Set-defined graph classes: $χ$-boundedness meets tropical algebra

What the math gives to ML

The paper turns feasibility of structured tropical linear inequalities into a finite mean-payoff game, where feasibility or infeasibility is witnessed by positional strategies and cycle-weight conditions. The transferable asset is not the graph-class application itself, but the conversion of global min/max constraints into local argmin/argmax policies whose stability can be checked through weighted cycles. This suggests a policy-based optimizer or routing controller that replaces unconstrained scalar updates with max-plus inequalities and uses strategy improvement to detect unstable feedback loops. The most promising first target is MoE routing, where expert-load and score constraints naturally form a small tropical system and can be solved or approximately repaired without backpropagating through a large constrained optimization problem.