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2008

Volume 38, Issue 5, pp. 1661-2112


Universal Arguments and their Applications

Boaz Barak and Oded Goldreich

SIAM J. Comput. 38, pp. 1661-1694 (34 pages)

Online Publication Date: December 19, 2008

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We put forward a new type of computationally sound proof system called universal arguments. Universal arguments are related but different from both CS proofs (as defined by Micali [SIAM J. Comput., 37 (2000), pp. 1253–1298]) and arguments (as defined by Brassard, Chaum, and Crépeau [J. Comput. System Sci., 37 (1988), pp. 156–189]. In particular, we adopt the instance-based prover-efficiency paradigm of CS proofs but follow the computational-soundness condition of argument systems (i.e., we consider only cheating strategies that are implementable by polynomial-size circuits). We show that universal arguments can be constructed based on standard intractability assumptions that refer to polynomial-size circuits (rather than based on assumptions that refer to subexponential-size circuits as used in the construction of CS proofs). Furthermore, these protocols have a constant number of rounds and are of the public-coin type. As an application of these universal arguments, we weaken the intractability assumptions used in the non–black-box zero-knowledge arguments of Barak [in Proceedings of the 42nd IEEE Symposiun on Foundations of Computer Science, 2001]. Specifically, we only utilize intractability assumptions that refer to polynomial-size circuits (rather than assumptions that refer to circuits of some “nice” superpolynomial size).

Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography

Dmitry Gavinsky, Julia Kempe, Iordanis Kerenidis, Ran Raz, and Ronald de Wolf

SIAM J. Comput. 38, pp. 1695-1708 (14 pages)

Online Publication Date: December 19, 2008

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We give an exponential separation between one-way quantum and classical communication protocols for a partial Boolean function (a variant of the Boolean hidden matching problem of Bar-Yossef et al.). Previously, such an exponential separation was known only for a relational problem. The communication problem corresponds to a strong extractor that fails against a small amount of quantum information about its random source. Our proof uses the Fourier coefficients inequality of Kahn, Kalai, and Linial. We also give a number of applications of this separation. In particular, we show that there are privacy amplification schemes that are secure against classical adversaries but not against quantum adversaries; and we give the first example of a key-expansion scheme in the model of bounded-storage cryptography that is secure against classical memory-bounded adversaries but not against quantum ones.

Graph Distances in the Data-Stream Model

Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang

SIAM J. Comput. 38, pp. 1709-1727 (19 pages) | Cited 1 time

Online Publication Date: December 19, 2008

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We explore problems related to computing graph distances in the data-stream model. The goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory. We are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the data-stream model. In particular, we are interested in the trade-offs between model parameters such as per-data-item processing time, total space, and the number of passes that may be taken over the stream. These trade-offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature. Our results include the following: (1) Spanner construction: There exists a single-pass, $\tilde{O}(tn^{1+1/t})$-space, $\tilde{O}(t^2n^{1/t})$-time-per-edge algorithm that constructs a $(2t+1)$-spanner. For $t=\Omega(\log n/{\log\log n})$, the algorithm satisfies the semistreaming space restriction of $O(n\operatorname{polylog}n)$ and has per-edge processing time $O(\operatorname{polylog}n)$. This resolves an open question from [J. Feigenbaum et al., Theoret. Comput. Sci., 348 (2005), pp. 207–216]. (2) Breadth-first-search (BFS) trees: For any even constant $k$, we show that any algorithm that computes the first $k$ layers of a BFS tree from a prescribed node with probability at least $2/3$ requires either greater than $k/2$ passes or $\tilde{\Omega}(n^{1+1/k})$ space. Since constructing BFS trees is an important subroutine in many traditional graph algorithms, this demonstrates the need for new algorithmic techniques when processing graphs in the data-stream model. (3) Graph-distance lower bounds: Any $t$-approximation of the distance between two nodes requires $\Omega(n^{1+1/t})$ space. We also prove lower bounds for determining the length of the shortest cycle and other graph properties. (4) Techniques for decreasing per-edge processing: We discuss two general techniques for speeding up the per-edge computation time of streaming algorithms while increasing the space by only a small factor.

