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The Graph Isomorphism Problem: Its Structural Complexity
Introduction Preliminaries Decision Problems, Search Problems, and Counting Problems NP-Completeness The Classes P and NP Reducibility. Reducing the Construction Problem to the Decision Problem
Optimal proof systems imply complete sets for promise classes
TLDR
The existence of (p-)optimal proof systems and the existence of complete problems for certain promise complexity classes like UP, NP ∩ Sparse, RP or BPP are shown.
On pseudorandomness and resource-bounded measure
TLDR
By using the Nisan–Wigderson design of a pseudorandom generator, it is unconditionally show the inclusion MA ⊆ ZPP NP and that MA ∩ coMA is low for ZPPNP.
On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties
TLDR
Focusing on dimensions $k=1,2$, this work investigates subgraph patterns whose counts are $k-WL invariant, and whose occurrence is $k$- WL invariants, and achieves a complete description of all such patterns for dimension k=1 and considerably extend the previous results known for k=2.
New Collapse Consequences of NP Having Small Circuits
TLDR
It is shown that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP(NP), which improves on the well-known result of Karp, Lipton, and Sipser (1980) stating a collapse of PH to its second level σ 2 P under the same assumption.
New Collapse Consequences of NP Having Small Circuits
TLDR
It is shown that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP (NP), and new collapse consequences are derived under the assumption that complexity classes like UP, FewP, and C=P have polynomials.
Completeness results for graph isomorphism
TLDR
It is proved that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC2 and that testing isomorphicism of two trees encoded as pointer lists is L-complete.
Is the Standard Proof System for SAT P-Optimal?
TLDR
This work investigates the question whether there is a (p-)optimal proof system for SAT or for TAUT and its relation to completeness and collapse results for nondeterministic function classes, and shows some relations between various completeness assumptions.
The Difference and Truth-Table Hierarchies for NP
Definition de deux hierarchies des classes de complexite. Localisation de la hierarchie de differences et de la hierarchie du tableau de verite pour NP. Exemples d'ensembles complets dans les deux
Complexity-Restricted Advice Functions
TLDR
The authors consider uniform subclasses of the nonuniform complexity classes defined by Karp and Lipton via the notion of advice functions, and it turns out that the $\NP$ reduction classes of bounded versions of this reducibility coincide with the odd levels of the boolean hierarchy.
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