Parameterized Complexity of Superstring Problems

In the Shortest Superstring problem we are given a set of strings $$S=\{s_1, \ldots, s_n\}$$S={s1,ź,sn} and integer $$\ell $$ℓ and the question is to decide whether there is a superstring s of length at most $$\ell $$ℓ containing all strings of S as substrings. We obtain several parameterized algorithms and complexity results for this problem. In particular, we give an algorithm which in time $$2^{\mathcal {O}(k)} {\text {poly}}(n)$$2O(k)poly(n) finds a superstring of length at most $$\ell $$ℓ containing at least k strings of S. We complement this by a lower bound showing that such a parameterization does not admit a polynomial kernel up to some complexity assumption. We also obtain several results about "below guaranteed values" parameterization of the problem. We show that parameterization by compression admits a polynomial kernel while parameterization "below matching" is hard.