Sliding Cubes in Parallel

2026-03-09

AI Summary

This study addresses the problem of parallel reconfiguration in the sliding cube model of three-dimensional programmable matter under connectivity constraints. By integrating computational complexity theory, combinatorial optimization, and input-sensitive algorithm design, it establishes for the first time that deciding the existence of a valid reconfiguration sequence remains NP-hard even when both the makespan and the symmetric difference between source and target configurations are constant. The work further proves log-APX-hardness for this problem under both parallel and sequential models. Additionally, it presents an asymptotically optimal reconfiguration algorithm with a worst-case makespan of O(n) and demonstrates that determining the optimality of schedules with makespan 1 or 2 is also NP-hard, thereby significantly strengthening existing complexity results.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the classic sliding cube model for programmable matter under parallel reconfiguration in three dimensions, providing novel algorithmic and surprising complexity results in addition to generalizing the best known bounds from two to three dimensions. In general, the problem asks for reconfiguration sequences between two connected configurations of $n$ indistinguishable unit cube modules under connectivity constraints; a connected backbone must exist at all times. The makespan of a reconfiguration sequence is the number of parallel moves performed. We show that deciding the existence of such a sequence is NP-hard, even for constant makespan and if the two input configurations have constant-size symmetric difference, solving an open question in [Akitaya et al., ESA 25]. In particular, deciding whether the optimal makespan is 1 or 2 is NP-hard. We also show log-APX-hardness of the problem in sequential and parallel models, strengthening the APX-hardness claim in [Akitaya et al., SWAT 22]. Finally, we outline an asymptotically worst-case optimal input-sensitive algorithm for reconfiguration. The produced sequence has length that depends on the bounding box of the input configurations which, in the worst case, results in a $O(n)$ makespan.

Problem

• sliding cubes
• programmable matter
• reconfiguration
• connectivity constraints
• makespan

Innovation

• sliding cubes
• programmable matter
• parallel reconfiguration
• NP-hardness
• input-sensitive algorithm