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59
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By d ..."
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Cited by 174 (28 self)
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We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 144 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Spectral Partitioning of Random Graphs
, 2001
"... Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all ..."
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Cited by 87 (3 self)
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Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident.
Recommendation Systems: A Probabilistic Analysis
 In Proc. IEEE Symp. on Foundations of Computer Science (FOCS
, 1998
"... A recommendation system tracks past actions of a group of users to make recommendations to individual members of the group. The growth of computermediated marketing and commerce has led to increased interest in such systems. We introduce a simple analytical framework for recommendation systems, inc ..."
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Cited by 53 (2 self)
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A recommendation system tracks past actions of a group of users to make recommendations to individual members of the group. The growth of computermediated marketing and commerce has led to increased interest in such systems. We introduce a simple analytical framework for recommendation systems, including a basis for defining the utility of such a system. We perform probabilistic analyses of algorithmic methods within this framework. These analyses yield insights into how much utility can be derived from the memory of past actions and on how this memory can be exploited. 1. Introduction Collaborative filtering (sometimes known as a recommendation system) is a process by which information on the preferences and actions of a group of users is tracked by a system which then, based on the patterns it observes, tries to make useful recommendations to individual users [10, 12, 18, 19, 20, 22, 23]. For instance, a book recommendation system might recommend Jules Verne to someone interested ...
Finding and Certifying a Large Hidden Clique in a SemiRandom Graph
, 1999
"... Alon, Krivelevich and Sudakov (Random Structures and Algorithms, 1998) designed an algorithm based on spectral techniques that almost surely finds a clique of size \Omega\Gamma p n) hidden in an otherwise random graph. We show that a different algorithm, based on the Lov'asz theta function, al ..."
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Cited by 47 (11 self)
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Alon, Krivelevich and Sudakov (Random Structures and Algorithms, 1998) designed an algorithm based on spectral techniques that almost surely finds a clique of size \Omega\Gamma p n) hidden in an otherwise random graph. We show that a different algorithm, based on the Lov'asz theta function, almost surely both finds the hidden clique and certifies its optimality. Our algorithm has an additional advantage of being more robust: it also works in a semirandom hidden clique model, in which an adversary can remove edges from the random portion of the graph. 1 Introduction A clique in a graph G is a subset of the vertices every two of which are connected by an edge. The maximum clique problem, that is, finding a clique of maximum size in a graph, is fundamental in the area of combinatorial optimization, and is closely related to the independent set problem (clique on the edge complement graph G), the vertex cover problem (the vertex complement of the independent set) and chromatic...
On The Quality Of Spectral Separators
 SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
, 1998
"... Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral met ..."
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Cited by 43 (3 self)
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Computing graph separators is an important step in many graph algorithms. A popular technique for finding separators involves spectral methods. However, there has not been much prior analysis of the quality of the separators produced by this technique; instead it is usually claimed that spectral methods "work well in practice." We present an initial attempt at such analysis. In particular, we consider two popular spectral separator algorithms, and provide counterexamples that show these algorithms perform poorly on certain graphs. We also consider a generalized definition of spectral methods that allows the use of some specified number of the eigenvectors corresponding to the smallest eigenvalues of the Laplacian matrix of a graph, and show that if such algorithms use a constant number of eigenvectors, then there are graphs for which they do no better than using only the second smallest eigenvector. Further, using the second smallest eigenvector of these graphs produces partitions that are poor with respect to bounds on the gap between the isoperimetric number and the cut quotient of the spectral separator. Even if a generalized spectral algorithm uses n # for 0 < # < 1 4 eigenvectors, there exist graphs for which the algorithm fails to find a separator with a cut quotient within n 1 4 #  1 of the isoperimetric number. We also introduce some facts about the structure of eigenvectors of certain types of Laplacian and symmetric matrices; these facts provide the basis for the analysis of the counterexamples. Finally, we discuss some developments in spectral partitioning that have occurred since these results first appeared.
Bounds On The Cover Time
 J. Theoretical Probab
, 1988
"... . Consider a particle that moves on a connected, undirected graph G with n vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. The cover time is the first time when the particle has visited all the vertices in the graph starting from ..."
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Cited by 42 (0 self)
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. Consider a particle that moves on a connected, undirected graph G with n vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. The cover time is the first time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time for a graph to the eigenvalues of the Markov chain that describes the random walk above. An interesting consequence is that regular expander graphs have expected cover time \Theta(n log n). iii 1. Introduction. Consider a particle moving on an undirected graph G = (V; E) from vertex to vertex according to the following rule: the probability of a transition from vertex i, of degree d i , to vertex j is 1=d i if (i; j) 2 E, and 0 otherwise. This stochastic process is a Markov chain; it is called a random walk on the graph G. In this paper we derive upper and lower bounds on the expected cover...
Algorithms for Graph Partitioning on the Planted Partition Model
, 1999
"... The NPhard graph bisection problem is to partition the nodes of an undirected graph into two equalsized groups so as to minimize the number of edges that cross the partition. The more general graph lpartition problem is to partition the nodes of an undirected graph into l equalsized groups so as ..."
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Cited by 36 (0 self)
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The NPhard graph bisection problem is to partition the nodes of an undirected graph into two equalsized groups so as to minimize the number of edges that cross the partition. The more general graph lpartition problem is to partition the nodes of an undirected graph into l equalsized groups so as to minimize the total number of edges that cross between groups. We present a simple, lineartime algorithm for the graph lpartition problem and analyze it on a random "planted lpartition" model. In this model, the n nodes of a graph are partitioned into l groups, each of size n=l; two nodes in the same group are connected by an edge with some probability p, and two nodes in different groups are connected by an edge with some probability r ! p. We show that if p \Gamma r n \Gamma1=2+ffl for some constant ffl, then the algorithm finds the optimal partition with probability 1 \Gamma exp(\Gamman \Theta(ffl) ). 1