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Traveling Salesman Problem: determine k-exchange feasibility

Given a current solution $S$ and a $k$-exchange move $(v_1, .., v_{2k+1})$ with $v_1 = v_{2k+1}$, $v_i \neq v_j$, $(v_i, v_{i+1}) \in E(S)$ iff $i$ odd, i.e. we remove all edges $(v_i, v_{i+1})$ for $...