Max-TSP

Find a maximum-weight Hamiltonian cycle in a complete weighted graph or complete directed graph. Nonnegative weights admit a cycle-cover patching half-approximation for Max-TSP without a triangle inequality. For integer weights, changing $c_{ij}$ to $M-c_{ij}$, with $M\ge \max c_{ij}$, changes an $n$-vertex tour's weight from $C$ to $nM-C$. Consequently the strict-threshold decision version is NP-complete: minimum-tour threshold $C\le L$ becomes maximum-tour threshold $C^{\prime }>nM-L-1$.

Solve the maximum-weight assignment problem forbidding fixed points, obtaining a cycle cover of a directed graph whose weight bounds every Hamiltonian cycle above. Remove a lightest edge from each graph cycle. Each graph cycle has at least two edges, so at least half its nonnegative weight survives. Connect the resulting disjoint graph paths cyclically; completeness supplies the connectors and their nonnegative weights cannot reduce the retained sum. The Hungarian algorithm and linear-time patching give a polynomial time approximation algorithm. Negative weights invalidate a universal multiplicative half guarantee: when every weight is $-1$, every tour has weight $-n<(-n)/2$.