On the number of P-isomorphism classes of NP-complete sets

1981 ◽  
Author(s):  
Stephen R. Mahaney
Keyword(s):  
Isomorphism Classes ◽  
Np Complete ◽  
Complete Sets

Unions of Disjoint NP-Complete Sets

2011 ◽  
pp. 240-251
Author(s):  
Christian Glaßer ◽  
John M. Hitchcock ◽  
A. Pavan ◽  
Stephen Travers
Keyword(s):  
Np Complete ◽  
Complete Sets

Relativized isomorphisms of NP-complete sets

1993 ◽  
Vol 3 (2) ◽  
pp. 186-205 ◽  
Author(s):  
Judy Goldsmith ◽  
Deborah Joseph
Keyword(s):  
Np Complete ◽  
Complete Sets

scholarly journals One-way functions and the nonisomorphism of NP-complete sets

1991 ◽  
Vol 81 (1) ◽  
pp. 155-163 ◽  
Author(s):  
Juris Hartmanis ◽  
Lane A. Hemachandra
Keyword(s):  
Np Complete ◽  
Complete Sets

scholarly journals THE INFORMATIONAL CONTENT OF CANONICAL DISJOINT NP-PAIRS

2009 ◽  
Vol 20 (03) ◽  
pp. 501-522 ◽  
Author(s):  
CHRISTIAN GLAßER ◽  
ALAN L. SELMAN ◽  
LIYU ZHANG
Keyword(s):  
Open Problem ◽  
Structural Complexity ◽  
Informational Content ◽  
Proof Systems ◽  
Canonical Pair ◽  
Propositional Proof Systems ◽  
Np Complete ◽  
Complete Sets ◽  
Propositional Proof ◽  
Almost All

We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between propositional proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: For which propositional proof systems f and g does the implication [Formula: see text] hold, and for which does it fail? Q2: For which propositional proof systems of different strengths are the canonical pairs equivalent? Q3: What do (non-)equivalent canonical pairs tell about the corresponding propositional proof systems? Q4: Is every NP-pair (A, B), where A is NP-complete, strongly many-one equivalent to the canonical pair of some propositional proof system? In short, we show that Q1 and Q2 can be answered with 'for almost all', which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the propositional proof systems are not "very similar," while equivalent, P -inseparable canonical pairs tell that they are not "very different." We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NP-complete sets are NP-complete. This demonstrates a new connection between propositional proof systems, disjoint NP-pairs, and unions of disjoint NP-complete sets.

Properties of NP‐Complete Sets

2006 ◽  
Vol 36 (2) ◽  
pp. 516-542 ◽  
Author(s):  
Christian Glaßer ◽  
A. Pavan ◽  
Alan L. Selman ◽  
Samik Sengupta
Keyword(s):  
Np Complete ◽  
Complete Sets

scholarly journals Building Graphs from Colored Trees

10.37236/433 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Rachel M. Esselstein ◽  
Peter Winkler
Keyword(s):  
Computational Complexity ◽  
Finite Graph ◽  
Realization Problem ◽  
Neighborhood Conditions ◽  
Unique Root ◽  
Isomorphism Classes ◽  
Colored Graphs ◽  
Infinite Tree ◽  
Np Complete ◽  
Unique Vertex

We will explore the computational complexity of satisfying certain sets of neighborhood conditions in graphs with various properties. More precisely, fix a radius $\rho$ and let $N(G)$ be the set of isomorphism classes of $\rho$-neighborhoods of vertices of $G$ where $G$ is a graph whose vertices are colored (not necessarily properly) by colors from a fixed finite palette. The root of the neighborhood will be the unique vertex at the "center" of the graph. Given a set $\mathcal{S}$ of colored graphs with a unique root, when is there a graph $G$ with $N(G)=\mathcal{S}$? Or $N(G) \subset \mathcal{S}$? What if $G$ is forced to be infinite, or connected, or both? If the neighborhoods are unrestricted, all these problems are recursively unsolvable; this follows from the work of Bulitko [Graphs with prescribed environments of the vertices. Trudy Mat. Inst. Steklov., 133:78–94, 274, 1973]. In contrast, when the neighborhoods are cycle free, all the problems are in the class $\mathtt{P}$. Surprisingly, if $G$ is required to be a regular (and thus infinite) tree, we show the realization problem is NP-complete (for degree 3 and higher); whereas, if $G$ is allowed to be any finite graph, the realization problem is in P.

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