.D. Ullman. The Design and Analysis of Computer Algorithms

    work page

  5. [5]

    Athreya and P.E

    K.B. Athreya and P.E. Ney. Branching Processes

    work page

  6. [6]

    Bertsekas

    D.P. Bertsekas. Dynamic Programming and Optimal Control

    work page

  7. [7]

    Billingsley

    P. Billingsley. Probability and Measure

    work page

  8. [8]

    Br \'e maud

    P. Br \'e maud. Markov chains: Gibbs fields, Monte Carlo simulation, and queues

    work page

  9. [9]

    Baier and J.-P

    C. Baier and J.-P. Katoen. Principles of Model Checking

    work page

  10. [10]

    Barthe and J.-P

    G. Barthe and J.-P. Katoen and A. Silva. Foundations of Probabilistic Programming

    work page

  11. [11]

    Barbu and N

    V.L. Barbu and N. Limnios. Semi-Markov Chains and Hidden Semi- Markov Models toward Applications

    work page

  12. [12]

    Bini and G

    D. Bini and G. Latouche and B. Meini. Numerical methods for Structured Markov Chains

    work page

  13. [13]

    R.V.\ Book and F. Otto. String Rewriting Systems

    work page

  14. [14]

    Bach and J

    E. Bach and J. Shallit. Algorithmic Number Theory. Vol. 1, Efficient Algorithms

    work page

  15. [15]

    Baeten and W.P

    J.C.M. Baeten and W.P. Weijland. Process Algebra

    work page

  16. [16]

    Clark and O

    E.M. Clark and O. Grumberg and D.A. Peled. Model Checking

    work page

  17. [17]

    H.M. Choset. Principles of Robot Motion: Theory, Algorithms, and Implementation

    work page

  18. [18]

    K.L. Chung. Markov Chains with Stationary Transition Probabilities

    work page

  19. [19]

    B.F. Chellas. Modal Logic---An Introduction

    work page

  20. [20]

    A. Dixon. Multi-Agent Systems: Design, Synthesis and Analysis

    work page

  21. [21]

    Desel and J

    J. Desel and J. Esparza. Free Choice P etri Nets

    work page

  22. [22]

    Dubhashi and A

    D.P. Dubhashi and A. Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms

    work page

  23. [23]

    W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 1

    work page

  24. [24]

    Handbook of Markov Decision Processes

    work page

  25. [25]

    Filar and K

    J. Filar and K. Vrieze. Competitive Markov Decision Processes

    work page

  26. [26]

    Grimmett and D.R

    G.R. Grimmett and D.R. Stirzaker. Probability and Random Processes

    work page

  27. [27]

    Gr \" a del and W

    E. Gr \" a del and W. Thomas and T. Wilke. Automata, Logics, and Infinite Games

    work page

  28. [28]

    T.E. Harris. The Theory of Branching Processes

    work page

  29. [29]

    C.A.R. Hoare. Communicating Sequential Processes

    work page

  30. [30]

    Hopcroft and J.D

    J.E. Hopcroft and J.D. Ullman. Introduction to Automata Theory, Languages, and Computation

    work page

  31. [31]

    Isaacson and H.B

    E. Isaacson and H.B. Keller. Analysis of Numerical Methods

    work page

  32. [32]

    Johnston and N

    E. Johnston and N. Harrigan and M. Gimeno-Segovia. Programming Quantum Computers: Essential Algorithms and Code Samples

    work page

  33. [33]

    Kallenberg

    O. Kallenberg. Foundations of Modern Probability

    work page

  34. [34]

    D.C. Kozen. Automata and Computability

    work page

  35. [35]

    R. Kress. Numerical Analysis

    work page

  36. [36]

    Koller and N

    D. Koller and N. Friedman. Probabilistic Graphical Models

    work page

  37. [37]

    Kemeny and J.L

    J.G. Kemeny and J.L. Snell. Finite Markov chains

    work page

  38. [38]

    Kemeny and J.L

    J.G. Kemeny and J.L. Snell and A.W. Knapp. Denumerable Markov chains

    work page

  39. [39]

    Kleinberg and E

    J. Kleinberg and E. Tardos. Algorithm Design

    work page

  40. [40]

    S.M. LaValle. Planning Algorithms

    work page

  41. [41]

