The Maximum TSP

Barvinok, Alexander; Гимади, Э. Х.; Serdyukov, A. N.

doi:10.1007/0-306-48213-4_12

Cited by 17 publications

(14 citation statements)

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“…The best known performance ratio for a polynomial approximation algorithm for such an instance of ATSP is 4/3 log 3 n ≈ 0.842 log 2 n [13]. When C is Hamiltonian symmetrical and satisfies the weak τ -triangle inequality (see Section 4), we observe that the performance ratio becomes min{4τ, 3 2 τ 2 } for τ ≥ 1 and when C satisfies the weak range inequality (see Section 4) the performance ratio becomes 4+δ 3 for δ ≥ 0.…”

Section: Lemma 12 [12]mentioning

confidence: 90%

“…For the maximization version of the TSP, the best known polynomial time approximation algorithm has a performance ratio of 2/3 for the ATSP [13] and 3/4 bound for the STSP [3]. Thus from Theorem 4.3, the maximization TSP when C is Hamiltonian symmetrical can be approximated by the 3/4-approximation scheme given in [24] for the STSP.…”

Section: Theorem 42 [9]mentioning

confidence: 99%

“…However, no corresponding results are known for Hamiltonian cycles. We give a complete characterization of DTC (3) and DPC(3) skewsymmetric cost matrices associated with complete digraphs. It is also shown that there are no skew symmetric DPC(2) matrices of size greater than 3.…”

Section: Lemma 12 [12]mentioning

confidence: 99%

“…The proof of this lemma is little bit long and we relegate this to the appendix. (3,4) and (4,5). Construct a tourĤ from H using Scheme 3 with arc (2, 3) as arc (i, j).…”

Section: Skew-symmetric Dtc(3) and Dpc(3) Matrices For Complete Digraphsmentioning

confidence: 99%

“…In addition, if C satisfies the triangle inequality this bound can be improved to 7/8 by using the algorithm of Hassin and Rubinstein [10] on the matrix (C + C T )/2. Several polynomially solvable cases of the symmetric TSP can be exploited (for both maximization and minimization versions) to solve the ATSP with cost matrix C whenever 1/2(C + C T ) satisfies the required conditions [3,11].…”

Section: Theorem 42 [9]mentioning

confidence: 99%

See 4 more Smart Citations

On cost matrices with two and three distinct values of Hamiltonian paths and cycles

Kabadi¹,

Punnen²

2006

SIAM J. Discrete Math.

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Polynomially testable characterization of cost matrices associated with a complete digraph on n nodes such that all the Hamiltonian cycles (tours) have the same cost is well known. Tarasov [25] obtained a characterization of cost matrices where tour costs take two distinct values. We provide a simple alternative characterization of such cost matrices that can be tested in O(n 2) time. We also provide analogous results where tours are replaced by Hamiltonian paths. When the cost matrix is skew-symmetric, we provide polynomially testable characterizations such that the tour costs take three distinct values. Corresponding results for the case of Hamiltonian paths are also given. Using these results, special instances of the asymmetric travelling salesman problem (ATSP) are identified that are solvable in polynomial time and that have improved ǫ-approximation schemes. In particular, we observe that the 3/2 performance guarantee of the Christofides algorithm extends to all metric Hamiltonian symmetric matrices. Further, we identify special classes of ATSP for which polynomial ǫ-approximation algorithms are available for ǫ ∈ {3/2, 4/3, 4τ, 3τ 2 2 , 4+δ 3 } where τ > 1/2 and δ ≥ 0

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Section: Lemma 12 [12]mentioning

confidence: 90%

Section: Theorem 42 [9]mentioning

confidence: 99%

Section: Lemma 12 [12]mentioning

confidence: 99%

“…The proof of this lemma is little bit long and we relegate this to the appendix. (3,4) and (4,5). Construct a tourĤ from H using Scheme 3 with arc (2, 3) as arc (i, j).…”

Section: Skew-symmetric Dtc(3) and Dpc(3) Matrices For Complete Digraphsmentioning

confidence: 99%