LWPP and WPP are not uniformly gap-definable

AbstractResolving an issue open since Fenner, Fortnow, and Kurtz raised it in [S. Fenner, L. Fortnow, S. Kurtz, Gap-definable counting classes, J. Comput. System Sci. 48 (1) (1994) 116–148], we prove that LWPP is not uniformly gap-definable and that WPP is not uniformly gap-definable. We do so in the context of a broader investigation, via the polynomial degree bound technique, of the lowness, Turing hardness, and inclusion relationships of counting and other central complexity classes

Average-case intractability vs. worst-case intractability

AbstractWe show that not all sets in NP (or other levels of the polynomial-time hierarchy) have efficient average-case algorithms unless the Arthur-Merlin classes MA and AM can be derandomized to NP and various subclasses of P/poly collapse to P. Furthermore, other complexity classes like P(PP) and PSPACE are shown to be intractable on average unless they are easy in the worst case

The Quantitative Structure of Exponential Time

Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed

A Quantum Algorithm for the Sub-Graph Isomorphism Problem

We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number of qubits that scales logarithmically with the number of vertices of the graphs; and (2) on a new Ansatz that can efficiently probe the permutation space. Simulations are then presented to showcase the approach on graphs up to 16 vertices, whereas, given the logarithmic scaling, the approach could be applied to realistic sub-graph isomorphism problem instances in the medium term.Comment: 30 pages, 14 figure

Some Facets of Complexity Theory and Cryptography: A Five-Lectures Tutorial

In this tutorial, selected topics of cryptology and of computational complexity theory are presented. We give a brief overview of the history and the foundations of classical cryptography, and then move on to modern public-key cryptography. Particular attention is paid to cryptographic protocols and the problem of constructing the key components of such protocols such as one-way functions. A function is one-way if it is easy to compute, but hard to invert. We discuss the notion of one-way functions both in a cryptographic and in a complexity-theoretic setting. We also consider interactive proof systems and present some interesting zero-knowledge protocols. In a zero-knowledge protocol one party can convince the other party of knowing some secret information without disclosing any bit of this information. Motivated by these protocols, we survey some complexity-theoretic results on interactive proof systems and related complexity classes.Comment: 57 pages, 17 figures, Lecture Notes for the 11th Jyvaskyla Summer Schoo

Complexity of certificates, heuristics, and counting types , with applications to cryptography and circuit theory

In dieser Habilitationsschrift werden Struktur und Eigenschaften von Komplexitätsklassen wie P und NP untersucht, vor allem im Hinblick auf: Zertifikatkomplexität, Einwegfunktionen, Heuristiken gegen NP-Vollständigkeit und Zählkomplexität. Zum letzten Punkt werden speziell untersucht: (a) die Komplexität von Zähleigenschaften von Schaltkreisen, (b) Separationen von Zählklassen mit Immunität und (c) die Komplexität des Zählens der Lösungen von ,,tally`` NP-Problemen

Data-driven exploration of mobility interaction patterns

In this thesis we propose an analysis framework for studying the interactions between moving objects, such as cars on roads, pedestrians in a square, etc. We follow a data mining approach, based on the computation of simple interaction events and on the extraction of complex patterns, describing frequent combinations of events that happen together and their evolution in time. The work includes two case studies on real datasets, respectively on cars and roads

Graph Isomorphism is Low for ZPP(NP) and other Lowness results

We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several group-theoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2

Graph isomorphism is low for ZPP^NP and other lowness results

N/