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arxiv: 2605.10917 · v1 · submitted 2026-05-11 · 💻 cs.LG · cs.MA· cs.RO

Optimal and Scalable MAPF via Multi-Marginal Optimal Transport and Schr\"odinger Bridges

Usman A. Khan , Joseph W. Durham This is my paper

Pith reviewed 2026-05-12 03:48 UTC · model grok-4.3

classification 💻 cs.LG cs.MAcs.RO
keywords multi-agent path findingmulti-marginal optimal transportSchrödinger bridgeslinear programmingtotally unimodularSinkhorn algorithmentropic regularizationanonymous assignment
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The pith

MAPF on graphs reduces to a polynomial-sized linear program via Markovian multi-marginal optimal transport.

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

The paper shows that anonymous multi-agent path finding, where robots must reach targets on a finite connected graph without assigned pairs, can be reformulated as a multi-marginal optimal transport problem that exploits the graph's Markovian transition structure. This structure collapses the usual exponential-size joint transport plan into a linear program whose size grows only polynomially with the graph. Under the anonymity condition and standard assumptions, the LP is feasible and totally unimodular, so its optimum is an integral assignment of non-overlapping paths in both space and time. For large instances the same model is relaxed through Schrödinger bridges to an entropically regularized problem solved iteratively by Sinkhorn methods, after which the resulting fractional plan templates a smaller LP that recovers near-optimal integral solutions.

Core claim

Anonymous MAPF is equivalent to a special Markovian MMOT on the graph, under which the joint transport plan over all agents reduces to a polynomial-sized LP. This LP is feasible and totally unimodular, producing minimum-cost integral {0,1} transports that never overlap in space or time. The Schrödinger-bridge relaxation further converts the problem to an entropic regularization of the MMOT that admits an iterative Sinkhorn-type algorithm; the resulting shadow fractional transport then serves as a template for a reduced LP that yields near-optimal integral solutions at substantially lower complexity.

What carries the argument

Markovian multi-marginal optimal transport (MMOT) on graphs, whose structure collapses the exponential joint plan to a polynomial linear program; the Schrödinger-bridge entropic regularization then supplies the scalable fractional template.

If this is right

  • The resulting LP produces min-cost integral transports with no space-time overlaps.
  • Schrödinger bridges yield an iterative Sinkhorn solution for the entropically regularized problem.
  • The fractional shadow transport serves as a template that reduces the final LP size while preserving near-optimality.
  • The overall approach scales to large graphs while retaining integral, collision-free solutions.

Where Pith is reading between the lines

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

  • Existing optimal-transport libraries could be repurposed to solve MAPF instances without writing custom path-finding code.
  • The Markovian reduction may extend to continuous-time or stochastic dynamics on the same graph by replacing the discrete transition matrix with the appropriate generator.
  • Seeding the reduced LP with the bridge solution could be tested on graphs with obstacles or time-varying edge costs to measure how often optimality is retained.

Load-bearing premise

The graph dynamics are Markovian and the assignment is anonymous, so that the multi-marginal transport reduces to a totally unimodular linear program whose integral solutions avoid all space-time overlaps.

What would settle it

An explicit graph, set of agents, and targets satisfying the stated conditions for which the LP optimum is fractional or any integral solution produced by the reduced bridge template contains a space-time overlap.

Figures

Figures reproduced from arXiv: 2605.10917 by Joseph W. Durham, Usman A. Khan.

