We consider the following (re)optimization problem: Given a minimum-cost Hamiltonian cycle of a complete non-negatively real weighted graph $G=(V,E,c)$ obeying the \emph{strengthened} triangle inequality (i.e., for some \emph{strength factor} $\frac{1}{2} \leq \beta <1$, we have that $\forall x,y,z \in V, c(x,y) \leq \beta(c(x,z)+c(y,z))$), and given a set of $k$ edge weight modifications producing a new weighted graph still obeying the strengthened triangle inequality, find a minimum-cost Hamiltonian cycle of the modified graph. This problem is known to be \np-hard already for a single edge weight modification. However, in this case, if both the input and the modified graph obey the strengthened triangle inequality and the respective strength factors are fixed (i.e., independent of $|V|$), then it has been shown that the problem admits a \ptas (which just consists of either returning the old optimal cycle, or instead computing --- for finitely many inputs --- a new optimal solution from scratch, depending on the required accuracy in the approximation). In this paper we first extend the analysis of the \ptas to show its applicability for all $k=O(1)$, and then we provide a large set of experiments showing that, in most practical circumstances, altering (uniformly at random) even several edge weights does not affect the goodness of the old optimal solution.

### Reoptimizing the Strengthened Metric TSP on Multiple Edge Weight Modifications

#### Abstract

We consider the following (re)optimization problem: Given a minimum-cost Hamiltonian cycle of a complete non-negatively real weighted graph $G=(V,E,c)$ obeying the \emph{strengthened} triangle inequality (i.e., for some \emph{strength factor} $\frac{1}{2} \leq \beta <1$, we have that $\forall x,y,z \in V, c(x,y) \leq \beta(c(x,z)+c(y,z))$), and given a set of $k$ edge weight modifications producing a new weighted graph still obeying the strengthened triangle inequality, find a minimum-cost Hamiltonian cycle of the modified graph. This problem is known to be \np-hard already for a single edge weight modification. However, in this case, if both the input and the modified graph obey the strengthened triangle inequality and the respective strength factors are fixed (i.e., independent of $|V|$), then it has been shown that the problem admits a \ptas (which just consists of either returning the old optimal cycle, or instead computing --- for finitely many inputs --- a new optimal solution from scratch, depending on the required accuracy in the approximation). In this paper we first extend the analysis of the \ptas to show its applicability for all $k=O(1)$, and then we provide a large set of experiments showing that, in most practical circumstances, altering (uniformly at random) even several edge weights does not affect the goodness of the old optimal solution.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/38151
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