The paper gives a 2.097-approximation for Prize-Collecting Ordered TSP and a 2.41-approximation for Prize-Collecting Multi-Path TSP.
On the approximation hardness of some generalizations of TSP
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Approximating Prize-Collecting Variants of TSP
The paper gives a 2.097-approximation for Prize-Collecting Ordered TSP and a 2.41-approximation for Prize-Collecting Multi-Path TSP.