Figure 1
Figure 1. Figure 1: A 10 × 10 grid with {50, 25} robots ▲ and targets •, and {0, 50} obstacles. ily placed obstacles alter the connectivity and diameter of the underlying graph, and it no longer remains a regular grid. We next demonstrate the Schrodinger shadow trans- ¨ port (P2) in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) Optimal P1; (Middle) Schrodinger shadow ¨ P2; (Right) Integral projection P3. Top (N = 20, T = 15): P1 cost 181; P2 cost 1053; P3 with 23% edges retained at cost 181, i.e., 0% degradation. Bottom (N = 80, T = 10): P1 cost 402; P2 cost 3160; P3 with 22% edges retained at cost 436, i.e., 8.5% degradation [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scalable MAPF: The vertical axis plots the cost degradation x, i.e., the cost of transport obtained from P3 is (1 + x)copt, where copt is the optimal min-cost from P1; the horizontal axis plots the % of edges retained from the full P1 transport [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Runtime scaling across 162 runs at 5% robot density, T = 30. (Left) Solve time of P1 (circles) and P2+P3 (squares) versus K; curves show power-law fits aKp + b. (Right) Speedup versus K; the green line connects averages, the shaded band shows ±1 std. dev., and red annotations indicate the average cost gap. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: examines the cost-gap-versus-speedup tradeoff from two perspectives. The left panel plots the cost gap against the speedup for each of the 162 individual runs, colored by the number of vertices K. The cluster structure confirms that larger instances achieve higher speedups at comparable or lower cost gaps, i.e., the shadow-based pruning becomes more effective as K grows. The right panel plots the average c… view at source ↗
Figure 6
Figure 6. Figure 6: Pipeline decomposition across 8 grid sizes at 5% robot density, T = 30; all values are averages over 14–25 independent instances. (Left) Solve time of P1 (gray bars) versus the Sinkhorn (P2, blue) and LP solve (P3, red) components of the pipeline; connected dots trace the scaling shape of each; italic percentages show the Sinkhorn share of the P2+P3 time. (Right) Number of LP variables in P1 (blue) versus … view at source ↗
Figure 7
Figure 7. Figure 7: Parameter sensitivity across 260 runs (13 instances, 20 combinations each) at K = 10,000, T = 30. (Left) Dual-axis plot at λ = 0: effective Sinkhorn sweeps (left axis, blue) and cost gap (%, right axis, red) versus ε; shaded bands show ±1 standard deviation across instances. (Middle) Cost gap versus edges kept (%) for all 233 feasible runs, colored by ε. (Right) Average cost gap versus ε for each λ. 19 [P… view at source ↗
Figure 8
Figure 8. Figure 8: (right) compares the uniform and non-uniform averages. Every solution is verified integral [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A 25 × 25 grid with {200} robots ▲, {200} targets •, and {225} obstacles. Every cell is either a robot, a target, or an obstacle. The robot trajectories come from the min-cost transport obtained by solving P1 over T = 12. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: A larger 40 × 40 grid with 300 robots ▲, 300 targets •, and 100 obstacles. The robot trajectories come from the min-cost transport obtained by solving P1 over T = 6. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A larger 40 × 40 grid with 800 robots ▲ and 800 targets •; every cell is either occupied by a robot or a target. The robot trajectories come from the min-cost transport obtained by solving P1 over T = 6. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Min-cost vs. Min-makespan: A 6 × 8 grid with 4 robots ▲, 4 targets •, and 16 obstacles. Edges in the gray shaded region have cost 10; rest follow our move-wait cost convention. With T = 10, the min-cost transport avoids the high cost interior and takes the robots from the boundary of the grid, with all one-cost moves for a total of 40. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Min-cost vs. Min-makespan: A 6 × 8 grid with 4 robots ▲, 4 targets •, and 16 obstacles. Edges in the gray shaded region have cost 10; rest follow our move-wait cost convention. With T = 9, all robots cannot travel on the boundary as that requires 10 moves; the min-cost transport therefore takes two robots from the boundary in 9 steps each, and two from the higher cost interior edges for a total cost of 82… view at source ↗
Figure 14
Figure 14. Figure 14: Min-cost vs. Min-makespan: A 6 × 8 grid with 4 robots ▲, 4 targets •, and 16 obstacles. Edges in the gray shaded region have cost 10; rest follow our move-wait cost convention. With T = 8, boundary paths are no longer feasible within the given time horizon. All robots must travel through the interior edges to reach the targets for a total cost of 132. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Min-cost vs. Min-makespan: A 6 × 8 grid with 4 robots ▲, 4 targets •, and 16 obstacles. Edges in the gray shaded region have cost 10; rest follow our move-wait cost convention. We choose the cost structure described in Assumption 3.5. P1 consequently provides the minimum makespan solution that terminates the robot motion in 5 steps, i.e., achieves the minimum makespan, when solved over a longer T = 10 hor… view at source ↗
Figure 16
Figure 16. Figure 16: A 40 × 40 grid with 80 robots ▲, 80 targets •, and T = 10. The robot paths are superimposed over the horizon T. The first figure is the Schrodinger shadow that shows the likely mass transport obtained by solving ¨ P2 after τ¯ Sinkhorn iterations in Appendix G; ε = 0.2, τ¯ = 50. The next figures show the integral projection P3 by keeping the highest-valued X% of edges from the shadow. Cost is compared with… view at source ↗
Figure 17
Figure 17. Figure 17: A 40 × 40 grid with 80 robots ▲, 80 targets •, and T = 10. The robot paths are superimposed over the horizon T. The first figure is the Schrodinger shadow that shows the likely mass transport obtained by solving ¨ P2 after τ¯ Sinkhorn iterations in Appendix G; ε = 0.5, τ¯ = 10. The next figures show the integral projection P3 by keeping the highest-valued X% of edges from the shadow. Cost is compared with… view at source ↗
Figure 18
Figure 18. Figure 18: A 40 × 40 grid with 80 robots ▲, 80 targets •, and T = 10. The robot paths are superimposed over the horizon T. The first figure is the Schrodinger shadow that shows the likely mass transport obtained by solving ¨ P2 after τ¯ Sinkhorn iterations in Appendix G; ε = 50, τ¯ = 5. The next figures show the integral projection P3 by keeping the highest-valued X% of edges from the shadow. Cost is compared with t… view at source ↗
read the original abstract

