FPT/XP algorithms for Simultaneous Interval Number parameterized by solution size

Determine whether the Simultaneous Interval Number problem is fixed-parameter tractable or slice-wise polynomial when parameterized solely by the target simultaneous interval number d of the input graph.

Background

The paper provides FPT algorithms for Simultaneous Interval Number when parameterized by cluster-modular cardinality plus solution size and by other related parameters, but leaves open the complexity of parameterizing by the solution size alone.

This is highlighted in the introduction as one of the central unresolved issues.

References

The existence of FPT or even XP algorithms for Thinness or Simultaneous Interval Number parameterized by the solution size (the numbers t and d, respectively) is open.

— Computing parameters that generalize interval graphs using restricted modular partitions (2512.22975 - Bonomo-Braberman et al., 28 Dec 2025) in Section 1 (Introduction)