다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

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

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of "word of mouth" in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target?We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63% of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks.We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform node-selection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.

/pdf/maximizing-the-spread-of-influence-through-a-social-network-47y1ehuvwi.pdf

Maximizing the Spread of Influence through a Social Network

Tables of integrals

1 Introduction 2 Map 3 The Data Stream Phenomenon 4 Data Streaming: Formal Aspects 5 Foundations: Basic Mathematical Ideas 6 Foundations: Basic Algorithmic Techniques 7 Foundations: Summary 8 Streaming Systems 9 New Directions 10 Historic Notes 11 Concluding Remarks Acknowledgements References

/pdf/data-streams-algorithms-and-applications-1046wbaqok.pdf

Data Streams: Algorithms and Applications

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k ≥ 3 and tree-minors in bounded-degree graphs The complexity of these algorithms is related to the distance of the graph from being Ck-minor free (resp, free from having the corresponding tree-minor) In particular, if the graph is Ω (1) -far from being cycle-free (ie, a constant fraction of the edges must be deleted to make the graph cycle-free), then the algorithm finds a cycle of polylogarithmic length in time , where N denotes the number of vertices This time complexity is optimal up to polylogarithmic factors

The foregoing results are the outcome of our study of the complexity of one-sided error property testing algorithms in the bounded-degree graphs model For example, we show that cycle-freeness of N -vertex graphs can be tested with one-sided error within time complexity , where e denotes the proximity parameter This matches the known query lower bound for one-sided error cycle-freeness testing, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on e We show that the same upper bound holds for testing whether the input graph has a simple cycle of length at least k, for any k ≥ 3 On the other hand, for any fixed tree T, we show that T -minor freeness has a one-sided error tester of query complexity that only depends on the proximity parameter e

Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges Such an extension is not possible with respect to finding tree-minors in complexity

Finding Cycles and Trees in Sublinear Time.

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $\Omega(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time $\tildeO(\sqrt{N})$, where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. 
The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity $\tildeO(\poly(1/\e)\cdot\sqrt{N})$. This matches the known $\Omega(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter $\e$. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter $\e$. 
Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.

Finding Cycles and Trees in Sublinear Time

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Two examples of our results are the following:• Motivated by the Unique Games Conjecture, Trevisan [FOCS '05] and Arora, Barak and Steurer [FOCS '10] showed that given a graph G, one can remove only 1% of G's edges and thus obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are (essentially) best possible even when one is allowed to remove 99% of G's edges. In particular, our results imply that the eigenspace enumeration approach of Arora-Barak-Steurer cannot give (even quasi-) polynomial time algorithms for unique games.• A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log n)1-o(1) such that not all graphs in the family have O(n) edges.The above results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.

/pdf/decomposing-a-graph-into-expanding-subgraphs-17cbeob9b8.pdf

Decomposing a graph into expanding subgraphs

The regularity lemma of Szemeredi gives a concise approximate description of a graph via a so called regular-partition of its vertex set. In this paper we address the following problem: can a graph have two "distinct" regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very "similar".

En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rodl and Schacht ([11]) and Lovasz and Szegedy ([9],[10]), from a previously known variant of the regularity lemma due to Alon et al. [2]. The proof also provides a deterministic polynomial time algorithm for finding such partitions.

/pdf/can-a-graph-have-distinct-regular-partitions-21gsuf4d3q.pdf

Can a graph have distinct regular partitions

The classical model of graph property testing, introduced by Goldreich, Goldwasser and Ron, assumes that the algorithm can obtain uniformly distributed vertices from the input graph. Goldreich introduced a more general model, called the Vertex-Distribution-Free model (or VDF for short) in which the testing algorithm obtains vertices drawn from an arbitrary and unknown distribution. The main motivation for this investigation is that it can allow one to give different weight/importance to different parts of the input graph, as well as handle situations where one cannot obtain uniformly selected vertices from the input. Goldreich proved that any property which is testable in this model must (essentially) be hereditary, and that several hereditary properties can indeed be tested in this model. He further asked which properties are testable in this model. In this paper we completely solve Goldreich’s problem by giving a precise characterization of the graph properties that are testable in the VDF model. Somewhat surprisingly this characterization takes the following clean form: say that a graph property P is extendable if given any graph G satisfying P, one can add one more vertex to G, and connect it to some of the vertices of G in a way that the resulting graph satisfies P. Then a property P is testable in the VDF model if and only ifP is hereditary and extendable.

Asaf Shapira

Papers

Finding Cycles and Trees in Sublinear Time.

Finding Cycles and Trees in Sublinear Time

Decomposing a graph into expanding subgraphs

Can a graph have distinct regular partitions

Testing Graphs against an Unknown Distribution.