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Issue 8 | 2010 | pp. 3441-3904
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2010

Volume 39, Issue 8, pp. 3441-3904


Bounded Independence Fools Halfspaces

Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola

SIAM J. Comput. 39, pp. 3441-3462 (22 pages) | Cited 2 times

Online Publication Date: August 19, 2010

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We show that any distribution on $\{-1,+1\}^n$ that is $k$-wise independent fools any halfspace (or linear threshold function) $h:\{-1,+1\}^n\to\{-1,+1\}$, i.e., any function of the form $h(x)=\operatorname{sign}(\sum_{i=1}^{n}w_{i}x_{i}-\theta)$, where the $w_1,\dots,w_n$ and $\theta$ are arbitrary real numbers, with error $\epsilon$ for $k=O(\epsilon^{-2}\log^2(1/\epsilon))$. Our result is tight up to $\log(1/\epsilon)$ factors. Using standard constructions of $k$-wise independent distributions, we obtain the first explicit pseudorandom generators $G:\{-1,+1\}^s\to\{-1,+1\}^n$ that fool halfspaces. Specifically, we fool halfspaces with error $\epsilon$ and seed length $s=k\cdot\log n=O(\log n\cdot\epsilon^{-2}\log^2(1/\epsilon))$. Our approach combines classical tools from real approximation theory with structural results on halfspaces by Servedio [Comput. Complexity, 16 (2007), pp. 180–209].

Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence

Anna Gál and Parikshit Gopalan

SIAM J. Comput. 39, pp. 3463-3479 (17 pages)

Online Publication Date: August 19, 2010

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We show that any deterministic streaming algorithm that makes a constant number of passes over the input and gives a constant factor approximation of the length of the longest increasing subsequence in a sequence of length $n$ must use space $\Omega(\sqrt{n})$. This proves a conjecture made by Gopalan et al. [Proceedings of the 18th Annual ACM–SIAM Symposium on Discrete Algorithms, 2007, pp. 318–327] who proved a matching upper bound. Our results yield asymptotically tight lower bounds for all approximation factors, thus resolving the main open problem from their paper. Our proof is based on analyzing a related communication problem and proving a direct sum type property for it.

Reaching and Distinguishing States of Distributed Systems

Robert M. Hierons

SIAM J. Comput. 39, pp. 3480-3500 (21 pages)

Online Publication Date: August 19, 2010

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Some systems interact with their environment at physically distributed interfaces, called ports, and in testing such a system it is normal to place a tester at each port. Each tester observes only the events at its port and it is known that this limited observational power introduces additional controllability and observability problems into testing. Given a multiport finite state machine (FSM) $M$, we consider the problems of defining strategies for the testers either to reach a given state of $M$ or to distinguish two states of $M$. These are important problems since most techniques for testing from a single-port FSM use sequences that reach and distinguish states. Both problems can be solved in low-order polynomial time for single-port FSMs but we prove that the corresponding decision problems are undecidable for multiport FSMs. However, we also show that they can be solved in low-order polynomial times for deterministic FSMs if we restrict our attention to controllable tests. These results have important ramifications for testing from a multiport FSM since they suggest that methods for testing from a single-port FSM cannot be easily adapted. In addition, two FSMs can be distinguished if and only if their initial states can be distinguished and so the results suggest that, in contrast to single-port FSMs, we cannot expect to produce general complete test generation methods for multiport FSMs.

Explicit Construction of a Small $\epsilon$-Net for Linear Threshold Functions

Yuval Rabani and Amir Shpilka

SIAM J. Comput. 39, pp. 3501-3520 (20 pages) | Cited 1 time

Online Publication Date: August 26, 2010

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We give explicit constructions of $\epsilon$-nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension $n$ and in $\frac{1}{\epsilon}$. To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in $n$ and a covering radius of $\frac{n}{2}-c\sqrt{n\log n}$ for any constant $c$. This improves upon the well-known construction of dual BCH codes that guarantee only a covering radius of $\frac{n}{2}-c\sqrt{n}$.

