Scalability of linear-program formulations for optimal gadget search

Determine whether the TSSW linear-program formulation for finding optimal gadget reductions from LIN-k to MAX-k can be made computationally tractable beyond the special cases in which auxiliary variables can be eliminated or substantially reduced.

Background

The paper compares its AlphaEvolve-based search for gadget reductions with the TSSW framework, which formulates optimal gadget discovery as a linear program. Eliminating existential quantifiers by canonicalizing auxiliary variables causes the resulting linear program to grow doubly exponentially in the number of satisfying assignments of the source predicate.

The authors note that special cases may permit a reduction in the number of auxiliary variables and therefore a more manageable formulation, but they leave unresolved whether such reductions are possible for general gadget-search instances. This is a concrete unresolved question about the applicability of the LP approach outside specially structured cases.

References

Sometimes (as is the case for our I_0 gadget for MAX{3}) it is possible to argue that not all auxiliary variables are required, leading to a more tractable LP, but it is not clear that this is possible outside of very special cases.

— Reinforced Generation of Combinatorial Structures: Applications to Complexity Theory  (2509.18057 - Nagda et al., 22 Sep 2025) in Section 5.3, “Comparison to other computational techniques”