We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.

/pdf/normalized-cuts-and-image-segmentation-2ru3ikxuj5.pdf

Normalized cuts and image segmentation

We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We have applied this approach to segmenting static images and found results very encouraging.

/pdf/normalized-cuts-and-image-segmentation-1q390dk21h.pdf

The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

/pdf/the-probabilistic-method-5g63k0itke.pdf

The Probabilistic Method

We present a polynomial-time algorithm that, given as a input the description of a game with incomplete information and any number of players, produces a protocol for playing the game that leaks no partial information, provided the majority of the players is honest. Our algorithm automatically solves all the multi-party protocol problems addressed in complexity-based cryptography during the last 10 years. It actually is a completeness theorem for the class of distributed protocols with honest majority. Such completeness theorem is optimal in the sense that, if the majority of the players is not honest, some protocol problems have no efficient solution [C].

/pdf/how-to-play-any-mental-game-4r22j29pmt.pdf

How to play ANY mental game

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set.

/pdf/computational-complexity-a-modern-approach-5gtk8urev9.pdf

Computational Complexity: A Modern Approach

Does co-NP have short interactive proofs?

Graph Bisection is the problem of partitioning the vertices of a graph into two equal-size pieces so as to minimize the number of edges between the two pieces. This paper presents an algorithm that will, for almost all graphs in a certain class, output the minimum-size bisection. Furthermore the algorithm will yield, for almost all such graphs, a proof that the bisection is optimal. The algorithm is based on computing eigenvalues and eigenvectors of matrices associated with the graph.

Eigenvalues and graph bisection: An average-case analysis

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(m
 s
 /(logm)2s
 ) for fixeds, and sizem
 Ω(logm) form/4]. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm)
 s
 ) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n
 1/4 · (logn)1/2)), improving a recent result of exp (Ω(n
 1/8-e)) due to Andreev.

http://www.thi.informatik.uni-frankfurt.de/~jukna/boolean/Alon-Boppana.pdf

The monotone circuit complexity of Boolean functions

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems. The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.

Approximating maximum independent sets by excluding subgraphs

Publisher Summary This chapter discusses the complexity of finite functions. A deterministic Turing machine consists of a finite control and a finite collection of tapes each with a head for reading and writing. The finite control is a finite collection of states. A tape is an infinite list of cells each containing a symbol. Initially, all tapes have blanks except for the first, which contains the input string. Once started, the machine goes from state to state, reading the symbols under the heads, writing new ones, and moving the heads. The exact action taken is governed by the current state, the symbols read, and the next-move function of the machine. This continues until a designated halt state is entered. The machine indicates its output by the halting condition of the tapes. In a nondeterministic Turing machine, the next-move function is multivalued. There can be several computations on a given input and several output values.

Ravi B. Boppana

Papers

Does co-NP have short interactive proofs?

Eigenvalues and graph bisection: An average-case analysis

The monotone circuit complexity of Boolean functions

Approximating maximum independent sets by excluding subgraphs

The Complexity of Finite Functions.