Randomized Self-Assembly for Exact Shapes

David Doty

SIAM J. Comput. 39, pp. 3521-3552 (32 pages)

Online Publication Date: September 29, 2010

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Working in Winfree's abstract tile assembly model, we show that a constant-sized tile assembly system can be programmed through relative tile concentrations to build an $n\times n$ square with high probability for any sufficiently large $n$. This answers an open question of Kao and Schweller [Automata, Languages and Programming, Lecture Notes in Comput. Sci. 5125, Springer, Berlin, 2008, pp. 370–384], who showed how to build an approximately $n\times n$ square using tile concentration programming and asked whether the approximation could be made exact with high probability. We show how this technique can be modified to answer another question of Kao and Schweller by showing that a constant-sized tile assembly system can be programmed through tile concentrations to assemble arbitrary finite scaled shapes, which are shapes modified by replacing each point with a $c\times c$ block of points for some integer $c$. Furthermore, we exhibit a smooth trade-off between specifying bits of $n$ via tile concentrations versus specifying them via hard-coded tile types, which allows tile concentration programming to be employed for specifying a fraction of the bits of “input” to a tile assembly system, under the constraint that concentrations can be specified to only a limited precision. Finally, to account for some unrealistic aspects of the tile concentration programming model, we show how to modify the construction to use only concentrations that are arbitrarily close to uniform.

Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász–Schrijver Hierarchy

Konstantinos Georgiou, Avner Magen, Toniann Pitassi, and Iannis Tourlakis

SIAM J. Comput. 39, pp. 3553-3570 (18 pages) | Cited 1 time

Online Publication Date: October 14, 2010

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Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. Probabilistically checkable proof (PCP)-based techniques of Dinur and Safra [Ann. of Math./ (2), 162 (2005), pp. 439–486] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. There is a widespread belief that semidefinite programming (SDP) techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [Theory Comput., 2 (2006), pp. 19–51], our aim is to show that a large family of linear programming (LP)- and SDP-based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166–190] introduced the systems $LS$ and $LS_+$ for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, $LS_+$ captures the celebrated SDP-based algorithms for Max Cut and Sparsest Cut mentioned above. We rule out polynomial-time SDP-based $2-\Omega(1)$ approximations for Vertex Cover using $LS_+$. In particular, for every $\epsilon>0$ we prove an integrality gap of $2-\epsilon$ for Vertex Cover SDPs obtained by tightening the standard LP relaxation with $\Omega(\sqrt{\log n/\log\log n})$ rounds of $LS_+$. While tight integrality gaps were known for Vertex Cover in the weaker $LS$ system [G. Schoenebeck, L. Trevisan, and M. Tulsiani, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2007, pp. 302–310], previous results did not rule out a $2-\Omega(1)$ approximation after even two rounds of $LS_+$.

Approximation Algorithms for Scheduling Parallel Jobs

Klaus Jansen and Ralf Thöle

SIAM J. Comput. 39, pp. 3571-3615 (45 pages)

Online Publication Date: October 26, 2010

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In this paper we study variants of the nonpreemptive parallel job scheduling problem in which the number of machines is polynomially bounded in the number of jobs. For this problem we show that a schedule with length at most $(1+\varepsilon)\,\mathrm{OPT}$ can be calculated in polynomial time. Unless $P=NP$, this is the best possible result (in the sense of approximation ratio), since the problem is strongly NP-hard. For the case where all jobs must be allotted to a subset of consecutive machines, a schedule with length at most $(1.5+\varepsilon)\,\mathrm{OPT}$ can be calculated in polynomial time. The previously best known results are algorithms with absolute approximation ratio 2. Furthermore, we extend both algorithms to the case of malleable jobs with the same approximation ratios.

