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2004

Volume 33, Issue 4, pp. 761-1009


Compaction, Retraction, and Constraint Satisfaction

Narayan Vikas

SIAM J. Comput. 33, pp. 761-782 (22 pages) | Cited 1 time

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In this paper, we show a very close relationship among the compaction, retraction, and constraint satisfaction problems in the context of reflexive and bipartite graphs. The compaction and retraction problems are special graph coloring problems, and the constraint satisfaction problem is well known to have an important role in artificial intelligence. The relationships we present provide evidence that, similar to the retraction problem, it is likely to be difficult to determine whether for every fixed reflexive or bipartite graph, the compaction problem is polynomial time solvable or NP-complete. In particular, the relationships that we present relate to a long-standing open problem concerning the equivalence of the compaction and retraction problems.

Strict Polynomial-Time in Simulation and Extraction

Boaz Barak and Yehuda Lindell

SIAM J. Comput. 33, pp. 783-818 (36 pages) | Cited 1 time

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The notion of efficient computation is usually identified in cryptography and complexity with (strict) probabilistic polynomial-time. However, until recently, in order to obtain \emph{constant-round} zero-knowledge proofs and proofs of knowledge, one had to allow simulators and knowledge extractors to run in time that is only polynomial on the average (i.e., expected polynomial-time). Recently Barak gave the first constant-round zero-knowledge argument with a strict (in contrast to expected) polynomial-time simulator. The simulator in his protocol is a nonblack-box simulator (i.e., it makes inherent use of the description of the code of the verifier).
In this paper, we further address the question of strict polynomial-time in constant-round zero-knowledge proofs and arguments of knowledge. First, we show that there exists a constant-round zero-knowledge argument of knowledge with a strict polynomial-time knowledge extractor. As in the simulator of Barak's zero-knowledge protocol, the extractor for our argument of knowledge is not black-box and makes inherent use of the code of the prover. On the negative side, we show that nonblack-box techniques are essential for both strict polynomial-time simulation and extraction. That is, we show that no (nontrivial) constant-round zero-knowledge proof or argument can have a strict polynomial-time black-box simulator. Similarly, we show that no (nontrivial) constant-round zero-knowledge proof or argument of knowledge can have a strict polynomial-time black-box knowledge extractor.

The Complexity of Three-Way Statistical Tables

Jesus De Loera and Shmuel Onn

SIAM J. Comput. 33, pp. 819-836 (18 pages) | Cited 4 times

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Multiway tables with specified marginals arise in a variety of applications in statistics and operations research. We provide a comprehensive complexity classification of three fundamental computational problems on tables: existence, counting, and entry-security.
One outcome of our work is that each of the following problems is intractable already for "slim" 3-tables, with constant number 3 of rows: (1) deciding existence of 3-tables with specified 2-marginals; (2) counting all 3-tables with specified 2-marginals; (3) deciding whether a specified value is attained in a specified entry by at least one of the 3-tables having the same 2-marginals as a given table. This implies that a characterization of feasible marginals for such slim tables, sought by much recent research, is unlikely to exist.
Another consequence of our study is a systematic efficient way of embedding the set of 3-tables satisfying any given 1-marginals and entry upper bounds in a set of slim 3-tables satisfying suitable 2-marginals with no entry bounds. This provides a valuable tool for studying multi-index transportation problems and multi-index transportation polytopes. Remarkably, it enables us to automatically recover a famous example due to Vlach of a "real-feasible integer-infeasible" collection of 2-marginals for 3-tables of smallest possible size (3,4,6).

On Multidimensional Packing Problems

Chandra Chekuri and Sanjeev Khanna

SIAM J. Comput. 33, pp. 837-851 (15 pages) | Cited 3 times

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We study the approximability of multidimensional generalizations of three classical packing problems: multiprocessor scheduling, bin packing, and the knapsack problem. Specifically, we study the vector scheduling problem, its dual problem, namely, the vector bin packing problem, and a class of packing integer programs. The vector scheduling problem is to schedule nd-dimensional tasks on m machines such that the maximum load over all dimensions and all machines is minimized. The vector bin packing problem, on the other hand, seeks to minimize the number of bins needed to schedule all n tasks such that the maximum load on any dimension across all bins is bounded by a fixed quantity, say, 1. Such problems naturally arise when scheduling tasks that have multiple resource requirements. Finally, packing integer programs capture a core problem that directly relates to both vector scheduling and vector bin packing, namely, the problem of packing a maximum number of vectors in a single bin of unit height. We obtain a variety of new algorithmic as well as inapproximability results for these three problems.

