## A dual ascent algorithm for the 1-tree relaxation of the symmetric traveling salesman problem |

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1-tree relaxation bounds produced branch-chord exchange combinatorial optimization Computational Results Crowder 11 current dual vector deﬁne the set deﬁnition direction 1 G R direction of ascent direction vector 1W dual ascent algorithm dual ascent method E E\T edge costs edge pair entry to AscentDirection faster ascent ﬁnding ﬁrst function AscentDirection Given the current Held and Karp heuristic introduced by Held involving the edge Jonker 13 Karp 9 Lagrangean Lemma lower bound minimum cost 1-tree minimum spanning tree number of vertices optimal 1-tree optimizing L(A optimum for L(A primitive-direction ascent recursive reinitiate AscentDirection resemble a tour run-times satisﬁes the sufficient scope for improvement step-size determination strategy subgradient algorithm Subgradient Bound subgradient method substitutes suﬁicient condition superﬂuous Symmetric Traveling Salesman Tarjan 16 test problems tour of G transmuter graph provides traveling salesman problem traveling salesman tour updating upper bounds vector 1W satisﬁes vertices with degree Wolfe and Crowder