The algorithm reduces the running time of an algorithm of Frieze from O(n^{1.5)) to O(n^(4/3 + o)... more The algorithm reduces the running time of an algorithm of Frieze from O(n^{1.5)) to O(n^(4/3 + o)). It also introduces the concept of admissible permutations that is used in algorithms for obtaining solutions to the AP and the TSP.
We improve proofs in "The Floyd-Warshall Algorithm, the AP and the TSP (III). We also simplify th... more We improve proofs in "The Floyd-Warshall Algorithm, the AP and the TSP (III). We also simplify the method for obtaining a good upper bound for an optimal solution.
Let M be an n X n symmetric cost matrix. Assume that D is a derangement in M, i.e.,a set of disjo... more Let M be an n X n symmetric cost matrix. Assume that D is a derangement in M, i.e.,a set of disjoint cycles consisting of edges that contains all of the n points of M. The modified Floyd-Warshall algorithm applied to (D')^-1(M^-)A^- (where A is an asymmetric cost matrix containing D', a derangement)yielded a solution to the Assignment Problem in O((n^2)logn) running time. Here, applying a variation of the modified F-W to (D^-1)M^-1, we can obtain D = D_FWABS, the smallest-valued derangement obtainable using the modified F-W. Let T_TSPOPT be an optimal tour in M. We give conditions for obtaining D_ABSOLUTE, the smallest-valued derangement obtainable in M, where |D_ABSOLUTE| <= |T_TSPOPT|.
Let M be an n X n symmetric cost matrix. Assume that D is a derangement of edges in M, i.e., a se... more Let M be an n X n symmetric cost matrix. Assume that D is a derangement of edges in M, i.e., a set of point-disjoint cycles containing all of the n points of M.The modified Floyd-Warshall algorithm applied to ((D')^-1)A^- (where A is an asymmetric cost matrix containing D', a derangement)yielded a solution to the Assignment Problem in O((n^2)logn) running time. Here, applying a variation of the modified F-W algorithm to D^-1)M^-, we may possibly obtain a smaller-valued derangement than D consisting of entries in M. A minimally-valued derangement would be of great value as a good and natural lower bound for an optimal tour in M.
This paper is example 5 in chapter 5. Let H be an n-cycle. A permutation s is H-admissible if Hs ... more This paper is example 5 in chapter 5. Let H be an n-cycle. A permutation s is H-admissible if Hs = H' where H' is an n-cycle. Here we define a 19 X 19 matrix, M, in the following way: We obtain the remainders modulo 100 of each of the smallest 342 odd primes. we obtain the remainders modulo 100 of each of the primes. They are placed in M according to the original value of each prime. Thus their placement depends on the the original ordinal values of the primes according to size. We use this ordering to place the primes in M. Let H_0 be an initial 19 cycles arbitrarily chosen. We apply a sequence of up to [ln(n)+1] H_0 3-cycles to obtain a 19-cycle of smaller value than H_0, call the new 19-cycle H_1. We follow this procedure to obtain H_1. We call [ln(n)] + 1 a chain. We add up the values of the 19-cycles in each chain. This procedure continues until we cannot obtain a chain the sum of whose values is not negative. COMMENT. I've renamed the document "Yhe General Traveling Salesman Problem, Version 5". I preciously named it "The Traveling Salesman, Version 5". Although the algorithms work on the GTSP, I thought that more people would google it if it was named "The Traveling Salesman Problem." Rhar qas because my work is only available through arxiv.org,
. Although the graph (digraph) becomes non-random as the algorithm proceeds, the probability for ... more . Although the graph (digraph) becomes non-random as the algorithm proceeds, the probability for success stays the same. We also give examples.
. We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be a... more . We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be an arbitrary graph(digraph). Then - in polynomial time - either an algorithm obtains a hamilton circuit(cycle)or else the algorithm points to at least one vertex that cannot belong to any hamilton circuit(cycle) of G. We give criteria for determining which vertices should be examined.
We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and rando... more We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.
The algorithm reduces the running time of an algorithm of Frieze from O(n^{1.5)) to O(n^(4/3 + o)... more The algorithm reduces the running time of an algorithm of Frieze from O(n^{1.5)) to O(n^(4/3 + o)). It also introduces the concept of admissible permutations that is used in algorithms for obtaining solutions to the AP and the TSP.
