We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

ing WS1S Systems to Verify Parameterized Networks . . . . . . . . . . . . 188 Kai Baukus, Saddek Bensalem, Yassine Lakhnech and Karsten Stahl FMona: A Tool for Expressing Validation Techniques over Infinite State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 J.-P. Bodeveix and M. Filali Transitive Closures of Regular Relations for Verifying Infinite-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bengt Jonsson and Marcus Nilsson Diagnostic and Test Generation Using Static Analysis to Improve Automatic Test Generation . . . . . . . . . . . . . 235 Marius Bozga, Jean-Claude Fernandez and Lucian Ghirvu Efficient Diagnostic Generation for Boolean Equation Systems . . . . . . . . . . . . 251 Radu Mateescu Efficient Model-Checking Compositional State Space Generation with Partial Order Reductions for Asynchronous Communicating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Jean-Pierre Krimm and Laurent Mounier Checking for CFFD-Preorder with Tester Processes . . . . . . . . . . . . . . . . . . . . . . . 283 Juhana Helovuo and Antti Valmari Fair Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Thomas A. Henzinger and Sriram K. Rajamani Integrating Low Level Symmetries into Reachability Analysis . . . . . . . . . . . . . 315 Karsten Schmidt Model-Checking Tools Model Checking Support for the ASM High-Level Language . . . . . . . . . . . . . . 331 Giuseppe Del Castillo and Kirsten Winter Table of

Tools and Algorithms for the Construction and Analysis of Systems. Proc. TACAS 2009

Journal of the ACM

Recently, as more and more social network data has been published in one way or another, preserving privacy in publishing social network data becomes an important concern. With some local knowledge about individuals in a social network, an adversary may attack the privacy of some victims easily. Unfortunately, most of the previous studies on privacy preservation can deal with relational data only, and cannot be applied to social network data. In this paper, we take an initiative towards preserving privacy in social network data. We identify an essential type of privacy attacks: neighborhood attacks. If an adversary has some knowledge about the neighbors of a target victim and the relationship among the neighbors, the victim may be re-identified from a social network even if the victim's identity is preserved using the conventional anonymization techniques. We show that the problem is challenging, and present a practical solution to battle neighborhood attacks. The empirical study indicates that anonymized social networks generated by our method can still be used to answer aggregate network queries with high accuracy.

/pdf/preserving-privacy-in-social-networks-against-neighborhood-jqqu4o7yv8.pdf

Preserving Privacy in Social Networks Against Neighborhood Attacks

Let $f:2^X \rightarrow \cal R_+$ be a monotone submodular set function, and let $(X,\cal I)$ be a matroid. We consider the problem ${\rm max}_{S \in \cal I} f(S)$. It is known that the greedy algorithm yields a $1/2$-approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73-87] for this problem. For certain special cases, e.g., ${\rm max}_{|S| \leq k} f(S)$, the greedy algorithm yields a $(1-1/e)$-approximation. It is known that this is optimal both in the value oracle model (where the only access to $f$ is through a black box returning $f(S)$ for a given set $S$) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177-188] and for explicitly posed instances assuming $P \neq NP$ [U. Feige, J. ACM, 45 (1998), pp. 634-652]. In this paper, we provide a randomized $(1-1/e)$-approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307-328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrak, Proceedings of the $38$th ACM Symposium on Theory of Computing, 2008, pp. 67-74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires $|X|$ to be exponential in the original problem size, we are able to achieve a $(1-1/e-o(1))$-approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.

/pdf/maximizing-a-monotone-submodular-function-subject-to-a-1ag1ags3m8.pdf

Maximizing a Monotone Submodular Function Subject to a Matroid Constraint

The Matrix Chain Ordering Problem is a well studied optimization problem, aiming at finding optimal parentheses assignment for minimizing the number of arithmetic operations required when computing a chain of matrix multiplications. Existing algorithms include the O(N^3) dynamic programming of Godbole (1973) and the faster O(NlogN) algorithm of Hu and Shing (1982). We show that both may result in sub-optimal parentheses assignment for modern machines as they do not take into account inter-processor communication costs that often dominate the running time. Further, the optimal solution may change when using fast matrix multiplication algorithms. We adapt the O(N^3) dynamic programming algorithm to provide optimal solutions for modern machines and modern matrix multiplication algorithms, and obtain an adaption of the O(NlogN) algorithm that guarantees a constant approximation.

Computation of Matrix Chain Products on Parallel Machines

We study the following geometric hypergraph coloring problem: given a planar point set and an integer k, we wish to color the points with k colors so that any axis-aligned strip containing sufficiently many points contains all colors. We show that if the strip contains at least 2k − 1 points, such a coloring can always be found. In dimension d, we show that the same holds provided the strip contains at least k(4 ln k + ln d) points. We also consider the dual problem of coloring a given set of axis-aligned strips so that any sufficiently covered point in the plane is covered by k colors. We show that in dimension d the required coverage is at most d(k − 1) + 1. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. From the computational point of view, we show that deciding whether a three-dimensional point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. This shows a big contrast with the planar case, for which this decision problem is easy.

Colorful Strips

General-purpose hard-error resiliency solutions such as checkpoint-restart severely degrade performance. For numerical linear algebra, more efficient solutions incur lower overhead. Current solutions require a significant increase in the number of processors. Further, they are based on distributed algorithms that guarantee good performance only when the matrices are large enough to fill all the local memories. Otherwise, their inter-processor communication costs are asymptotically larger than the lower bounds dictate. We obtain fault tolerant parallel matrix multiplication algorithms that reduce the resource overhead by minimizing both the number of additional processors and the communication costs. In particular, we reduce the number of additional processors from Θ( √ P ) to 1 (or from Θ(h √ P ) to h, where h is the maximum number of simultaneous faults), and we save a Θ(logP ) factor of the latency costs. Further, for local memories larger then the minimum required to store the input and output, we obtain fault tolerant adaptations of the 2.5D algorithm that significantly reduce the communication costs, with very few additional processors.

Fault Tolerant Resource Efficient Matrix Multiplication.

Funding information Israel Science Foundation, Grant/Award Number: 863/15, 1878/14, 1901/14; Ministry of Science and Technology, Grant/Award Number: 3-10891; Einstein Foundation and the Minerva Foundation; Intel Collaborative Research Institute for Computational Intelligence (ICRI-CI); United States-Israel Binational Science Foundation (BSF); HUJI Cyber Security Research Center Summary Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache-efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One algorithm that we present performs more arithmetic than the nonrecursive version, but this allows it to benefit from calling highly optimized matrix multiplication routines; the other performs the same number of operations as the nonrecursive version, suing custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments demonstrate that the blocked and recursive versions are much faster than the previous algorithms and that the inertia strongly influences their relative performance, as predicted by our analysis.

High‐performance direct algorithms for computing the sign function of triangular matrices

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks in algorithms for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One of the novel algorithms that we present performs more arithmetic than the non-recursive version, but this allows it to benefit from calling highly-optimized matrix-multiplication routines; the other performs the same number of operations as the non-recursive version, but it uses custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments show that the blocked and recursive versions are much faster than the previous algorithms, and that the inertia strongly influences their relative performance, as predicted by our analysis.

Oded Schwartz

Papers

Computation of Matrix Chain Products on Parallel Machines

Colorful Strips

Fault Tolerant Resource Efficient Matrix Multiplication.

High‐performance direct algorithms for computing the sign function of triangular matrices

High-Performance Algorithms for Computing the Sign Function of Triangular Matrices