Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

In this paper the generality and wide applicability of Zero-knowledge proofs, a notion introduced by Goldwasser, Micali, and Rackoff is demonstrated. These are probabilistic and interactive proofs that, for the members of a language, efficiently demonstrate membership in the language without conveying any additional knowledge. All previously known zero-knowledge proofs were only for number-theoretic languages in NP fl CONP. Under the assumption that secure encryption functions exist or by using "physical means for hiding information, '' it is shown that all languages in NP have zero-knowledge proofs. Loosely speaking, it is possible to demonstrate that a CNF formula is satisfiable without revealing any other property of the formula, in particular, without yielding neither a satis@ing assignment nor properties such as whether there is a satisfying assignment in which xl = X3 etc. It is also demonstrated that zero-knowledge proofs exist "outside the domain of cryptography and number theory. " Using no assumptions. it is shown that both graph isomorphism and graph nonisomor- phism have zero-knowledge interactive proofs. The mere existence of an interactive proof for graph nonisomorphism is interesting, since graph nonisomorphism is not known to be in NP and hence no efficient proofs were known before for demonstrating that two graphs are not isomorphic.

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Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems

As most ‘real-world’ data is structured, research in kernel methods has begun investigating kernels for various kinds of structured data. One of the most widely used tools for modeling structured data are graphs. An interesting and important challenge is thus to investigate kernels on instances that are represented by graphs. So far, only very specific graphs such as trees and strings have been considered.

On Graph Kernels: Hardness Results and Efficient Alternatives

We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shockgraph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint.

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Shock graphs and shape matching

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

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 Counting versus Deciding for Graph Isomorphism Uniqueness of the Solution Random Reductions Promise Problems Reducing Multiple Questions to One Quantifiers, Games, and Interactive Proofs The Polynomial-Time Hierarchy Interactive Proof Systems The Class IP Zero-Knowledge Probabilistic Classes Probability Amplification The BP-Operator Arthur-Merlin Games Lowness and Collapses Circuits and Sparse Sets Polynomial Size Circuits Circuits for NP Circuits for Graph Isomorphism Reductions to Sparse Sets Counting Properties Decision Reduces to Parity Graph Isomorphism is low for PP The Reconstruction Conjecture Bibliography Index

The Graph Isomorphism Problem : Its Structural Complexity

This report documents the program and the outcomes of Dagstuhl Seminar 15511 “The Graph Isomorphism Problem”. The goal of the seminar was to bring together researchers working on the numerous topics closely related to the Isomorphism Problem to foster their collaboration. To this end the participants of the seminar included researchers working on the theoretical and practical aspects of isomorphism ranging from the fields of algorithmic group theory, finite model theory, combinatorial optimization to algorithmics. A highlight of the conference was the presentation of a new quasi-polynomial time algorithm for the Graph Isomorphism Problem, providing the first improvement since 1983. Seminar December 13–18, 2015 – http://www.dagstuhl.de/15511 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory

Uwe Schöning

Papers

The Graph Isomorphism Problem : Its Structural Complexity

The Graph Isomorphism Problem

Graph isomorphism is in the low hierarchy

A deterministic (2 - 2/( k + 1)) n algorithm for k -SAT based on local search

A low and a high hierarchy within NP