We improve proofs in "The Floyd-Warshall Algorithm, the AP and the TSP (III). We also simplify th... more We improve proofs in "The Floyd-Warshall Algorithm, the AP and the TSP (III). We also simplify the method for obtaining a good upper bound for an optimal solution.
Let M be an n X n symmetric cost matrix. Assume that D is a derangement in M, i.e.,a set of disjo... more Let M be an n X n symmetric cost matrix. Assume that D is a derangement in M, i.e.,a set of disjoint cycles consisting of edges that contains all of the n points of M. The modified Floyd-Warshall algorithm applied to (D')^-1(M^-)A^- (where A is an asymmetric cost matrix containing D', a derangement)yielded a solution to the Assignment Problem in O((n^2)logn) running time. Here, applying a variation of the modified F-W to (D^-1)M^-1, we can obtain D = D_FWABS, the smallest-valued derangement obtainable using the modified F-W. Let T_TSPOPT be an optimal tour in M. We give conditions for obtaining D_ABSOLUTE, the smallest-valued derangement obtainable in M, where |D_ABSOLUTE| <= |T_TSPOPT|.
Let M be an n X n symmetric cost matrix. Assume that D is a derangement of edges in M, i.e., a se... more Let M be an n X n symmetric cost matrix. Assume that D is a derangement of edges in M, i.e., a set of point-disjoint cycles containing all of the n points of M.The modified Floyd-Warshall algorithm applied to ((D')^-1)A^- (where A is an asymmetric cost matrix containing D', a derangement)yielded a solution to the Assignment Problem in O((n^2)logn) running time. Here, applying a variation of the modified F-W algorithm to D^-1)M^-, we may possibly obtain a smaller-valued derangement than D consisting of entries in M. A minimally-valued derangement would be of great value as a good and natural lower bound for an optimal tour in M.
This paper is example 5 in chapter 5. Let H be an n-cycle. A permutation s is H-admissible if Hs ... more This paper is example 5 in chapter 5. Let H be an n-cycle. A permutation s is H-admissible if Hs = H' where H' is an n-cycle. Here we define a 19 X 19 matrix, M, in the following way: We obtain the remainders modulo 100 of each of the smallest 342 odd primes. we obtain the remainders modulo 100 of each of the primes. They are placed in M according to the original value of each prime. Thus their placement depends on the the original ordinal values of the primes according to size. We use this ordering to place the primes in M. Let H_0 be an initial 19 cycles arbitrarily chosen. We apply a sequence of up to [ln(n)+1] H_0 3-cycles to obtain a 19-cycle of smaller value than H_0, call the new 19-cycle H_1. We follow this procedure to obtain H_1. We call [ln(n)] + 1 a chain. We add up the values of the 19-cycles in each chain. This procedure continues until we cannot obtain a chain the sum of whose values is not negative. COMMENT. I've renamed the document "Yhe General Traveling Salesman Problem, Version 5". I preciously named it "The Traveling Salesman, Version 5". Although the algorithms work on the GTSP, I thought that more people would google it if it was named "The Traveling Salesman Problem." Rhar qas because my work is only available through arxiv.org,
. Although the graph (digraph) becomes non-random as the algorithm proceeds, the probability for ... more . Although the graph (digraph) becomes non-random as the algorithm proceeds, the probability for success stays the same. We also give examples.
. We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be a... more . We've changed Conjectures 1.1 and 1.2 so that they cover arbitrary graphs(digraphs). Let G be an arbitrary graph(digraph). Then - in polynomial time - either an algorithm obtains a hamilton circuit(cycle)or else the algorithm points to at least one vertex that cannot belong to any hamilton circuit(cycle) of G. We give criteria for determining which vertices should be examined.
We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and rando... more We give polynomial-time algorithms for obtaining hamilton circuits in random graphs, G, and random directed graphs, D. If n is finite, we assume that G or D contains a hamilton circuit. If G is an arbitrary graph containing a hamilton circuit, we conjecture that Algorithm G always obtains a hamilton circuit in polynomial time.
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