It is well known that given a finite set of strings of equal length, the Consensus String problem—the problem of deciding whether or not there exists a consensus string whose distance is at most r from every string in the given set—is proven to be NP-complete. A similar problem called the Closest Substring problem asks whether there exists a string w of length l such that each string in a given set L has a substring whose distance is at most r (called radius) from w. As the Closest Substring problem is a generalized version of the Consensus String problem, it is obvious that the problem is NP-hard for a finite set of strings. We show that the Closest Substring problem for regular languages represented by nondeterministic finite automata (NFAs) is PSPACE-complete. The main difference from the previous work is that we consider an infinite set of strings, which is recognized by an NFA as input instead of a finite set of strings. We also prove that the Closest Substring problem for acyclic NFAs lies in the second level of the polynomial-time hierarchy (formula presented) and is both NP-hard and coNP-hard.
|Title of host publication||Developments in Language Theory - 22nd International Conference, DLT 2018, Proceedings|
|Editors||Mizuho Hoshi, Shinnosuke Seki|
|Number of pages||12|
|Publication status||Published - 2018|
|Event||22nd International Conference on Developments in Language Theory, DLT 2018 - Tokyo, Japan|
Duration: 2018 Sep 10 → 2018 Sep 14
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Other||22nd International Conference on Developments in Language Theory, DLT 2018|
|Period||18/9/10 → 18/9/14|
Bibliographical noteFunding Information:
Acknowledgements. This work was supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (2018-0-00255, Autonomous digital companion framework and application).
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)