Private Approximation of Search Problems

Amos Beimel, Paz Carmi, Kobbi Nissim, and Enav Weinreb

SIAM J. Comput. 38, pp. 1728-1760 (33 pages)

Online Publication Date: December 19, 2008

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Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

Approximating Minimum Max-Stretch Spanning Trees on Unweighted Graphs

Yuval Emek and David Peleg

SIAM J. Comput. 38, pp. 1761-1781 (21 pages)

Online Publication Date: December 19, 2008

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Given a graph \(G\) and a spanning tree \(T\) of \(G\), we say that \(T\) is a tree \(t\)-spanner of \(G\) if the distance between every pair of vertices in \(T\) is at most \(t\) times their distance in \(G\). The problem of finding a tree \(t\)-spanner minimizing \(t\) is referred to as the Minimum Max-Stretch spanning Tree (MMST) problem. This paper concerns the MMST problem on unweighted graphs. The problem is known to be NP-hard, and the paper presents an \(O(\log n)\)-approximation algorithm for it. Furthermore, it is established that unless \(\mathrm{P}=\mathrm{NP}\), the problem cannot be approximated additively by any \(o(n)\) term.

The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell)

Libor Barto, Marcin Kozik, and Todd Niven

SIAM J. Comput. 38, pp. 1782-1802 (21 pages) | Cited 3 times

Online Publication Date: January 09, 2009

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Bang-Jensen and Hell conjectured in 1990 (using the language of graph homomorphisms) a constraint satisfaction problem (CSP) dichotomy for digraphs with no sources or sinks. The conjecture states that the CSP for such a digraph is tractable if each component of its core is a cycle and is $NP$-complete otherwise. In this paper we prove this conjecture and, as a consequence, a conjecture of Bang-Jensen, Hell, and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons, and Krokhin in 2005.

Spanners of Complete $k$-Partite Geometric Graphs

Prosenjit Bose, Paz Carmi, Mathieu Couture, Anil Maheshwari, Pat Morin, and Michiel Smid

SIAM J. Comput. 38, pp. 1803-1820 (18 pages)

Online Publication Date: January 09, 2009

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We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a “small” stretch factor and “few” edges. We present two algorithms for this problem. The first algorithm computes a $(5+\epsilon)$-spanner of $K$ with $O(n)$ edges in $O(n\log n)$ time. The second algorithm computes a $(3+\epsilon)$-spanner of $K$ with $O(n\log n)$ edges in $O(n \log n)$ time. The latter result is optimal: We show that for any $2\leq k\leq n-\Theta(\sqrt{n\log n})$, spanners with $O(n\log n)$ edges and stretch factor less than 3 do not exist for all complete $k$-partite geometric graphs.

Profiles of Tries

Gahyun Park, Hsien-Kuei Hwang, Pierre Nicodème, and Wojciech Szpankowski

SIAM J. Comput. 38, pp. 1821-1880 (60 pages)

Online Publication Date: January 09, 2009

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Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to the (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fill-up level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. The analysis of profiles is surprisingly arduous, but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles varies. Near the root, the external profiles tend to zero at an exponential rate, and then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically and then polynomially; they then tend polynomially to zero again. Furthermore, the expected profiles of asymmetric tries are oscillating in a range where profiles grow polynomially, while symmetric tries are nonoscillating, in contrast to most shape parameters of random tries studied previously. Such a periodic behavior for asymmetric tries implies that the depth satisfies a central limit theorem but not a local limit theorem of the usual form. Also the widest levels in symmetric tries contain a linear number of nodes, differing from the order $n/\sqrt{\log n}$ for asymmetric tries, $n$ being the size of the trees. Finally, it is observed that profiles satisfy central limit theorems when the variance goes unbounded, while near the height they are distributed according to Poisson laws. As a consequence of these results we find typical behaviors of the height, shortest path, fill-up level, and depth. These results are derived here by methods of analytic algorithmics such as generating functions, Mellin transform, Poissonization and de-Poissonization, the saddle-point method, singularity analysis, and uniform asymptotic analysis.

On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs

Umberto Straccia, Manuel Ojeda-Aciego, and Carlos V. Damásio

SIAM J. Comput. 38, pp. 1881-1911 (31 pages)

Online Publication Date: January 09, 2009

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Unlike monotone single-valued functions, multivalued mappings may have zero, one, or (possibly infinitely) many minimal fixed-points. The contribution of this work is twofold. First, we overview and investigate the existence and computation of minimal fixed-points of multivalued mappings, whose domain is a complete lattice and whose range is its power set. Second, we show how these results are applied to a general form of logic programs, where the truth space is a complete lattice. We show that a multivalued operator can be defined whose fixed-points are in one-to-one correspondence with the models of the logic program.