We consider anonymous multi-agent path finding (MAPF) where a set of robots is tasked to travel to a set of targets on a finite, connected graph. We show that MAPF can be cast as a special class of multi-marginal optimal transport (MMOT) problems with an underlying Markovian structure, under which the exponentially large MMOT collapses to a linear program (LP) polynomial in size. Focusing on the anonymous setting, we establish conditions under which the corresponding LP is feasible, totally unimodular, and consequently, yields min-cost, integral $(\{0,1\})$ transports that do not overlap in both space and time. To adapt the approach to large-scale problems, we cast the MAPF-MMOT in a probabilistic framework via Schr\"odinger bridges. Under standard assumptions, we show that the Schr\"odinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schr\"odinger bridge, being a probabilistic framework, provides a shadow (fractional) transport that we use as a template to solve a reduced LP and demonstrate that it results in near-optimal, integral transports at a significant reduction in complexity. Extensive experiments highlight the optimality and scalability of the proposed approaches.

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 claims that anonymous multi-agent path finding (MAPF) on a finite connected graph can be cast as a Markovian multi-marginal optimal transport (MMOT) problem. Under this structure the exponentially large MMOT reduces to a polynomial-sized linear program (LP). For the anonymous setting the authors establish conditions under which the LP is feasible and totally unimodular, yielding min-cost integral {0,1} transports with no space-time overlaps. To scale to large instances they recast the problem via Schrödinger bridges, showing that the bridge reduces to entropic regularization of the MMOT solvable by Sinkhorn iterations; the resulting fractional transport serves as a support template for a reduced exact LP that produces near-optimal integral solutions. Experiments are reported to confirm both optimality and improved scalability.

Significance. If the claimed reductions, feasibility conditions, and total-unimodularity arguments hold, the work supplies a principled, polynomial-complexity route to exact optimal anonymous MAPF together with a practical near-optimal extension for large graphs. The explicit use of the Markovian factorization to collapse the MMOT and the subsequent Schrödinger-bridge reduction are genuine strengths; they connect optimal-transport theory to multi-agent planning in a way that could influence both theoretical analyses and deployed robotics systems.

minor comments (2)
  1. [Abstract] The abstract states that the LP is 'polynomial in size' and that the Schrödinger-bridge reduction holds 'under standard assumptions,' but does not name the precise scaling (e.g., O(T|E|)) or list the positivity/finite-cost conditions required for the entropic equivalence. The main text should make both explicit, ideally with a dedicated assumptions paragraph or theorem statement.
  2. The description of the reduced LP that uses the Schrödinger-bridge shadow transport as a template would benefit from a short complexity comparison (variables/constraints before and after reduction) and from pseudocode or a clear algorithmic outline.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, as well as the recommendation for minor revision. The assessment correctly identifies the key contributions regarding the Markovian MMOT formulation for anonymous MAPF, the total unimodularity result, and the Schrödinger bridge scaling approach.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins by modeling anonymous MAPF as a Markov-structured MMOT on the time-expanded graph, where the joint transport plan factors explicitly into consecutive marginals due to the Markov property; this yields an LP whose variables and constraints scale linearly with the graph size, a direct consequence of flow conservation and the chosen factorization rather than any self-referential definition. Total unimodularity follows from the resulting constraint matrix being a network matrix (after standard vertex splitting for capacities), which is a known graph-theoretic fact independent of the paper. The Schrödinger-bridge step invokes the standard entropic-regularization equivalence under positivity and finite-cost assumptions on the reference measure, a result from prior optimal-transport literature that does not depend on the current MAPF instance or any fitted parameters. The shadow-transport template is used only heuristically to prune the exact LP, with no claim that the reduced solution equals the original by construction. No self-citation is load-bearing for the core claims, no ansatz is smuggled, and no quantity is fitted to data then renamed a prediction. The chain is therefore self-contained against external benchmarks in MMOT and network flows.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on domain assumptions from graph theory and optimal transport; no new free parameters or invented entities are introduced in the abstract.

axioms (3)
  • domain assumption The environment is a finite, connected graph.
    Stated explicitly as the setting for the MAPF problem.
  • domain assumption The multi-marginal transport possesses an underlying Markovian structure.
    Invoked to collapse the exponential MMOT to a polynomial LP.
  • domain assumption Standard assumptions hold for the Schrödinger-bridge formulation.
    Required for the entropic regularization and Sinkhorn iteration to apply.

pith-pipeline@v0.9.0 · 5533 in / 1543 out tokens · 47930 ms · 2026-05-12T03:48:43.832045+00:00 · methodology

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