Counting Complexity of Solvable Black-Box Group Problems

N. V. Vinodchandran

SIAM J. Comput. 33, pp. 852-869 (18 pages)

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We place many computational problems over solvable black-box groups in the counting complexity classes SPP or LWPP\@. The classes SPP and LWPP are considered classes of low counting complexity. In particular, SPP is low (powerless when used as oracles) for all gap-definable counting classes (PP\@, C$_=$P\@, Mod$_k$P\@, etc.) and LWPP is low for PP and C$_=$P\@. The results improve the upper bounds for these problems proved in [Arvind and Vinodchandran, Theoret. Comput. Sci., 180 (1997), pp. 17--45], where the authors place these problems in randomized versions of SPP and LWPP. Because of the randomization, upper bounds in that paper implied lowness only for the class PP. The results in this paper favor the belief that these problems are unlikely to be complete for NP.

Time of Deterministic Broadcasting in Radio Networks with Local Knowledge

Dariusz R. Kowalski and Andrzej Pelc

SIAM J. Comput. 33, pp. 870-891 (22 pages) | Cited 8 times

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We consider broadcasting in radio networks, modeled as undirected graphs, whose nodes know only their own label and labels of their neighbors. In every step every node acts either as a transmitter or as a receiver. A node acting as a transmitter sends a message which can potentially reach all of its neighbors. A node acting as a receiver in a given step gets a message if and only if exactly one of its neighbors transmits in this step.
Bar-Yehuda, Goldreich, and Itai [J. Comput. System Sci., 45 (1992), pp. 104--126] considered broadcasting in this model. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks of diameter 3. This claim turns out to be incorrect in this model (although it is valid in a more pessimistic model [R. Bar-Yehuda, O. Goldreich, and A. Itai, Errata Regarding "On the time complexity of broadcast in radio networks: An exponential gap between determinism and randomization," http://www.wisdom.weizmann.ac.il/mathusers/oded/p\_bgi.html, 2002]). We construct an algorithm that broadcasts in logarithmic time on all graphs from the Bar-Yehuda, Goldreich, and Itai paper (BGI). Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter $o(\log \log n)$. On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time $\Omega(\sqrt[4]{n})$ on these graphs. In view of the randomized algorithm from BGI, running in expected time ${\cal O}(D \log n + \log ^2 n)$ on all $n$-node graphs of diameter D (cf. also a recent ${\cal O}(D \log (n/D) + \log ^2 n)$-time algorithm from [D. Kowalski and A. Pelc, Proceedings of the 22nd Annual ACM Symposium on Principles of Distributed Computing, Boston, 2003, pp. 73--82; A. Czumaj and W. Rytter, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, Cambridge, MA, 2003, pp. 492--501]), our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting, under the considered model of radio communication.

The Parameterized Complexity of Counting Problems

Jörg Flum and Martin Grohe

SIAM J. Comput. 33, pp. 892-922 (31 pages) | Cited 13 times

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We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W$[t]$, for $t\ge 1$, that corresponds to Downey and Fellows's W-hierarchy [R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999] and we show that a few central W-completeness results for decision problems translate to \#W-completeness results for the corresponding counting problems.
Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W$[1]-complete. This makes it highly unlikely that these problems are fixed-parameter tractable, even though their decision versions are fixed-parameter tractable. More explicitly, our result shows that most likely there is no $f(k) \cdot n^c$-algorithm for counting cycles or paths of length k in a graph of size n for any computable function $f: \mathbb{N} \to \mathbb{N}$ and constant c, even though there is a $2^{O(k)} \cdot n^{2.376}$ algorithm for finding a cycle or path of length k [N. Alon, R. Yuster, and U. Zwick, J. ACM, 42 (1995), pp. 844--856].

