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Continuous Multi-agent Path Finding via Satisfiability Modulo Theories (SMT)

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Part of the Lecture Notes in Computer Science book series (LNAI,volume 12613)


We address multi-agent path finding (MAPF) with continuous movements and geometric agents, i.e. agents of various geometric shapes moving smoothly between predefined positions. We analyze a new solving approach based on satisfiability modulo theories (SMT) that is designed to obtain optimal solutions with respect to common cumulative objectives. The standard MAPF is a task of navigating agents in an undirected graph from given starting vertices to given goal vertices so that agents do not collide with each other in vertices or edges of the graph. In the continuous version (MAPF\(^\mathcal {R}\)), agents move in an n-dimensional Euclidean space along straight lines that interconnect predefined positions. Agents themselves are geometric objects of various shapes occupying certain volume of the space - circles, polygons, etc. We develop concepts for circular omni-directional agents having constant velocities in the 2D plane but a generalization for different shapes is possible. As agents can have different shapes/sizes and are moving smoothly along lines, a movement along certain lines done with small agents can be non-colliding while the same movement may result in a collision if performed with larger agents. Such a distinction rooted in the geometric reasoning is not present in the standard MAPF. The SMT-based approach for MAPF\(^\mathcal {R}\) called SMT-CBS\(^\mathcal {R}\) reformulates previous Conflict-based Search (CBS) algorithm in terms of SMT. Lazy generation of constraints is the key idea behind the previous algorithm SMT-CBS. Each time a new conflict is discovered, the underlying encoding is extended with new to eliminate the conflict. SMT-CBS\(^\mathcal {R}\) significantly extends this idea by generating also the decision variables lazily. Generating variables on demand is needed because in the continuous case the number of possible decision variables is potentially uncountable hence cannot be generated in advance as in the case of SMT-CBS. We compared SMT-CBS\(^\mathcal {R}\) and adaptations of CBS for the continuous variant of MAPF experimentally.


  • Multi-agent path finding (MAPF)
  • Satisfiability modulo theory (SMT)
  • Continuous time
  • Continuous space
  • Makespan optimal solutions
  • Sum-of-costs optimal solutions
  • Geometric agents

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  • DOI: 10.1007/978-3-030-71158-0_19
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  1. 1.

    Different versions of MAPF permit entering of a vertex being simultaneously vacated by another agent excluding the trivial case when agents swap their position across an edge.

  2. 2.

    In our current implementation we followed a more cautious definition of the collision - it occurs even if agents appear too close to each other.

  3. 3.

    The formal details of the theory \(T_{ MAPF ^\mathcal {R}}\) are not relevant from the algorithmic point of view. Nevertheless let us note that the signature of \(T_{ MAPF ^\mathcal {R}}\) consists of non-logical symbols describing agents’ positions at a time such as at(aut) - agent a at vertex u at time t.

  4. 4.

    The complete source codes will be made available to enable reproducibility of presented results on the author’s website:

  5. 5.

    All experiments were run on a system with Ryzen 7 3.0 GHz, 16 GB RAM, under Ubuntu Linux 18.

  6. 6.

    To enable reproducibility of presented results we will provide complete source code of our solvers on author’s web:

  7. 7.

    All experiments were run on a system with Ryzen 7 3.0 GHz, 16 GB RAM, under Ubuntu Linux 18.


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This work has been supported by GAČR - the Czech Science Foundation, grant registration number 19-17966S.

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Correspondence to Pavel Surynek .

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Surynek, P. (2021). Continuous Multi-agent Path Finding via Satisfiability Modulo Theories (SMT). In: Rocha, A.P., Steels, L., van den Herik, J. (eds) Agents and Artificial Intelligence. ICAART 2020. Lecture Notes in Computer Science(), vol 12613. Springer, Cham.

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