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A valid computation sequence of length N can use no more than N tape cells, since at the very worst m moves right one tape cell in each step. Thus there are at most N time units and N tape cells to consider. We will encode computations of M on input x as truth assignments to various arrays of Boolean variables, which describe things like where the read head is at time i, which symbol is occupying cell j at time i, etc. We will write down clauses involving these variables that will describe legal moves of the machine and unique legal starting and accepting configurations of M on x.
G4 : at time 0, computation is in the initial configuration of reading input x = x0 x1 . . xn−1 . • G5 : by time N , M has entered state qy and accepted x Q[N, y]. , the action is faithful. Then the final formula ϕ is simply the conjunction of the Gi . It is clear that M has a satisfying assignment if and only if ϕ is satisfiable. We also know that 3S AT is NP-complete by a reduction: given a clause c of size m, like (x1 + x 3 + x5 + x 7 + x8 ), we replace it by introducing new variables zic , and then we use this conjunction of clauses (x1 + x 3 + z1c )(zc1 + x5 + z2c )(zc2 + x 7 + x8 ).
2 Shrinking the Search Tree In some cases, it is possible to significantly improve the constants in the algorithms. We only look at the example of V ERTEX C OVER, which, despite its apparent simplicity, has a rich structure and admits many parameterized algorithms. Fix k and consider a graph G. Now if G does not have a vertex of degree 3 or more, then it is a fairly trivial graph consisting of a collection of paths and cycles and isolated vertices. If such a G has more than 2k edges then it cannot have a size k vertex cover.