On Worst-Case Robin Hood Hashing

Luc Devroye, Pat Morin, and Alfredo Viola

SIAM J. Comput. 33, pp. 923-936 (14 pages)

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We consider open addressing hashing and implement it by using the Robin Hood strategy; that is, in case of collision, the element that has traveled the farthest can stay in the slot. We hash $\sim \alpha n$ elements into a table of size $n$ where each probe is independent and uniformly distributed over the table, and $\alpha < 1$ is a constant. Let $M_n$ be the maximum search time for any of the elements in the table. We show that with probability tending to one, $M_n \in [ \log_2 \log n + \sigma, \log_2 \log n + \tau ]$ for some constants $\sigma, \tau$ depending upon $\alpha$ only. This is an exponential improvement over the maximum search time in case of the standard FCFS (firstcome first served) collision strategy and virtually matches the performance of multiple-choice hash methods.

Online Routing in Triangulations

Prosenjit Bose and Pat Morin

SIAM J. Comput. 33, pp. 937-951 (15 pages) | Cited 7 times

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We consider online routing algorithms for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms, one that works for all Delaunay triangulations and the other that works for all regular triangulations; (2) a randomized memoryless algorithm that works for all triangulations; (3) an O(1) memory algorithm that works for all convex subdivisions; (4) an O(1) memory algorithm that approximates the shortest path in Delaunay triangulations; and (5) theoretical and experimental results on the competitiveness of these algorithms.

Distributional Results for Costs of Partial Match Queries in Asymmetric K-Dimensional Tries

Werner Schachinger

SIAM J. Comput. 33, pp. 952-983 (32 pages)

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In this paper we study the costs CN of partial match retrievals in K-dimensional tries (K-d tries), constructed from N records. The probabilistic model that we assume is the asymmetric Bernoulli model: keys are sequences of independently and identically distributed random variables, which assume the values 0 and 1 with probability $p \neq \frac{1}{2}$ and 1-p, and are pairwise independent. We determine the extremal asymptotic orders that the sequence of expectations $(\mathbb{E}\,C_N)_{N \ge 0}$ may have for different fixed queries, as well as the narrow region that contains $(\mathbb{E}\,C_N)_{N \geq 0}$ for almost every query. Furthermore we show that $(\mathbb{E}\,C_N)_{N \geq 0}$ and $({\rm Var}\,C_N)_{N \ge 0}$ have the same asymptotics up to a logarithmic factor and, employing a central limit theorem for martingale difference arrays, we prove asymptotic normality of $\frac{C_N - \mathbb{E}\,C_N}{\sqrt{{\rm Var}\,C_N}}$. For random queries, assumed to be independent of the keys and having their specified components distributed according to the same Bernoulli model, no limiting distribution for CN exists, but we can prove asymptotic normality of $\ln C_N$, when appropriately normalized, and determine ${\rm Var}\,C_N$ up to a logarithmic factor, where now $(\mathbb{E}\,C_N)^2 = o({\rm Var}\,C_N)$.

On Proving Circuit Lower Bounds against the Polynomial-Time Hierarchy

Jin-Yi Cai and Osamu Watanabe

SIAM J. Comput. 33, pp. 984-1009 (26 pages) | Cited 1 time

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We consider the problem of proving circuit lower bounds against the polynomial-time hierarchy. We give both positive and negative results. For the positive side, for any fixed integer k>>0, we give an explicit $\Sigma^{\rm p}_2$ language, acceptable by a $\Sigma^{\rm p}_2$ machine with running time O(nk2+k), that requires circuit size >nk. This provides a constructive version of an existence theorem of R. Kannan [Inform. and Control, 55 (1982), pp. 40--56]. Our main theorem is on the negative side. We give evidence that it is infeasible to give relativizable proofs that any single language in the polynomial-time hierarchy requires superpolynomial circuit size. Our proof techniques are based on the decision tree version of the Switching Lemma for constant depth circuits and the Nisan--Wigderson pseudorandom generator. We also take this opportunity to publish some previously unpublished older results of the first author on constant depth circuits, both straight lower bounds and inapproximability results based on decision tree--type Switching Lemmas.
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