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2008

Volume 38, Issue 2, pp. vii-752

* Special Issue Dedicated to the Thirty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2005)

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Special Issue Dedicated to the Thirty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2005)

Ronald Fagin, Anupam Gupta, Ravi Kumar, and Ryan O'Donnell

SIAM J. Comput. 38, pp. vii-vii ( pages)

Online Publication Date: May 23, 2008

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This volume comprises the polished and fully refereed versions of a selection of papers presented at the Thirty-Seventh Annual ACM Symposium on Theory of Computing (STOC 2005), held in Baltimore, Maryland, May 22–24, 2005. Unrefereed preliminary versions of the papers presented at the symposium appeared in the proceedings of the meeting, published by ACM. The symposium was sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT).
The STOC 2005 Program Committee consisted of Gerth Stølting Brodal, Harry Buhrman, Jin-Yi Cai, Cynthia Dwork, Ronald Fagin (chair), Martin Farach-Colton, Anupam Gupta, Sariel Har-Peled, Russell Impagliazzo, Kamal Jain, Adam Tauman Kalai, David Karger, Claire Kenyon, Subhash Khot, Ravi Kumar, Moni Naor, Ryan O'Donnell, Toniann Pitassi, Tim Roughgarden, Alistair Sinclair, and Amnon Ta-Shma.
Out of 290 “Extended Abstracts” submitted to the STOC 2005 Program Committee, 84 were selected for presentation at the symposium. The present volume includes 9 of these papers that were invited to this volume. All papers were refereed in accordance with the SIAM Journal on Computing's stringent standards, and these papers 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.
Ronald Fagin, the Program Chair of the 2005 STOC Conference, invited three other members of the Program Committee (Anupam Gupta, Ravi Kumar, and Ryan O'Donnell) to assist in editing this special issue of the of the SIAM Journal on Computing, and all agreed. We feel that it was an honor to edit this issue.

An $O(\logn \log\logn)$ Space Algorithm for Undirected st-Connectivity

Vladimir Trifonov

SIAM J. Comput. 38, pp. 449-483 (35 pages)

Online Publication Date: May 23, 2008

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We present a deterministic $O(\log n \log \log n)$ space algorithm for undirected st-connectivity. It is based on a space-efficient simulation of the deterministic EREW algorithm of Chong and Lam [J. Algorithms, 18 (1995), pp. 378–402], an approach suggested by Prof. Vijaya Ramachandran, and uses the universal exploration sequences for trees constructed by Koucký in [Proceedings of the 16th Annual IEEE Conference on Computational Complexity, 2001, pp. 21–27]. Our result improves the $O(\log^{4/3} n)$ bound of Armoni et al. in [Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1997, pp. 230–239] and is a big step towards the optimal $O(\log n)$. Independently of our result and using a different set of techniques, the optimal bound was achieved by Reingold in [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 2005, pp. 376–385].

The Mixing Time of the Thorp Shuffle

Ben Morris

SIAM J. Comput. 38, pp. 484-504 (21 pages)

Online Publication Date: May 23, 2008

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The Thorp shuffle is defined as follows. Cut a deck of cards into two equal piles. Drop the first card from the left pile or the right pile according to the outcome of a fair coin flip, then drop from the other pile. Continue this way until both piles are empty. We show that the mixing time for the Thorp shuffle with $2^d$ cards is polynomial in $d$.

Every Monotone Graph Property Is Testable

Noga Alon and Asaf Shapira

SIAM J. Comput. 38, pp. 505-522 (18 pages) | Cited 3 times

Online Publication Date: May 23, 2008

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A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on $\epsilon$. This result unifies several previous results in the area of property testing and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi's regularity lemma. The main ideas behind this application may be useful in characterizing all testable graph properties and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph theory, is that for any monotone graph property ${\cal P}$, any graph that is $\epsilon$-far from satisfying ${\cal P}$ contains a subgraph of size depending on $\epsilon$ only, which does not satisfy ${\cal P}$. Finally, we prove the following compactness statement: If a graph $G$ is $\epsilon$-far from satisfying a (possibly infinite) set of monotone graph properties ${\cal P}$, then it is at least $\delta_{{\cal P}}(\epsilon)$-far from satisfying one of the properties.

The Round Complexity of Two-Party Random Selection

Saurabh Sanghvi and Salil Vadhan

SIAM J. Comput. 38, pp. 523-550 (28 pages)

Online Publication Date: May 23, 2008

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We study the round complexity of two-party protocols for generating a random $n$-bit string such that the output is guaranteed to have bounded “bias,” even if one of the two parties deviates from the protocol (possibly using unlimited computational resources). Specifically, we require that the output's statistical difference from the uniform distribution on $\{0,1\}^n$ is bounded by a constant less than 1. We present a protocol for the above problem that has $2 \log^* n + O(1)$ rounds, improving a previous $2n$-round protocol of Goldreich, Goldwasser, and Linial (FOCS '91). Like the GGL Protocol, our protocol actually provides a stronger guarantee, ensuring that the output lands in any set $T \subseteq \{0,1\}^n$ of density $\mu$ with probability at most $O(\sqrt{\mu + \delta})$, where $\delta$ may be an arbitrarily small constant. We then prove a nearly matching lower bound, showing that any protocol guaranteeing bounded statistical difference requires at least $\log^* n - \log^* \log^* n - O(1)$ rounds. We also prove several results for the case when the output's bias is measured by the maximum multiplicative factor by which a party can increase the probability of a set $T \subseteq \{0,1\}^n$.

