Collapsing Superstring Conjecture
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In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 211/23-approximation in polynomial time. While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. The Greedy Algorithm repeatedly merges two strings with the largest intersection into one, until only one string remains. We develop a graph-theoretic framework for studying approximation algorithms for SCS. In this framework, we give a (stronger) counterpart to the Greedy Conjecture: We conjecture that the presented in this paper Greedy Hierarchical Algorithm gives a 2-approximation for SCS. This algorithm is almost as simple as the standard Greedy Algorithm, and we suggest a combinatorial approach for proving this conjecture. We support the conjecture by showing that the Greedy Hierarchical Algorithm gives a 2-approximation in the case when all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjecture on tens of thousands of instances of SCS. Except for its conjectured good approximation ratio, the Greedy Hierarchical Algorithm finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2.
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