Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1

We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spot-checks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the P-NP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...

We consider the problem of establishing jam-resistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.

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. Finding a natural meeting ground between the highly developed complexity theory of computer science ---with its historical roots in logic and the discrete mathematics of the integers--- and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ---with its rich history and longstanding traditions in the continuous mathematics of analysis--- presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the Letts-Villard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...

This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both the "approximation method" of Razborov [13] and the "topological approach" of Sipser [15] for proving circuit lower bounds. In Karchmer's setting the lower bound prover shows that a given circuit C is too small for computing a given function f by contradiction, in the following way. She tries to combine (or 'fuse', as we propose calling it) correct accepting computations of inputs in f \Gamma1 (1) by C into an incorrect accepting computation of an input in f \Gamma1 (0). It turns out that this "Fusion Method" reduces the dynamic computation of f by C into a static combinatorial cover problem, which provides the lower bound. Moreover, different restrictions on how we can fuse computations ...

Distribution A, Approved for public release, distribution unlimited Abstract. We consider the problem of establishing jam-resistant, wireless, omnidirectional communication channels when there is no initial shared secret. No existing system achieves this. We propose a general algorithm for this problem, the BBC algorithm, and give several instantiations of it. We develop and analyze this algorithm within the framework of a new type of code, concurrent codes, which are those superimposed codes that allow efficient decoding. Finally, we propose the Universal Concurrent Code algorithm, and prove that it covers all possible concurrent codes, and give connections between its theory and that of monotone Boolean functions.

Let SYM + denote the class of Boolean functions computable by depth-two size-n log O(1) n circuits with a symmetric-function gate at the root and AND gates of fan-in log O(1) n at the next level, or equivalently, the class of Boolean functions f such that f(x1 ; : : : ; xn) can be expressed as f(x1 ; : : : ; xn) = hn (pn(x1 ; : : : ; xn)) for some polynomial pn over Z of degree log O(1) n and norm (the sum of the absolute values of its coefficients) n log O(1) n and some function hn : Z ! f0; 1g. Building on work of Yao [Yao90], Beigel and Tarui [BT91] showed that ACC ` SYM + , where ACC is the class of Boolean functions computable by constant-depth polynomial-size circuits with NOT, AND, OR, and MODm gates for some fixed m. In this paper, we consider augmenting the power of ACC circuits by allowing randomness and allowing an exact-threshold gate as the output gate (an exact-threshold gate outputs 1 if exactly k of its inputs are 1, where k is a parameter; it outputs 0...

I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld