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Improved lower bounds on the approximability of the traveling salesman problem. (English) Zbl 0971.68075

Summary: This paper deals with lower bounds on the approximability of different subproblems of the Traveling Salesman Problem (TSP) which is known not to admit any polynomial time approximation algorithm in general (unless \(\text{P}=\text{NP}\)). First of all, we present an improved lower bound for the TS with Triangle Inequality, \(\Delta\)-TSP for short. Moreover our technique, an extension of the method of L. Engebretsen [\((*)\) Lect. Notes Comput. Sci. 1563, 373-382 (1999)], also applies to the case of relaxed and sharpened triangle inequality, respectively, denoted \(\Delta_\beta\)-TSP for an appropriate \(\beta\). In case of the \(\Delta\)-TSP, we obtain a lower bound of \({3813\over 3812}- \varepsilon\) on the polynomial-time approximability (for any small \(\varepsilon>0\)), compared to the previous bound of \({5381\over 5380}- \varepsilon\) in \((*)\). In case of the \(\Delta_\beta\)-TSP, for the relaxed case \((\beta> 1)\) we present a lower bound of \({3803+ 10\beta\over 3804+ 8\beta}- \varepsilon\), and for the sharpened triangle inequality \(({1\over 2}< \beta<1)\), the lower bound is \({7611+ 10\beta^2+5\beta\over 7612+8\beta^2+ 4\beta}- \varepsilon\). The latter result is of interest especially since it shows that the TSP is \({\mathcal A}{\mathcal P}{\mathcal X}\)-hard even if one comes arbitrarily close to the trivial case that all edges have the same cost.

MSC:

68Q25 Analysis of algorithms and problem complexity
68R10 Graph theory (including graph drawing) in computer science
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References:

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