A PTAS for the Min-Max Euclidean Multiple TSP
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We present a polynomial-time approximation scheme (PTAS) for the min-max multiple TSP problem in Euclidean space, where multiple traveling salesmen are tasked with visiting a set of n points and the objective is to minimize the maximum tour length. For an arbitrary ε > 0, our PTAS achieves a (1 + ε)-approximation in time O (n ((1/ε) log (n/ε))^O(1/ε)). Our approach extends Sanjeev Arora's dynamic-programming (DP) PTAS for the Euclidean TSP (https://doi.org/10.1145/290179.290180). Our algorithm introduces a rounding process to balance the allocation of path lengths among the multiple salesman. We analyze the accumulation of error in the DP to prove that the solution is a (1 + ε)-approximation.
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