TSP - State Design

$dp[\text{mask}][\text{last}]$ = minimum cost to visit exactly the cities in mask, ending at city last. Base case: $dp[1][0] = 0$. You start at city $0$, having visited only city $0$, with zero travel cost.

The mask $= 1$ means only bit $0$ is set. Goal: Visit all cities (mask $= 2^n - 1$), then return to city $0$. The answer is $\min_{\text{last}} (dp[2^n - 1][\text{last}] + \text{cost}[\text{last}][0])$. You check all possible final cities and add the return flight cost.