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Given a graph G = (V, E), construct the following instance of the TSP: the set of cities is the same as V , and the distance between cities u and v is 1 if {u, v} is an edge of G and 1 + otherwise, for some > 1 to be determined The budget of the TSP instance is equal to the number of nodes, |V | It is easy to see that if G has a Rudrata cycle, then the same cycle is also a tour within the budget of the TSP instance; and that conversely, if G has no Rudrata cycle, then there is no solution: the cheapest possible TSP tour has cost at least n + (it must use at least one edge of length 1 + , and the total length of all n 1 others is at least n 1) Thus R UDRATA CYCLE reduces to TSP In this reduction, we introduced the parameter because by varying it, we can obtain two interesting results If = 1, then all distances are either 1 or 2, and so this instance of the TSP satis es the triangle inequality: if i, j, k are cities, then d ij + djk dik (proof: a + b c holds for any numbers 1 a, b, c 2) This is a special case of the TSP which is of practical importance and which, as we shall see in 9, is in a certain sense easier, because it can be ef ciently approximated If on the other hand is large, then the resulting instance of the TSP may not satisfy the triangle inequality, but has another important property: either it has a solution of cost n or less, or all its solutions have cost at least n + (which now can be arbitrarily larger than n) There can be nothing in between! As we shall see in 9, this important gap property implies that, unless P = NP, no approximation algorithm is possible
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We have reduced SAT to the various search problems in Figure 87 Now we come full circle and argue that all these problems and in fact all problems in NP reduce to SAT In particular, we shall show that all problems in NP can be reduced to a generalization of SAT which we call CIRCUIT SAT In CIRCUIT SAT we are given a (Boolean) circuit (see Figure 813, and recall Section 77), a dag whose vertices are gates of ve different types:
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