Impossibility Results and Lower Bounds for Consensus under Link Failures

Ulrich Schmid, Bettina Weiss, and Idit Keidar

SIAM J. Comput. 38, pp. 1912-1951 (40 pages)

Online Publication Date: January 09, 2009

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We provide a suite of impossibility results and lower bounds for the required number of processes and rounds for synchronous consensus under transient link failures. Our results show that consensus can be solved even in the presence of $O(n^2)$ moving omission and/or arbitrary link failures per round, provided that both the number of affected outgoing and incoming links of every process is bounded. Providing a step further toward the weakest conditions under which consensus is solvable, our findings are applicable to a variety of dynamic phenomena such as transient communication failures and end-to-end delay variations. We also prove that our model surpasses alternative link failure modeling approaches in terms of assumption coverage.

Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

Kiran S. Kedlaya and Sergey Yekhanin

SIAM J. Comput. 38, pp. 1952-1969 (18 pages) | Cited 1 time

Online Publication Date: January 14, 2009

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A $k$-query locally decodable code (LDC) encodes an $n$-bit message $x$ as an $N$-bit codeword $C(x)$, such that one can probabilistically recover any bit $x_i$ of the message by querying only $k$ bits of the codeword $C(x)$, even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently vast improvements in upper bounds for the length of LDCs were achieved via constructions that rely on existence of certain special (“nice”) subsets of finite fields. In this work we extend the constructions of LDCs from “nice” subsets. We argue that further progress on upper bounds for LDCs via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number $m=2^t-1$ that has a prime factor $p>m^\gamma$ yields a family of $k(\gamma)$-query LDCs of length $\exp(n^{1/t})$. Conversely, if for some fixed $k$ and all $\epsilon>0$ one can use the “nice” subsets technique to obtain a family of $k$-query LDCs of length $\exp(n^\epsilon)$, then infinitely many Mersenne numbers have prime factors larger than currently known.

The Complexity of Weighted Boolean CSP

Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum

SIAM J. Comput. 38, pp. 1970-1986 (17 pages) | Cited 1 time

Online Publication Date: January 14, 2009

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This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set $\mathcal{F}$ of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is $\text{{\sf FP}}^{\text{{\sf\#P}}}$-complete unless either (1) every function in $\mathcal{F}$ is of “product type,” or (2) every function in $\mathcal{F}$ is “pure affine.” In the remaining cases, computing the partition function is in P.

On the Complexity of Numerical Analysis

Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen

SIAM J. Comput. 38, pp. 1987-2006 (20 pages) | Cited 2 times

Online Publication Date: January 14, 2009

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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum–Shub–Smale model of computation over the reals; and (b) a problem we call the “generic task of numerical computation,” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer $N$, decide whether $N>0$. In the Blum–Shub–Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean traveling salesman problem lies in the counting hierarchy—the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if $n!$ is not ultimately easy, then ACIT has subexponential complexity.

Interval Completion Is Fixed Parameter Tractable

Yngve Villanger, Pinar Heggernes, Christophe Paul, and Jan Arne Telle

SIAM J. Comput. 38, pp. 2007-2020 (14 pages)

Online Publication Date: January 30, 2009

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We present an algorithm with runtime $O(k^{2k}n^3m)$ for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph $G$ on $n$ vertices and $m$ edges, can we obtain an interval graph by adding at most $k$ new edges to $G$? This resolves the long-standing open question [H. Kaplan, R. Shamir, and R. E. Tarjan, SIAM J. Comput., 28 (1999), pp. 1906–1922; R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999; M. Serna and D. Thilikos, Bull. Eur. Assoc. Theory Comput. Sci. EATCS, 86 (2005), pp. 41–65; G. Gutin, S. Szeider, and A. Yeo, in Proceedings IWPEC 2006, Lecture Notes in Comput. Sci. 4169, Springer-Verlag, Berlin, 2006, pp. 60–71], first posed by Kaplan, Shamir, and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in profile minimization for sparse matrix computations [J. A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981; R. E. Tarjan, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 3–22], and our results show tractability for the case of a small number $k$ of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.

Optimizing Schema Languages for XML: Numerical Constraints and Interleaving

Wouter Gelade, Wim Martens, and Frank Neven

SIAM J. Comput. 38, pp. 2021-2043 (23 pages) | Cited 2 times

Online Publication Date: January 30, 2009

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The presence of a schema offers many advantages in processing, translating, querying, and storage of XML data. Basic decision problems such as equivalence, inclusion, and nonemptiness of intersection of schemas form the basic building blocks for schema optimization and integration, and algorithms for static analysis of transformations. It is thereby paramount to establish the exact complexity of these problems. Most common schema languages for XML can be adequately modeled by some kind of grammar with regular expressions at right-hand sides. In this paper, we observe that, apart from the usual regular operators of union, concatenation, and Kleene-star, schema languages also allow numerical occurrence constraints and interleaving operators. Although the expressiveness of these operators remains within the regular languages, the presence or absence of these operators has a significant impact on the complexity of the basic decision problems. We present a complete overview of the complexity of the basic decision problems for DTDs, XSDs, and Relax NG with regular expressions incorporating numerical occurrence constraints and interleaving. We also discuss chain regular expressions and the complexity of the schema simplification problem incorporating the new operators.

