Remember linear programs? E.g.<p>min 10x + 5y
s.t.   x &lt;= 10
       y &gt;= 5
       x+y = 8<p>(Not very useful example)<p>Now we can extend normal linear programs to what is known as Mixed-integer linear programming (MILP). This allows us to specify the domain of variables, i.e. we can say variable x must be a positive integer or variable y must be binary.<p>(This extension instantly makes solving generic MILPs NP-hard because obviously we can transform any e.g. 3-SAT problem into an equivalent MILP problem)<p>MILPs can also be used to solve the Traveling Salesman (TSP) problem. You use a binary variable for every possible pair of cities or edge the salesman could travel on that specifies if the edge is used or not. Since you want a loop, you also add a constraint for each city that it is entered and exited exactly once using the edge variables.<p>At this point, you could think we are done: we enter and exit each city once, so we have a loop and since we minimise the sum of edge costs, we also have the optimal loop.<p>Unfortunately, our formulation is incomplete because it allows a loophole for the solver: doing multiple tours or <i>subtours</i>.<p>If we have a TSP with 5 cities, the solver could choose the edges such that you have two loops or subtours, e.g. one with 3 and another with 2 cities. That would be perfectly consistent with our formulation since each city is still entered and exited exactly once. It&#x27;s not a valid solution for the TSP however since we require one single loop, not two unconnected ones.<p>To fix this problem, we need to add what are known as &quot;subtour elimination constraints&quot;. These check that for every partition of our city set into two sets A and B, the cities in A and B are connected, i.e. we have one edge leaving A for B and another returning from B to A (or vice versa).<p>Unfortunately, since we need to do this for every possible combination of sets A and B, we would have to add exponentially many of these subtour elimination constraints, and of course for very large TSPs even <i>enumerating</i> all sets A and B is already infeasible.<p>So what is done in practice is to first leave out all subtour elimination constraints, since this is what causes the exponential explosion (so far, our problem was polynomial in size). You then solve the linear relaxation of the MILP, i.e. you pretend that the binary constraints on the edge variables do not exist and use Simplex to solve the resulting linear problem.<p>Then you create a weighted graph: the cities become vertices, and the edge weights are given by the (fractional) values of the edge variables. For this graph, compute the minimal cut using your favourite polynomial time algorithm. This gives you three things: a partition of the cities into two sets A and B, and a value for the flow between these two sets. If the flow is less than 2, you have found a violation of the subtour elimination constraint for sets A and B. You add that one constraint to the solver (<i>cutting</i> the invalid solution) and start anew.<p>This process can be fully integrated into a branch &amp; bound or branch &amp; cut solver for MILPs.<p>(I&#x27;m sure wolfgke can give a much better explanation, though)