Short PCPs with Polylog Query Complexity

Eli Ben-Sasson and Madhu Sudan

SIAM J. Comput. 38, pp. 551-607 (57 pages) | Cited 2 times

Online Publication Date: May 23, 2008

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We give constructions of probabilistically checkable proofs (PCPs) of length $n \cdot polylog n$ proving satisfiability of circuits of size $n$ that can be verified by querying $polylog n$ bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping $n$ information bits to $n\cdot polylog n$ bit long codewords that are testable with $polylog n$ queries. Our constructions rely on new techniques revolving around properties of codes based on relatively high-degree polynomials in one variable, i.e., Reed–Solomon codes. In contrast, previous constructions of short PCPs, beginning with [L. Babai, L. Fortnow, L. Levin, and M. Szegedy, Checking computations in polylogarithmic time, in Proceedings of the 23rd ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 21–31] and until the recent [E. Ben-Sasson, O. Goldreich, P. Harsha, M. Sudan, and S. Vadhan, Robust PCPs of proximity, shorter PCPs, and applications to coding, in Proceedings of the 36th ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 13–15], relied extensively on properties of low-degree polynomials in many variables. We show how to convert the problem of verifying the satisfaction of a circuit by a given assignment to the task of verifying that a given function is close to being a Reed–Solomon codeword, i.e., a univariate polynomial of specified degree. This reduction also gives an alternative to using the “sumcheck protocol” [C. Lund, L. Fortnow, H. Karloff, and N. Nisan, J. ACM, 39 (1992), pp. 859–868]. We then give a new PCP for the special task of proving that a function is close to being a Reed–Solomon codeword. The resulting PCPs are not only shorter than previous ones but also arguably simpler. In fact, our constructions are also more natural in that they yield locally testable codes first, which are then converted to PCPs. In contrast, most recent constructions go in the opposite direction of getting locally testable codes from PCPs.

Lower-Stretch Spanning Trees

Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng

SIAM J. Comput. 38, pp. 608-628 (21 pages) | Cited 1 time

Online Publication Date: May 23, 2008

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We show that every weighted connected graph $G$ contains as a subgraph a spanning tree into which the edges of $G$ can be embedded with average stretch $O (\log^{2} n \log \log n)$. Moreover, we show that this tree can be constructed in time $O (m \log n + n \log^2 n)$ in general, and in time $O (m \log n)$ if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from $ m 2^{(O (\sqrt{\log n\log\log n})) }$ to $m \log^{O (1)}n$, and to $O ( n \log^{2} n \log \log n)$ when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

Improved Approximation Algorithms for Minimum Weight Vertex Separators

Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee

SIAM J. Comput. 38, pp. 629-657 (29 pages) | Cited 1 time

Online Publication Date: May 23, 2008

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We develop the algorithmic theory of vertex separators and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into $L_1$ (and even Euclidean embeddings) are insufficient but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an $O(\sqrt{\log n})$ approximation for minimum ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be $\Theta(\sqrt{\log n})$. We also prove an optimal $O(\log k)$-approximate max-flow/min-vertex-cut theorem for arbitrary vertex-capacitated multicommodity flow instances on $k$ terminals. For uniform instances on any excluded-minor family of graphs, we improve this to $O(1)$, and this yields a constant-factor approximation for minimum ratio vertex cuts in such graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best known ratio was $O(\log n)$. These results have a number of applications. We exhibit an $O(\sqrt{\log n})$ pseudoapproximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of $O(\sqrt{\log {opt}})$, where ${opt}$ is the size of an optimal separator, improving over the previous best bound of $O(\log {opt})$. Likewise, we obtain improved approximation ratios for treewidth: in any graph of treewidth $k$, we show how to find a tree decomposition of width at most $O(k \sqrt{\log k})$, whereas previous algorithms yielded $O(k \log k)$. For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constant-factor approximation for the treewidth. This in turn can be used to obtain polynomial-time approximation schemes for several problems in such graphs.

Tree-Walking Automata Do Not Recognize All Regular Languages

MikoŁaj Bojańczyk and Thomas Colcombet

SIAM J. Comput. 38, pp. 658-701 (44 pages)

Online Publication Date: May 23, 2008

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Tree-walking automata are a natural sequential model for recognizing tree languages. It is well known that every tree language recognized by a tree-walking automaton is regular. We show that the converse does not hold.

New and Improved Constructions of Nonmalleable Cryptographic Protocols

Rafael Pass and Alon Rosen

SIAM J. Comput. 38, pp. 702-752 (51 pages)

Online Publication Date: May 23, 2008

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We present a new constant-round protocol for nonmalleable zero-knowledge. Using this protocol as a subroutine, we obtain a new constant-round protocol for nonmalleable commitments. Our constructions rely on the existence of (standard) collision-resistant hash functions. Previous constructions either relied on the existence of trapdoor permutations and hash functions that are collision resistant against subexponential-sized circuits or required a superconstant number of rounds. Additional results are the first construction of a nonmalleable commitment scheme that is statistically hiding (with respect to opening) and the first nonmalleable commitments that satisfy a strict polynomial-time simulation requirement. Our approach differs from the approaches taken in previous works in that we view nonmalleable zero-knowledge as a building block rather than an end goal. This gives rise to a modular construction of nonmalleable commitments and results in a somewhat simpler analysis.
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