Parameterized Complexity of Superstring Problems

TL;DR

This paper explores the parameterized complexity of the Shortest Superstring problem, providing algorithms and hardness results for various parameterizations, including fixed-parameter tractability and kernelization bounds.

Contribution

It introduces new parameterized algorithms and complexity bounds for the Superstring problem, including fixed-parameter algorithms and kernelization results.

Findings

An FPT algorithm for superstring length with parameter k

A polynomial kernel for the compression parameterization

Hardness results for below matching parameterization

Abstract

In the Shortest Superstring problem we are given a set of strings $S=\{s_1, \ldots, s_n\}$ and integer $\ell$ and the question is to decide whether there is a superstring $s$ of length at most $\ell$ 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 $2^{O(k)} \operatorname{poly}(n)$ finds a superstring of length at most $\ell$ containing at least $k$ strings of $S$. We complement this by the 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.