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2008

Volume 37, Issue 6, pp. vii-1952

* Special Issue on Foundations of Computer Science

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Special Issue on Foundations of Computer Science

Irit Dinur and Eva Tardos

SIAM J. Comput. 37, pp. vii-vii ( pages)

Online Publication Date: March 26, 2008

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This volume comprises the polished and fully refereed versions of a selection of papers presented at the Forty-Sixth Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), held in Pittsburgh, Pennsylvania, October 23–25, 2005. Unrefereed preliminary versions of the papers presented at the symposium appeared in the proceedings of the meeting, published by IEEE.
The FOCS 2005 Program Committee consisted of Ziv Bar-Yossef, Paul Beame, Ran Canetti, Irit Dinur, Ashish Goel, Venkatesan Guruswami, Sariel Har-Peled, Michael Kearns, Richard Lipton, Frank McSherry, Satish Rao, Omer Reingold, Eva Tardos, Mikkel Thorup, Berthold Voecking, John Watrous, Mihalis Yannakakis, and David Zuckerman. Of 276 extended abstracts submitted to the FOCS 2005 Program Committee, 62 were selected for presentation at the symposium. Eight of those 62 papers are included in this volume. This collection encompasses a wide variety of topics and methods in theoretical computer science, often shedding new light on entire areas with a fresh approach. The topics include algorithms, learning, property testing, combinatorics, cryptography, and distributed and quantum computing. All papers were refereed in accordance with SICOMP's stringent standards, and most were substantially updated in the process.
We take this opportunity to thank all the referees whose anonymous work has significantly contributed to the value of this volume. It was an honor to edit this special section in SIAM Journal on Computing.

A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

Noga Alon and Asaf Shapira

SIAM J. Comput. 37, pp. 1703-1727 (25 pages) | Cited 6 times

Online Publication Date: March 26, 2008

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The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property ${\cal P}$ has an oblivious one-sided error tester if and only if ${\cal P}$ is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable.

An Algorithmic Version of the Hypergraph Regularity Method

P. E. Haxell, B. Nagle, and V. Rödl

SIAM J. Comput. 37, pp. 1728-1776 (49 pages) | Cited 2 times

Online Publication Date: March 26, 2008

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Extending the Szemerédi regularity lemma for graphs, P. Frankl and V. Rödl [Random Structures Algorithms, 20 (2002), pp. 131–164] established a 3-graph regularity lemma triple systems ${\cal G}_n$ admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion counting lemma [B. Nagle and V. Rödl, Random Structures Algorithms, 23 (2003), pp. 264–332] allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular” environment created by the 3-graph regularity lemma. Combined applications of these lemmas are known as the 3-graph regularity method. In this paper, we provide an algorithmic version of the 3-graph regularity lemma which, as we show, is compatible with a counting lemma. We also discuss some applications.

Agnostically Learning Halfspaces

Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio

SIAM J. Comput. 37, pp. 1777-1805 (29 pages) | Cited 2 times

Online Publication Date: March 26, 2008

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We give a computationally efficient algorithm that learns (under distributional assumptions) a halfspace in the difficult agnostic framework of Kearns, Schapire, and Sellie [Mach. Learn., 17 (1994), pp. 115–141], where a learner is given access to a distribution on labelled examples but where the labelling may be arbitrary (similar to malicious noise). It constructs a hypothesis whose error rate on future examples is within an additive $\epsilon$ of the optimal halfspace, in time poly$(n)$ for any constant $\epsilon>0$, for the uniform distribution over $\{-1,1\}^n$ or unit sphere in $\mathbb R^n,$ as well as any log-concave distribution in $\mathbb R^n$. It also agnostically learns Boolean disjunctions in time $2^{\tilde{O}(\sqrt{n})}$ with respect to any distribution. Our algorithm, which performs $L_1$ polynomial regression, is a natural noise-tolerant arbitrary-distribution generalization of the well-known “low-degree” Fourier algorithm of Linial, Mansour, and Nisan. We observe that significant improvements on the running time of our algorithm would yield the fastest known algorithm for learning parity with noise, a challenging open problem in computational learning theory.