Strict Cost Sharing Schemes for Steiner Forest

Lisa Fleischer, Jochen Könemann, Stefano Leonardi, and Guido Schäfer

SIAM J. Comput. 39, pp. 3616-3632 (17 pages) | Cited 1 time

Online Publication Date: October 26, 2010

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Gupta et al. [J. ACM, 54 (2007), article 11] and Gupta, Kumar, and Roughgarden [in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 365–372] recently developed an elegant framework for the development of randomized approximation algorithms for rent-or-buy network design problems. The essential building block of this framework is an approximation algorithm for the underlying network design problem that admits a strict cost sharing scheme. Such cost sharing schemes have also proven to be useful in the development of approximation algorithms in the context of two-stage stochastic optimization with recourse. The main contribution of this paper is to show that the Steiner forest problem admits cost shares that are 3-strict and 4-group-strict. As a consequence, we derive surprisingly simple approximation algorithms for the multicommodity rent-or-buy and the multicast rent-or-buy problems with approximation ratios 5 and 6, improving over the previous best approximation ratios of 6.828 and 12.8, respectively. We also show that no approximation ratio better than 4.67 can be achieved using the sample-and-augment framework in combination with the currently best known Steiner forest approximation algorithms. In the context of two-stage stochastic optimization, our result leads to a 6-approximation algorithm for the stochastic Steiner tree problem in the black-box model and a 5-approximation algorithm for the stochastic Steiner forest problem in the independent decision model.

A General Approach for Incremental Approximation and Hierarchical Clustering

Guolong Lin, Chandrashekhar Nagarajan, Rajmohan Rajaraman, and David P. Williamson

SIAM J. Comput. 39, pp. 3633-3669 (37 pages)

Online Publication Date: November 04, 2010

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We present a general framework and algorithmic approach for incremental approximation algorithms. The framework handles cardinality constrained minimization problems, such as the $k$-median and $k$-MST problems. Given some notion of ordering on solutions of different cardinalities $k$, we give solutions for all values of $k$ such that the solutions respect the ordering and such that for any $k$, our solution is close in value to the value of an optimal solution of cardinality $k$. For instance, for the $k$-median problem, the notion of ordering is set inclusion, and our incremental algorithm produces solutions such that for any $k$ and $k'$, $k<k'$, our solution of size $k$ is a subset of our solution of size $k'$. We show that our framework applies to this incremental version of the $k$-median problem (introduced by Mettu and Plaxton [R. R. Mettu and C. G. Plaxton, SIAM J. Comput., 32 (2003), pp. 816–832]) and incremental versions of the $k$-MST problem, $k$-vertex cover problem, $k$-set cover problem, as well as the uncapacitated facility location problem (which is not cardinality-constrained). For these problems we get either new incremental algorithms or improvements over what was previously known. We also show that the framework applies to hierarchical clustering problems. In particular, we give an improved algorithm for a hierarchical version of the $k$-median problem introduced by Plaxton [C. G. Plaxton, J. Comput. System Sci., 72 (2006), pp. 425–443].

Extension Preservation Theorems on Classes of Acyclic Finite Structures

David Duris

SIAM J. Comput. 39, pp. 3670-3681 (12 pages)

Online Publication Date: November 11, 2010

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A class of structures satisfies the extension preservation theorem if, on this class, every first-order sentence is preserved under extensions iff it is equivalent to an existential sentence. We consider different acyclicity notions for hypergraphs ($\gamma$-, $\beta$-, and $\alpha$-acyclicity and also acyclicity on hypergraph quotients) and estimate their influence on the validity of the extension preservation theorem on classes of finite structures. More precisely, we prove that the extension preservation theorem is satisfied for classes of finite structures having a $\gamma$-acyclic $k$-quotient that are closed under induced substructures and disjoint unions. We show that this is not the case for classes of $\beta$-acyclic structures. To achieve this, we make logical reductions from finite structures to their $k$-quotients and from $\gamma$-acyclic hypergraphs to acyclic graphs.

Quantified Equality Constraints

Manuel Bodirsky and Hubie Chen

SIAM J. Comput. 39, pp. 3682-3699 (18 pages)

Online Publication Date: November 17, 2010

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An equality template is a relational structure with infinite universe whose relations can be defined by Boolean combinations of equalities. We prove a complexity classification for quantified constraint satisfaction problems (QCSPs) over equality templates: These problems are in L (decidable in logarithmic space), NP-hard, or coNP-hard. To establish our classification theorem we combine methods from universal algebra with concepts from model theory.

