Computing the epistasis variance of large-scale traveling salesman problems

The interaction among variables of an optimization problem is known as epistasis, and its degree is an important measure for the nonlinearity of the problem. We address the problem of enormous time complexity for computing Davidor's epistasis variance of the traveling salesman problem (TSP). To reduce the complexity, we introduce the concept of schema-linear problem (SLP), show that TSP is a SLP, and present a relevant lemma, called Summation Rule. Using the Summation Rule, we provide a closed formula for epistasis that reduces the time complexity from O(n(n)) to O(n(2)). Additionally, we propose a new more scalable measure of epistasis by a careful derivation from the original.