Lower Bounds for the Noisy Broadcast Problem

Navin Goyal, Guy Kindler, and Michael Saks

SIAM J. Comput. 37, pp. 1806-1841 (36 pages)

Online Publication Date: March 26, 2008

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We prove the first nontrivial (superlinear) lower bound in the noisy broadcast model, defined by El Gamal in [Open problems presented at the $1984$ workshop on Specific Problems in Communication and Computation sponsored by Bell Communication Research, in Open Problems in Communication and Computation, T. M. Cover and B. Gopinath, eds., Springer-Verlag, New York, 1987, pp. 60–62]. In this model there are $n+1$ processors $P_0,P_1,\ldots,P_n$, each of which is initially given a private input bit $x_i$. The goal is for $P_0$ to learn the value of $f(x_1,\ldots,x_n)$, for some specified function $f$, using a series of noisy broadcasts. At each step a designated processor broadcasts one bit to all of the other processors, and the bit received by each processor is flipped with fixed probability (independently for each recipient). In 1988, Gallager [IEEE Trans. Inform. Theory, 34 (1988), pp. 176–180] gave a noise-resistant protocol that allows $P_0$ to learn the entire input with constant probability in $O(n\log\log n)$ broadcasts. We prove that Gallager's protocol is optimal, up to a constant factor. Our lower bound follows by reduction from a lower bound for generalized noisy decision trees, a new model which may be of independent interest. For this new model we show a lower bound of $\Omega(n \log n)$ on the depth of a tree that learns the entire input. While the above lower bound is for an $n$-bit function, we also show an $\Omega(n\log\log n)$ lower bound for the number of broadcasts required to compute certain explicit boolean-valued functions, when the correct output must be attained with probability at least $1-n^{-\alpha}$ for a constant parameter $\alpha>0$ (this bound applies to all threshold functions as well as any other boolean-valued function with linear sensitivity). This bound also follows by reduction from a lower bound of $\Omega(n\log n)$ on the depth of generalized noisy decision trees that compute the same functions with the same error. We also show a (nontrivial) $\Omega(n)$ lower bound on the depth of generalized noisy decision trees that compute such functions with small constant error. Finally, we show the first protocol in the noisy broadcast model that computes the Hamming weight of the input using a linear number of broadcasts.

The Symmetric Group Defies Strong Fourier Sampling

Cristopher Moore, Alexander Russell, and Leonard J. Schulman

SIAM J. Comput. 37, pp. 1842-1864 (23 pages) | Cited 1 time

Online Publication Date: March 26, 2008

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The dramatic exponential speedups of quantum algorithms over their best existing classical counterparts were ushered in by the technique of Fourier sampling, introduced by Bernstein and Vazirani and developed by Simon and Shor into an approach to the hidden subgroup problem. This approach has proved successful for abelian groups, leading to efficient algorithms for factoring, extracting discrete logarithms, and other number-theoretic problems. We show, however, that this method cannot resolve the hidden subgroup problem in the symmetric groups, even in the weakest, information-theoretic sense. In particular, we show that the Graph Isomorphism problem cannot be solved by this approach. Our work implies that any quantum approach based upon the measurement of coset states must depart from the original framework by using entangled measurements on multiple coset states.

Cryptography in the Bounded-Quantum-Storage Model

Ivan B. DamgÅrd, Serge Fehr, Louis Salvail, and Christian Schaffner

SIAM J. Comput. 37, pp. 1865-1890 (26 pages) | Cited 1 time

Online Publication Date: March 26, 2008

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We initiate the study of two-party cryptographic primitives with unconditional security, assuming that the adversary's quantum memory is of bounded size. We show that oblivious transfer and bit commitment can be implemented in this model using protocols where honest parties need no quantum memory, whereas an adversarial player needs quantum memory of size at least $n/2$ in order to break the protocol, where $n$ is the number of qubits transmitted. This is in sharp contrast to the classical bounded-memory model, where we can only tolerate adversaries with memory of size quadratic in honest players' memory size. Our protocols are efficient and noninteractive and can be implemented using today's technology. On the technical side, a new entropic uncertainty relation involving min-entropy is established.

Concurrent Nonmalleable Commitments

Rafael Pass and Alon Rosen

SIAM J. Comput. 37, pp. 1891-1925 (35 pages)

Online Publication Date: March 26, 2008

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We present a nonmalleable commitment scheme that retains its security properties even when concurrently executed a polynomial number of times. That is, a man-in-the-middle adversary who is simultaneously participating in multiple concurrent commitment phases of our scheme, both as a sender and as a receiver, cannot make the values to which he commits depend on the values to which he receives commitments. Our result is achieved without assuming an a priori bound on the number of executions and without relying on any setup assumptions. Our construction relies on the existence of standard claw-free permutations and requires only a constant number of communication rounds.

A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Philip N. Klein

SIAM J. Comput. 37, pp. 1926-1952 (27 pages)

Online Publication Date: March 26, 2008

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We give an algorithm requiring $O(c^{1/\epsilon^2}n)$ time to find an $\epsilon$-optimal traveling salesman tour in the shortest-path metric defined by an undirected planar graph with nonnegative edge-lengths. For the case of all lengths equal to 1, the time required is $O(c^{1/\epsilon} n)$.
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