Semi-Feasible Algorithms
Adapted from the introduction of the
book “Theory of Semi-Feasible Algorithms,” by
Lane Hemaspaandra
and
Leen Torenvliet:
The focus of complexity theory is the computational complexity of
sets. However, it is an underappreciated
fact that sets may have various
types of complexity, and not all types are harmony
with each other. For example, sets that are
complex in terms of
deterministic time may nonetheless be simple in other natural
senses. Unifying and making more widely accessible a vibrant stream
of research—semi-feasible computation—that perfectly showcases
this point is the primary goal of this book.
The semi-feasible sets, which are most commonly referred to as the
P-selective sets, are those sets for which there is a
deterministic polynomial-time algorithm that, given as input any two
strings, outputs the one in if exactly one is in
.
The reason we say that the semi-feasible sets
showcase the above distinction is that it
is well-known that the semi-feasible sets are arbitrarily complex
in terms of the deterministic time it takes to recognize them, yet
they are
simple in a wide range of other senses. In
particular, they have small circuits, they are in the extended low
hierarchy,
and
they cannot be
NP-complete
unless P=NP.
We find the semi-feasible sets to be fascinating for many
reasons. First, as mentioned above, they
showcase the fact that mere
deterministic time complexity is not the only potential type of
complexity in the world of computation; sets that are complex in
terms of deterministic time may nonetheless be simple in many other
computationally natural senses.
A
second reason that the semi-feasible sets are interesting
is that they crisply
capture the complexity of (left cuts of) real
numbers, and
recently a refinement of the semi-feasible sets has
been shown to capture the
complexity of complexity-bounded
real numbers.
A third and more historical reason for interest in the semi-feasible
sets is that they are the
complexity-theoretic analog of
a key notion from recursive function theory; the semi-feasible sets
are exactly what one gets when one alters the definition of
the semi-recursive
sets
by changing the selector function from “recursive”
to “polynomial-time computable.”
In the late 1960s the semi-recursive sets yielded great insight into
distinguishing the power
of reductions in the recursion-theoretic context,
and in 1979 Selman launched
a program that used—successfully, in the
context of
structural connections to exponential time—semi-feasible sets to understand
the structure of polynomial-time reductions.
A fourth and somewhat surprising reason to
study semi-feasible sets is that
semi-feasible sets (in their recently-defined nondeterministic
version) have been shown to conditionally resolve
Selman's important question as to whether NP machines can
cull down to one the large number of potential solutions of
satisfiable formulas; in particular, it is now known that NP lacks
such “unique solutions” unless the polynomial hierarchy
collapses.
- 1
-
This is a list of selected papers, from or related to this project,
by University of Rochester authors. Links to essentially all Lane's
conference and journal papers (and also his arXiv.org technical reports) can
be found via the pointers from the related entries within
Lane's entry at the DBLP
project.
Additionally, here is a link to Lane's complete
publication list
(note: that
list does not itself have links to papers).
- 2
-
E. Allender and L. Hemachandra.
Lower bounds for the low hierarchy.
Journal of the ACM, 39(1):234–251, 1992.
- 3
-
J.-Y. Cai, V. Chakaravarthy, L. Hemaspaandra, and M. Ogihara.
Competing provers yield improved Karp–Lipton collapse results.
Information and Computation, 198(1):1–23, 2005.
- 4
-
D. Denny-Brown, Y. Han, L. Hemaspaandra, and L. Torenvliet.
Semi-membership algorithms: Some recent advances.
SIGACT News, 25(3):12–23, 1994.
- 5
-
D. Eisenstat.
Simpler proofs of the power of one query to a P-selective set.
Technical Report TR-883, Department of Computer Science, University
of Rochester, Rochester, NY, October 2005.
- 6
-
P. Faliszewski and L. Hemaspaandra.
Advice for semifeasible sets and the complexity-theoretic
cost(lessness) of algebraic properties.
International Journal of Foundations of Computer Science,
16(5):913–928, 2005.
- 7
-
P. Faliszewski and L. Hemaspaandra.
Open questions in the theory of semifeasible computation.
SIGACT News, 37(1):47–65, 2006.
- 8
-
P. Faliszewski and L. Hemaspaandra.
The consequences of eliminating NP solutions.
Computer Science Review, 2(1):40–54, 2008.
- 9
-
E. Hemaspaandra, L. Hemaspaandra, T. Tantau, and O. Watanabe.
On the complexity of kings.
In Proceedings of the 16th International Symposium on
Fundamentals of Computation Theory, pages 328–340. Springer-Verlag Lecture
Notes in Computer Science #4639, August 2007.
- 10
-
E. Hemaspaandra, L. Hemaspaandra, T. Tantau, and O. Watanabe.
On the complexity of kings.
Theoretical Computer Science, 411(4–5):783–798, 2010.
- 11
-
L. Hemaspaandra.
Beautiful structures: An appreciation of the contributions of Alan
Selman.
SIGACT News, 45(3):54–70, 2014.
- 12
-
L. Hemaspaandra.
Complexity classes.
In K. Rosen, editor, Handbook of Discrete and Combinatorial
Mathematics, pages 1308–1314. CRC Press, 2nd edition, 2018.
- 13
-
L. Hemaspaandra.
The power of self-reducibility: Selectivity, information, and
approximation.
In D.-Z. Du and J. Wang, editors, Complexity and Approximation,
pages 19–47. Springer, 2020.
- 14
-
L. Hemaspaandra, H. Hempel, and A. Nickelsen.
Algebraic properties for selector functions.
SIAM Journal on Computing, 33(6):1309–1337, 2004.
- 15
-
L. Hemaspaandra, A. Hoene, A. Naik, M. Ogiwara, A. Selman, T. Thierauf, and
J. Wang.
Nondeterministically selective sets.
International Journal of Foundations of Computer Science,
6(4):403–416, 1995.
- 16
-
L. Hemaspaandra, A. Hoene, and M. Ogihara.
Reducibility classes of P-selective sets.
Theoretical Computer Science, 155(2):447–457, 1996.
Erratum appears in the same journal, 234(1–2):323.
- 17
-
L. Hemaspaandra and Z. Jiang.
P-selectivity: Intersections and indices.
Theoretical Computer Science, 145(1–2):371–380, 1995.
- 18
-
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.
- 19
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L. Hemaspaandra, C. Nasipak, and K. Parkins.
A note on linear-nondeterminism, linear-sized, Karp–Lipton
advice for the P-selective sets.
Journal of Universal Computer Science, 4(8):670–674, 1998.
- 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, 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.
- 22
-
L. Hemaspaandra and L. Torenvliet.
Optimal advice.
Theoretical Computer Science, 154(2):367–377, 1996.
- 23
-
L. Hemaspaandra and L. Torenvliet.
Theory of Semi-Feasible Algorithms.
Springer-Verlag, 2003.
- 24
-
L. Hemaspaandra and L. Torenvliet.
P-selectivity, immunity, and the power of one bit.
In Proceedings of the 32nd International Conference on Current
Trends in Theory and Practice of Computer Science, pages 323–331.
Springer-Verlag Lecture Notes in Computer Science #3881, January 2006.
- 25
-
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.
(Last modified: February 16, 2023.)
Lane A. Hemaspaandra
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