Optimality of Generated Minimum Birkhoff Decompositions

Determine whether the feasible decompositions generated for the minimum Birkhoff decomposition benchmark instances are optimal, that is, whether they use the minimum possible number of permutation matrices.

Background

The paper constructs benchmark instances for the minimum Birkhoff decomposition problem from feasible decompositions of doubly stochastic matrices. For sparse instances, the construction provides an upper bound of at most n permutation matrices, while the universal Birkhoff bound is (n−1)2+1 permutation matrices. The minimum number required is the optimization objective, and establishing optimality is generally difficult because the problem is NP-hard.

Although the construction guarantees feasibility, it does not establish that the generated decomposition uses the smallest possible number of permutation matrices. Consequently, the benchmark instances may have unknown optimal objective values, limiting the interpretation of classical and quantum results as optimality claims.

References

However, it is not known whether these feasible solutions are optimal.

— Quantum Optimization Benchmarking Library - The Intractable Decathlon  (2504.03832 - Koch et al., 4 Apr 2025) in Section 4, subsection “Minimum Birkhoff Decomposition,” subsection “Instances” (Section label: subsub:birk_instances)