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Counting Classes

UR-CS Participating Faculty: Lane A. Hemaspaandra (= Lane A. Hemachandra)

Project Description

This project studies counting classes. The term ``counting classes'' has come to refer to a certain collection of classes--such as #P, SPP, probabilistic classes, parity-based classes, etc.--that are defined in terms of the number of accepting paths of nondeterministic machines. (We're also interested in limited nondeterminism classes, such as the levels of the beta hierarchy, since such classes indirectly limit the number of accepting--and indeed total--paths of nondeterministic machines.)

Among the questions central to this project are:

  1. How powerful are counting classes?
  2. What are the properties of counting classes?
  3. How robust are the definitions of counting classes?

A more recent interest of this project is counting the frequency with which heuristic algorithms can, and cannot (unless shocking complexity class collapses occur) be correct regarding concrete problems hard for complexity classes, most especially problems related to elections.

Bibliography

1
This is a list of selected journal (except when the work has not yet appeared in journal/book form, plus in some cases some conference articles) papers, from or related to this project, by University of Rochester authors. Essentially all the papers listed below can be found, in their full technical report versions, in the UR-CS Technical Report Archive's theory section. Here is Lane's complete publication list and links to essentially all his conference and journal papers (and also his arXiv.org technical reports) can be found via the ``EE'' (electronic edition) links at Lane's entry at the DBLP project.

2
J. Cai and L. Hemachandra.
On the power of parity polynomial time.
Mathematical Systems Theory, 23(2):95-106, 1990.

3
L. Hemachandra and S. Rudich.
On the complexity of ranking.
Journal of Computer and System Sciences, 41(2):251-271, 1990.

4
R. Beigel, L. Hemachandra, and G. Wechsung.
Probabilistic polynomial time is closed under parity reductions.
Information Processing Letters, 37(2):91-94, 1991.

5
J. Cai and L. Hemachandra.
A note on enumerative counting.
Information Processing Letters, 38(4):215-219, 1991.

6
J. Goldsmith, L. Hemachandra, D. Joseph, and P. Young.
Near-testable sets.
SIAM Journal on Computing, 20(3):506-523, 1991.

7
L. Hemachandra and A. Hoene.
On sets with efficient implicit membership tests.
SIAM Journal on Computing, 20(6):1148-1156, 1991.

8
D. Eppstein, L. Hemachandra, J. Tisdall, and B. Yener.
Simultaneous strong separations of probabilistic and unambiguous complexity classes.
Mathematical Systems Theory, 25(1):23-36, 1992.

9
J. Goldsmith, L. Hemachandra, and K. Kunen.
Polynomial-time compression.
Computational Complexity, 2(1):18-39, 1992.

10
L. Hemachandra and M. Ogiwara.
Is #P closed under subtraction?
In G. Rozenberg and A. Salomaa, editors, Current Trends in Theoretical Computer Science: Essays and Tutorials, pages 523-536. World Scientific, 1993.

11
M. Ogiwara and L. Hemachandra.
A complexity theory for feasible closure properties.
Journal of Computer and System Sciences, 46(3):295-325, 1993.

12
L. Hemaspaandra and H. Vollmer.
The Satanic notations: Counting classes beyond #P and other definitional adventures.
SIGACT News, 26(1):2-13, 1995.

13
L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman.
Computing solutions uniquely collapses the polynomial hierarchy.
SIAM Journal on Computing, 25(4):697-708, 1996.

14
L. Hemaspaandra, M. Zaki, and M. Zimand.
Polynomial-time semi-rankable sets.
In Journal of Computing and Information, 2(1), Special Issue: Proceedings of the 8th International Conference on Computing and Information, pages 50-67. 1996.
CD-ROM ISSN 1201-8511/V2/#1.

15
S. Fischer, L. Hemaspaandra, and L. Torenvliet.
Witness-isomorphic reductions and local search.
In A. Sorbi, editor, Complexity, Logic, and Recursion Theory, pages 207-223. Marcel Dekker, Inc., 1997.

16
L. Hemaspaandra and M. Ogihara.
Universally serializable computation.
Journal of Computer and System Sciences, 55(3):547-560, 1997.

17
L. Hemaspaandra, H. Hempel, and G. Wechsung.
Self-specifying machines.
International Journal of Foundations of Computer Science, 10(3):263-276, 1999.

18
B. Borchert, L. Hemaspaandra, and J. Rothe.
Restrictive acceptance suffices for equivalence problems.
London Mathematical Society Journal of Computation and Mathematics, 3:86-95, 2000.

19
L. Hemaspaandra and J. Rothe.
A second step towards complexity-theoretic analogs of Rice's Theorem.
Theoretical Computer Science, 244(1-2):205-217, 2000.

20
L. Hemaspaandra, M. Ogihara, and G. Wechsung.
Reducing the number of solutions of NP functions.
Journal of Computer and System Sciences, 64(2):311-328, 2002.

21
L. Hemaspaandra and M. Thakur.
Lower bounds and the hardness of counting properties.
Theoretical Computer Science, 326(1-3):1-28, 2004.

22
H. Spakowski, M. Thakur, and R. Tripathi.
Quantum and classical complexity classes: Separations, collapses, and closure properties.
Information and Computation, 200(1):1-34, 2005.

23
L. Hemaspaandra, M. Ogihara, M. Zaki, and M. Zimand.
The complexity of finding top-Toda-equivalence-class members.
Theory of Computing Systems, 39(5):669-684, 2006.

24
L. Hemaspaandra, C. Homan, S. Kosub, and K. Wagner.
The complexity of computing the size of an interval.
SIAM Journal on Computing, 36(5):1264-1300, 2006-2007.

25
L. Hemaspaandra, C. Homan, and S. Kosub.
Cluster computing and the power of edge recognition.
Information and Computation, 205(8):1274-1293, 2007.

26
P. Faliszewski and L. Hemaspaandra.
The consequences of eliminating NP solutions.
Computer Science Review, 2(1):40-54, 2008.

27
M. Zuckerman, P. Faliszewski, Y. Bachrach, and E. Elkind.
Manipulating quota value in weighted voting games.
In Proceedings of the 23rd AAAI Conference on Artificial Intelligence, pages 215-220. AAAI Press, July 2008.

28
G. Erdélyi, L. Hemaspaandra, J. Rothe, and H. Spakowski.
Frequency of correctness versus average polynomial time.
Information Processing Letters, 109(16):946-949, 2009.

29
G. Erdélyi, L. Hemaspaandra, J. Rothe, and H. Spakowski.
Generalized juntas and NP-hard sets.
Theoretical Computer Science, 410(38-40):3995-4000, 2009.

30
P. Faliszewski and L. Hemaspaandra.
The complexity of power-index comparison.
Theoretical Computer Science, 410(1):101-107, 2009.

31
C. Homan and L. Hemaspaandra.
Guarantees for the success frequency of an algorithm for finding Dodgson-election winners.
Journal of Heuristics, 15(4):403-423, 2009.

32
P. Faliszewski, E. Hemaspaandra, and L. Hemaspaandra.
The complexity of manipulative attacks in nearly single-peaked electorates.
In Proceedings of the 13th Conference on Theoretical Aspects of Rationality and Knowledge, pages 228-237. ACM Digital Library, July 2011.
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(Last modified: February 1, 2012.)


Lane A. Hemaspaandra