## Abstract

The assignment problem arises in multi-robot task-allocation scenarios. Inspired by existing techniques that employ task exchanges between robots, this paper introduces an algorithm for solving the assignment problem that has several appealing features for online, distributed robotics applications. The method may start with any initial matching and incrementally improve the current solution to reach the global optimum, producing valid assignments at any intermediate point. It is an any-time algorithm with a performance profile that is attractive: quality improves linearly with stages (or time). Additionally, the algorithm is comparatively straightforward to implement and is efficient both theoretically (complexity of \(O(n^3\lg n)\) is better than many widely used solvers) and practically (comparable to the fastest implementation, for up to hundreds of robots/tasks). The algorithm generalizes “swap” primitives used by existing task exchange methods already used in the robotics community but, uniquely, is able to obtain global optimality via communication with only a subset of robots during each stage. We present a centralized version of the algorithm and two decentralized variants that trade between computational and communication complexity. The centralized version turns out to be a computational improvement and reinterpretation of the little-known method of Balinski–Gomory proposed half a century ago. Thus, deeper understanding of the relationship between approximate swap-based techniques—developed by roboticists—and combinatorial optimization techniques, e.g., the Hungarian and Auction algorithms—developed by operations researchers but used extensively by roboticists—is uncovered.

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## Notes

- 1.
The algorithm was first presented in the 2012 Robotics: Science and Systems conference (Liu and Shell 2012a).

- 2.
Every traversed entry on the path segments, no matter whether it is assigned or unassigned, must connect to a new entry in other rows, requiring a message be passed. The number is approximately half of all the traversed entries since each entry is counted twice for the analysis of communication complexity.

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Liu, L., Shell, D.A. An anytime assignment algorithm: From local task swapping to global optimality.
*Auton Robot* **35, **271–286 (2013). https://doi.org/10.1007/s10514-013-9351-2

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### Keywords

- Multi-robot task allocation
- Decentralized assignment
- Anytime algorithms
- Task swapping