Fast Convergence to Wardrop Equilibria by Adaptive Sampling Methods

Simon Fischer, Harald Räcke, and Berthold Vöcking

SIAM J. Comput. 39, pp. 3700-3735 (36 pages)

Online Publication Date: December 07, 2010

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We study the question of whether a large population of agents in a traffic network is able to converge to an equilibrium quickly. To that end, we consider a round-based variant of the Wardrop model. Every agent is allowed to reroute its traffic once in a while with the aim of finding a path with minimal latency. As a first result we find that using a replication policy which allows agents to imitate others gives rise to a bicriterial approximate equilibrium very quickly. In particular, the time bound depends logarithmically on the ratio between minimum and maximum latency but is otherwise independent of the network size. In the single-commodity case, this bicriteria approximate equilibrium has an intuitive interpretation as a state in which almost all agents are almost happy. This kind of approximate equilibrium, however, is transient. In order to reach a global approximation, we need to add an exploration component which enables the agents to explore the strategy space independently of the other agents. Although it can be shown that, when used exclusively, exploration policies imply an exponential lower bound, applying exploration carefully allows the population to approximate the global Wardrop equilibrium in polynomial time. Since the distributed and concurrent fashion of our policies bears the risk of oscillating behavior, we must take into account the steepness of the latency functions. We show that the relevant parameter is elasticity, a parameter closely related to the polynomial degree. This improves significantly over earlier results which depend on the absolute slope and therefore have a pseudopolynomial flavor.

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

Gábor Ivanyos, Marek Karpinski, and Nitin Saxena

SIAM J. Comput. 39, pp. 3736-3751 (16 pages)

Online Publication Date: December 08, 2010

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We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e., the problem of assigning values to the variables in a given symbolic matrix to maximize the resulting matrix rank. Matrix completion is one of the fundamental problems in computational complexity. It has numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [N. J. A. Harvey, D. R. Karger, and K. Murota, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 489–498; N. J. A. Harvey, D. R. Karger, and S. Yekhanin, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 1103–1111]. We design efficient deterministic algorithms for common generalizations of the results of Lovász and Geelen on this problem by allowing linear polynomials in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. Our methods are algebraic and quite different from those of Lovász and Geelen. We look at the problem of matrix completion in the more general setting of linear spaces of linear transformations and find a maximum rank element there using a greedy method. Matrix algebras and modules play a crucial role in the algorithm. We show (hardness) results for special instances of matrix completion naturally related to matrix algebras; i.e., in contrast to computing isomorphisms of modules (for which there is a known deterministic polynomial time algorithm), finding a surjective or an injective homomorphism between two given modules is as hard as the general matrix completion problem. The same hardness holds for finding a maximum dimension cyclic submodule (i.e., generated by a single element). For the “dual” task, i.e., finding the minimal number of generators of a given module, we present a deterministic polynomial time algorithm. The proof methods developed in this paper apply to fairly general modules and could also be of independent interest.

Fast Access to Distributed Atomic Memory

Partha Dutta, Rachid Guerraoui, Ron R. Levy, and Marko Vukolić

SIAM J. Comput. 39, pp. 3752-3783 (32 pages)

Online Publication Date: December 14, 2010

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We study efficient and robust implementations of an atomic read-write data structure over an asynchronous distributed message-passing system made of reader and writer processes, as well as a number of servers implementing the data structure. We determine the exact conditions under which every read and write involves one round of communication with the servers. These conditions relate the number of readers to the tolerated number of faulty servers and the nature of these failures.

Tightening Nonsimple Paths and Cycles on Surfaces

Éric Colin de Verdière and Jeff Erickson

SIAM J. Comput. 39, pp. 3784-3813 (30 pages) | Cited 1 time

Online Publication Date: December 14, 2010

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We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity $n$, genus $g\geq2$, and no boundary, we construct in $O(gn\log n)$ time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity $k$ in $O(gnk)$ time, or the shortest cycle homotopic to a given cycle of complexity $k$ in $O(gnk\log(nk))$ time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.

Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

David Eppstein, Michael T. Goodrich, and Darren Strash

SIAM J. Comput. 39, pp. 3814-3829 (16 pages)

Online Publication Date: December 14, 2010

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We provide linear-time algorithms for geometric graphs with sublinearly many edge crossings. That is, we provide algorithms running in $O(n)$ time on connected geometric graphs having $n$ vertices and $k$ pairwise crossings, where $k$ is smaller than $n$ by an iterated logarithmic factor. Specific problems that we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that “zeros in” on edge crossings, together with methods for applying planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having $n$ segments and $k$ crossings in linear time, for the case when $k$ is sublinear in $n$ by an iterated logarithmic factor.