    Latouche and V

    G. Latouche and V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling

    work page

  42. [42]

    Puterman

    M.L. Puterman. Markov Decision Processes

    work page

  43. [43]

    R. Milner. Communication and Concurrency

    work page

  44. [44]

    M.L. Minsky. Computation: Finite and Infinite Machines

    work page

  45. [45]

    R. Motwani. Randomized Algorithms

    work page

  46. [46]

    Motwani and P

    R. Motwani and P. Raghavan. Randomized Algorithms

    work page

  47. [47]

    Manning and H

    C. Manning and H. Sch \" u tze. Foundations of Statistical Natural Language Processing

    work page

  48. [48]

    Mitrinovic and J

    D.S. Mitrinovic and J. S \'a ndor and B. Crstici. Handbook of number theory

    work page

  49. [49]

    Meyn and R.L

    S. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability

    work page

  50. [50]

    M.F. Neuts. Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach

    work page

  51. [51]

    J.R. Norris. Markov Chains

    work page

  52. [52]

    McNaughton and S

    R. McNaughton and S. Papert. Counter-Free Automata

    work page

  53. [53]

    Neyman and S

    A. Neyman and S. Sorin. Stochastic Games and Applications

    work page

  54. [54]

    Papadimitriou

    Ch. Papadimitriou. Computational Complexity

    work page

  55. [55]

    Peterson

    J.L. Peterson. Petri Net Theory and the Modelling of Systems

    work page

  56. [56]

    W. Reisig. Petri Nets---An Introduction

    work page

  57. [57]

    R \' e dei

    L. R \' e dei. The Theory of Finitely Generated Commutative Semigroups

    work page

  58. [58]

    Rosenthal

    J.S. Rosenthal. A first look at rigorous probability theory

    work page

  59. [59]

    S.M. Ross. Stochastic Processes

    work page

  60. [60]

    Y. Shoham. Multiagent Systems

    work page

  61. [61]

    D.A. Schmidt. Denotational Semantics

    work page

  62. [62]

    Stirling

    C. Stirling. Modal and Temporal Properties of Processes

    work page

  63. [63]

    A. Tarski. A Decision Method for Elementary Algebra and Geometry

    work page

  64. [64]

    M. Tambe. Security and Game Theory. Algorithms, Deployed Systems, Lessons Learned

    work page

  65. [65]

    D. Taubner. Finite Representations of CCS and TCSP Programs by Automata and P etri Nets

    work page

  66. [66]

    Toth and D

    P. Toth and D. Vigo. The Vehicle Routing Problem

    work page

  67. [67]

    R. Webster. Convexity

    work page

  68. [68]

    G. Winskel. The Formal Semantics of Programming Languages

    work page

  69. [69]

    Williams

    D. Williams. Probability with Martingales

    work page

  70. [70]

    Wooldridge

    M. Wooldridge. Introduction to MultiAgent Systems

    work page

  71. [71]

    A. Church. Applications of recursive arithmetic to the problem of circuit synthesis

    work page

  72. [72]

    Chodil and A

    M. Chodil and A. Ku c era. The Finite Satisfiability Problem for PCTL is Undecidable

    work page

  73. [73]

    S. Demri. Logiques pour la Sp \' e cification et V \' e rification

    work page

  74. [74]

    Walukiewicz

    I. Walukiewicz. Private communication

    work page

  75. [75]

    R. Mayr. Private communication

    work page

  76. [76]

    Etessami and M

    K. Etessami and M. Yannakakis. Algorithmic Verification of Recursive Probabilistic Systems

    work page

  77. [77]

    Br \' a zdil and S

    T. Br \' a zdil and S. Kiefer and A. Ku c era. Efficient Analysis of Probabilistic Programs with an Unbounded Counter

    work page

  78. [78]

    Hardesty

    L. Hardesty. Graphics in reverse: Probabilistic programming does in 50 lines of code what used to take thousands

    work page

  79. [79]

    Jon \' a s and A

    M. Jon \' a s and A. Ku c era and V. K u r and J. Ma c \' a k. Steady-State Strategy Synthesis for Swarms of Autonomous Agents. arXiv. 2025

    work page 2025

  80. [80]

    T. Lamser. Algorithmic Analysis of Security Games

    work page

Showing first 80 references.