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In the last lecture, we discussed FPT algorithms for the k-vertex cover problem. By doing backtracking and analyzing different cases, we can get a recurrence with different cases, such as T (k) ≤ max{T (k − 1) + T (k − 4), T (k − 2) + T (k − 4),...} + poly(n), and such recurrences will solve to something of the form T (k) = O(c k · poly(n)), for a constant c ∈ (1, 2). Using more clever backtracking, one can develop even more complex recurrences and a running time of O(1.27 k · poly(n)). For the minimum vertex cover problem, we can therefore solve it on arbitrary graphs in time O ∗ (1.27 n). 1 In fact, one can get a slightly better running time for arbitrary graphs, via the following trick. Notice that if the vertex cover is of size at least n − δn for some δ < 1/2, we can solve the problem in O ∗ ( () n δn) time, by trying all vertex subsets of size at least n − δn. This bound is at most O ∗ (2 H(δ)n), where H is the binary entropy function, i.e., H(δ) = δ · log 2(1/δ) + (1 − δ) · log 2(1/(1 − δ)). On the other hand, if the vertex cover is of size at most n − δn, then we get O ∗ (1.27 n−δn) time. By calculation, w

Year: 2013

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