### Abstract

In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very flexible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces to a sparse set, then P = NP. We also show that any set A that reduces to some sparse set (via various types of reductions) in fact reduces by the same type of reduction to a sparse set that is simple relative to A. We give a complete characterization of the sets of low instance complexity in terms of reductions to tally sets; it follows that if P ≠ NP, then no set of low instance complexity can be complete for NP with respect to disjunctive reductions or conjunctive reductions.

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 19th International Colloquium, Proceedings |

Editors | Werner Kuich |

Publisher | Springer Verlag |

Pages | 162-173 |

Number of pages | 12 |

ISBN (Print) | 9783540557197 |

DOIs | |

State | Published - Jan 1 1992 |

Externally published | Yes |

Event | 19th International Colloquium on Automata, Languages, and Programming, ICALP 1992 - Wien, Austria Duration: Jul 13 1992 → Jul 17 1992 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 623 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 19th International Colloquium on Automata, Languages, and Programming, ICALP 1992 |
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Country | Austria |

City | Wien |

Period | 7/13/92 → 7/17/92 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - 19th International Colloquium, Proceedings*(pp. 162-173). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 623 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-55719-9_72

**Reductions to sets of low information content.** / Arvind, V.; Han, Y.; Hemachandra, L.; Köbler, J.; Lozano, A.; Mundhenk, M.; Ogihara, Mitsunori; Schöning, U.; Silvestri, R.; Thierauf, T.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automata, Languages and Programming - 19th International Colloquium, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 623 LNCS, Springer Verlag, pp. 162-173, 19th International Colloquium on Automata, Languages, and Programming, ICALP 1992, Wien, Austria, 7/13/92. https://doi.org/10.1007/3-540-55719-9_72

}

TY - GEN

T1 - Reductions to sets of low information content

AU - Arvind, V.

AU - Han, Y.

AU - Hemachandra, L.

AU - Köbler, J.

AU - Lozano, A.

AU - Mundhenk, M.

AU - Ogihara, Mitsunori

AU - Schöning, U.

AU - Silvestri, R.

AU - Thierauf, T.

PY - 1992/1/1

Y1 - 1992/1/1

N2 - In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very flexible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces to a sparse set, then P = NP. We also show that any set A that reduces to some sparse set (via various types of reductions) in fact reduces by the same type of reduction to a sparse set that is simple relative to A. We give a complete characterization of the sets of low instance complexity in terms of reductions to tally sets; it follows that if P ≠ NP, then no set of low instance complexity can be complete for NP with respect to disjunctive reductions or conjunctive reductions.

AB - In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very flexible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces to a sparse set, then P = NP. We also show that any set A that reduces to some sparse set (via various types of reductions) in fact reduces by the same type of reduction to a sparse set that is simple relative to A. We give a complete characterization of the sets of low instance complexity in terms of reductions to tally sets; it follows that if P ≠ NP, then no set of low instance complexity can be complete for NP with respect to disjunctive reductions or conjunctive reductions.

UR - http://www.scopus.com/inward/record.url?scp=84976776860&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976776860&partnerID=8YFLogxK

U2 - 10.1007/3-540-55719-9_72

DO - 10.1007/3-540-55719-9_72

M3 - Conference contribution

AN - SCOPUS:84976776860

SN - 9783540557197

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 162

EP - 173

BT - Automata, Languages and Programming - 19th International Colloquium, Proceedings

A2 - Kuich, Werner

PB - Springer Verlag

ER -