New Insights into the Role of Visit-to-Visit Glycemic Variability and Blood Pressure Variability in Cardiovascular Disease Risk

Zhou, Jin J.; Nuyujukian, Daniel S.; Reaven, Peter D.

doi:10.1007/s11886-021-01454-x

The multiple traveling salesman problem (mTSP) is an important variant of metric TSP where a set of k salespeople together visit a set of n cities while minimizing the total cost of the k routes under a given cost metric. The mTSP problem has applications to many real-life problems such as vehicle routing. Rothkopf [14] introduced another variant of TSP called many-visits TSP (MV-TSP) where a request r(v) is given for each city v and a single salesperson needs to visit each city r(v) times and return to his starting point. We note that in MV-TSP the cost of loops is positive, so a TSP solution cannot be trivially extended (without an increase in cost) to a MV-TSP solution by consecutively visiting each vertex to satisfy the visit requirements. A combination of mTSP and MV-TSP, called many-visits multiple TSP (MV-mTSP) was studied by Berczi, Mnich, and Vincze [3] where the authors give approximation algorithms for various variants of MV-mTSP. In this work, we show a simple linear programming (LP) based reduction that converts a mTSP LP-based algorithm to an LP-based algorithm for MV-mTSP with the same approximation factor. We apply this reduction to improve or match the current best approximation factors of several variants of the MV-mTSP. Our reduction shows that the addition of visit requests r(v) to mTSP does not make the problem harder to approximate even when r(v) is exponential in the number of vertices. To apply our reduction, we either use existing LP-based algorithms for mTSP variants or show that several existing combinatorial algorithms for mTSP variants can be interpreted as LP-based algorithms. This allows us to apply our reduction to these combinatorial algorithms while achieving improved guarantees.

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