Stream Order and Order Statistics: Quantile Estimation in Random-Order Streams

Sudipto Guha and Andrew McGregor

SIAM J. Comput. 38, pp. 2044-2059 (16 pages)

Online Publication Date: January 30, 2009

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When trying to process a data stream in small space, how important is the order in which the data arrive? Are there problems that are unsolvable when the ordering is worst case, but that can be solved (with high probability) when the order is chosen uniformly at random? If we consider the stream as if ordered by an adversary, what happens if we restrict the power of the adversary? We study these questions in the context of quantile estimation, one of the most well studied problems in the data-stream model. Our results include an $O($polylog $n)$-space, $O(\log\log n)$-pass algorithm for exact selection in a randomly ordered stream of $n$ elements. This resolves an open question of Munro and Paterson [Theoret. Comput. Sci., 23 (1980), pp. 315–323]. We then demonstrate an exponential separation between the random-order and adversarial-order models: using $O($polylog $n)$ space, exact selection requires $\Omega(\log n/\log\log n)$ passes in the adversarial-order model. This lower bound, in contrast to previous results, applies to fully general randomized algorithms and is established via a new bound on the communication complexity of a natural pointer-chasing style problem. We also prove the first fully general lower bounds in the random-order model: finding an element with rank $n/2\pm n^{\delta}$ in the single-pass random-order model with probability at least $9/10$ requires $\Omega(\sqrt{n^{1-3\delta}/\log n})$ space.

Sampling Algorithms and Coresets for $\ell_p$ Regression

Anirban Dasgupta, Petros Drineas, Boulos Harb, Ravi Kumar, and Michael W. Mahoney

SIAM J. Comput. 38, pp. 2060-2078 (19 pages) | Cited 2 times

Online Publication Date: February 06, 2009

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The $\ell_p$ regression problem takes as input a matrix $A\in\mathbb{R}^{n\times d}$, a vector $b\in\mathbb{R}^n$, and a number $p\in[1,\infty)$, and it returns as output a number ${\cal Z}$ and a vector $x_{\text{{\sc opt}}}\in\mathbb{R}^d$ such that ${\cal Z}=\min_{x\in\mathbb{R}^d}\|Ax-b\|_p=\|Ax_{\text{{\sc opt}}}-b\|_p$. In this paper, we construct coresets and obtain an efficient two-stage sampling-based approximation algorithm for the very overconstrained ($n\gg d$) version of this classical problem, for all $p\in[1, \infty)$. The first stage of our algorithm nonuniformly samples $\hat{r}_1=O(36^p d^{\max\{p/2+1,p\}+1})$ rows of $A$ and the corresponding elements of $b$, and then it solves the $\ell_p$ regression problem on the sample; we prove this is an 8-approximation. The second stage of our algorithm uses the output of the first stage to resample $\hat{r}_1/\epsilon^2$ constraints, and then it solves the $\ell_p$ regression problem on the new sample; we prove this is a $(1+\epsilon)$-approximation. Our algorithm unifies, improves upon, and extends the existing algorithms for special cases of $\ell_p$ regression, namely, $p = 1,2$ [K. L. Clarkson, in Proceedings of the 16th Annual ACM–SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2005, pp. 257–266; P. Drineas, M. W. Mahoney, and S. Muthukrishnan, in Proceedings of the 17th Annual ACM–SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2006, pp. 1127–1136]. In the course of proving our result, we develop two concepts—well-conditioned bases and subspace-preserving sampling—that are of independent interest.

Hierarchical Unambiguity

Holger Spakowski and Rahul Tripathi

SIAM J. Comput. 38, pp. 2079-2112 (34 pages)

Online Publication Date: February 06, 2009

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We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to generalize known constructs involving relativized unambiguity based complexity classes (UP and Promise-UP) to new constructs involving arbitrary higher levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical arrangement of unambiguous nondeterministic polynomial-time Turing machines, and so they differ substantially, in applicability and in nature, from standard methods (such as the switching lemma [J. Håstad, Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, 1987]), which play roles in carrying out similar generalizations. Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč in [Complexity Theory, Cambridge University Press, Cambridge, UK, 1993, pp. 101–146], on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP.
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