Some connections between bounded query classes and non-uniform complexity

Amir, Amihood; Beigel, Richard; Gasarch, William

doi:10.1016/s0890-5401(03)00091-9

Cited by 11 publications

(22 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A language A is p-selective if there is a polynomial time algorithm that acts as follows: On input of two words x 1 , x 2 it outputs a y ∈ {x 1 , x 2 }, and if at least one of x 1 , x 2 is in A, then y is in A. Another prominent example is membership comparable languages (due to [1,10]). For a positive integer m a language A is m-membership-comparable if there is a polynomial time algorithm that on input of m words x 1 , .…”

Section: Introductionmentioning

confidence: 99%

One query reducibilities between partial information classes

Bab

Nickelsen

2005

Theoretical Computer Science

View full text Add to dashboard Cite

A partial information algorithm for a language A computes for m input words (x 1 , . . . , x m ) a set of bitstrings containing A (x 1 , . . . , x m ). For a family D of sets of bitstrings of length m, A ∈ P[D] if there is a polynomial time partial information algorithm that always outputs a set from D. For the case m = 2 we investigate whether for families D 1 and D 2 the languages in P[D 1 ] are reducible to languages in P X [D 2 ] for some X in the Polynomial Hierarchy PH or in EXP. Several non-reducibilities follow from known structural properties of classes P[D]. Beigel et al. [Membership comparable and pselective sets, Technical Report 2002-006N, NEC Research Institute, 2002] showed non-reducibility from strongly 2-membership comparable languages to p-selective languages. They also showed one query (1-tt, for short) reducibility from 2-cheatable languages to p-selective languages. A proof of Tantau [Combinatorial representations of partial information classes and their truth-table closures, Master's Thesis, TU Berlin, Germany, 1999] yields a 1-tt reducibility from 2-cheatable languages to languages in P p 2 [MIN 2 ]. We achieve results for all remaining non-trivial pairs of classes P[D] for m = 2. Our positive results all show 1-tt reducibilities. Our negative results even hold if the reducing machines as well as the partial information algorithms for the languages we try to reduce to have access to oracles in EXP. We show: 1. The 2-cheatable languages are 1-tt p 2 -reducible to languages in p 2 [MIN 2 ]. 2. Languages in P[SEL 2 ∪ {xor 2 }] are 1-tt p 2 -reducible to p 2 -selective languages.

show abstract

Section: Introductionmentioning

confidence: 99%

One query reducibilities between partial information classes

Bab

Nickelsen

2005

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…This leads us to "enumerability" as a measure of complexity. The concept of enumerability in computational complexity theory was introduced independently by Beigel [Bei87a] and by Cai and Hemachandra [CH89] then later modified by Amir, Beigel and Gasarch [ABG90].…”

Section: I Can Be Computed In Time Nmentioning

confidence: 99%

“…We are now ready to prove the main result of this section. The techniques used in this proof and in Lemma 4.4 are derived from results on enumerability and self-reducibility by Amir, Beigel and Gasarch [ABG90].…”

Section: #Ga and N -Enumerabilitymentioning

confidence: 99%

“…In general, we would like to prove stronger non-enumerability results for #GA. For example, Amir, Beigel and Gasarch [ABG90] were able to prove that #SAT is not 2 n -enumerable for < 1 unless the Polynomial Hierarchy collapses. We cannot use their machinery for the case of #GA, because it turns out that #GA is actually exp(O( √ n log n))-enumerable.…”

Section: #Ga and Poly-enumerabilitymentioning

confidence: 99%

“…For example, Cai and Hemachandra showed that unless P = PP, the function #SAT is not n -enumerable for < 1. 6 This result was improved independently by Cai and Hemachandra [CH91] and by Amir, Beigel and Gasarch [ABG90], who showed that P = PP if and only if #SAT is p(n)-enumerable for some polynomial p. Moreover, Amir, Beigel and Gasarch [ABG90] proved that unless the Polynomial Hierarchy collapses to its fourth level, #SAT is not 2 n -enumerable for < 1. Since #SAT is clearly 2 n -enumerable, these results show tight upper and lower bounds on the enumerability of #SAT assuming that PH does not collapse.…”

mentioning

confidence: 95%

See 2 more Smart Citations

On finding the number of graph automorphisms

Chang¹,

Gasarch²,

Torán³

Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference

View full text Add to dashboard Cite

In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function which can restrict the output of f (x) to one of b(n) possible values. This paper investigates #GA, the function which computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI.

show abstract

Bounded Queries, Approximations, and the Boolean Hierarchy

Chang¹

2001

Information and Computation

View full text Add to dashboard Cite

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NP-hard approximation problems. The results in this paper take advantage of this machine-based model to prove that in many cases, NP-approximation problems have the upward collapse property. That is, a reduction between NP-approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if MaxClique reduces to (log n)-approximating MaxClique using many-one reductions, then the Traveling Salesman Problem (TSP) is equivalent to MaxClique under many-one reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean Hierarchy and nondeterministic bounded query classes.

show abstract

Some connections between bounded query classes and non-uniform complexity

Cited by 11 publications

References 56 publications

One query reducibilities between partial information classes

One query reducibilities between partial information classes

On finding the number of graph automorphisms

Bounded Queries, Approximations, and the Boolean Hierarchy

Contact Info

Product

Resources

About