Distributed complexity of radius-constrained assignments

Determine the distributed complexity of finding assignments for matchable semi-matching instances that satisfy the radius condition excluding improving alternating paths involving at most a prescribed number of clients.

Background

The paper characterizes the exact locality gaps for matchable semi-matchings when no improving reassignment of at most r clients exists, equivalently when no improving alternating path moves at most r clients. Its results are existential and approximation-theoretic: they quantify the quality of assignments satisfying this radius condition but do not provide a distributed algorithm or complexity classification for finding such assignments. The unresolved algorithmic issue is therefore to determine the distributed computational complexity of producing radius-constrained assignments.

References

We leave two extensions for future work. An optimum with load above one replaces the one-parent comparison graph by a graph with larger indegrees, so the present ancestor state no longer applies directly. A separate algorithmic question is the distributed complexity of finding assignments satisfying the radius condition.

— Exact Locality Gaps for Matchable Semi-Matchings  (2610.01648 - Gałązka et al., 1 Oct 2026) in Concluding remarks