Correctness of Gossip-Based Membership under Message Loss

Maxim Gurevich and Idit Keidar

SIAM J. Comput. 39, pp. 3830-3859 (30 pages)

Online Publication Date: December 14, 2010

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Due to their simplicity and effectiveness, gossip-based membership protocols have become the method of choice for maintaining partial membership in large peer-to-peer systems. A variety of gossip-based membership protocols were proposed. Some were shown to be effective empirically, lacking analytic understanding of their properties. Others were analyzed under simplifying assumptions, such as lossless and delayless network. It is not clear whether the analysis results hold in dynamic networks, where both nodes and network links can fail. In this paper we try to bridge this gap. We first enumerate the desirable properties of a gossip-based membership protocol, such as view uniformity, independence, and load balance. We then propose a simple send & forget protocol, and show that even in the presence of message loss, it achieves the desirable properties.

Analysis of Delays Caused by Local Synchronization

Julia Lipman and Quentin F. Stout

SIAM J. Comput. 39, pp. 3860-3884 (25 pages)

Online Publication Date: December 22, 2010

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Synchronization is often necessary in parallel computing, but it can create delays whenever the receiving processor is idle, waiting for the information to arrive. This is especially true for barrier, or global, synchronization, in which every processor must synchronize with every other processor. Nonetheless, barriers are the only form of synchronization explicitly supplied in OpenMP, and they occur whenever collective communication operations are used in MPI. Many applications do not actually require global synchronization; local synchronization, in which a processor synchronizes only with those processors from or to which information or resources are needed, is often adequate. However, when tasks take varying amounts of time, the behavior of a system under local synchronization is more difficult to analyze since processors do not start tasks at the same time. We show that when the synchronization dependencies form a directed cycle and the task times are geometrically distributed with $p=0.5$, then as the number of processors tends to infinity the processors are working $2-\sqrt{2}\approx0.59\%$ of the time. Under global synchronization, however, the time to complete each task is unbounded, increasing logarithmically with the number of processors. Similar results apply for $p\neq0.5$. We also present some of the combinatorial properties of the synchronization problem with geometrically distributed tasks on an undirected cycle. Nondeterministic synchronization is also examined, where processors decide randomly at the beginning of each task which neighbors(s) to synchronize with. We show that the expected number of task dependencies for random synchronization on an undirected cycle is the same as for deterministic synchronization on a directed cycle. Simulations are included to extend the analytic results. They show that more heavy-tailed distributions can actually create fewer delays than less heavy-tailed ones if the number of processors is small for some random-neighbor synchronization models. The results also show the rate of convergence to the steady state for various task distributions and synchronization graphs.

Lower Bounds for Randomized Consensus under a Weak Adversary

Hagit Attiya and Keren Censor-Hillel

SIAM J. Comput. 39, pp. 3885-3904 (20 pages)

Online Publication Date: December 22, 2010

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This paper studies the inherent trade-off between termination probability and total step complexity of randomized consensus algorithms. It shows that for every integer $k$, the probability that an $f$-resilient randomized consensus algorithm of $n$ processes does not terminate with agreement within $k(n-f)$ steps is at least $\frac{1}{c^k}$, for some constant $c$. A corresponding result is proved for Monte-Carlo algorithms that may terminate in disagreement. The lower bound holds for asynchronous systems, where processes communicate either by message passing or through shared memory, under a very weak adversary that determines the schedule in advance, without observing the algorithm's actions. This complements algorithms of Kapron et al. [Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), ACM, New York, SIAM, Philadelphia, 2008, pp. 1038–1047] for message-passing systems, and of Aumann [Proceedings of the 16th Annual ACM Symposium on Principles of Distributed Computing (PODC), ACM, New York, 1997, pp. 209–218] and Aumann and Bender [Distrib. Comput., 17 (2005), pp. 191–207] for shared-memory systems.
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