the second of the three formulations looks very impractical it says these conditions are not enough to characterize a tour even though the are restricted to be integers in the interval since sub tours like those in fig iv omitted also satisfy the conditions however if so called loop conditions like are imposed as added constraints as required these will rule out integer solutions which are not admissible this is vague about how many of these as required there are but it gives the impression that one of these has to be imposed for every possible invalid subtour which gives a combinatorially large number of these loop conditions i'll take a look at the first one of dantzig's