1,068 research outputs found
Approximating the Expansion Profile and Almost Optimal Local Graph Clustering
Spectral partitioning is a simple, nearly-linear time, algorithm to find
sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for
the quality of the approximation found by the algorithm. Local graph
partitioning algorithms [ST08,ACL06,AP09] run in time that is nearly linear in
the size of the output set, and their approximation guarantee is worse than the
guarantee provided by the Cheeger inequalities by a polylogarithmic
factor. It has been a long standing open problem to design
a local graph clustering algorithm with an approximation guarantee close to the
guarantee of the Cheeger inequalities and with a running time nearly linear in
the size of the output.
In this paper we solve this problem; we design an algorithm with the same
guarantee (up to a constant factor) as the Cheeger inequality, that runs in
time slightly super linear in the size of the output. This is the first
sublinear (in the size of the input) time algorithm with almost the same
guarantee as the Cheeger's inequality. As a byproduct of our results, we prove
a bicriteria approximation algorithm for the expansion profile of any graph.
Let . There is a polynomial
time algorithm that, for any , finds a set of measure
, and expansion . Our proof techniques also provide a simpler
proof of the structural result of Arora, Barak, Steurer [ABS10], that can be
applied to irregular graphs.
Our main technical tool is that for any set of vertices of a graph, a
lazy -step random walk started from a randomly chosen vertex of , will
remain entirely inside with probability at least . This
itself provides a new lower bound to the uniform mixing time of any finite
states reversible markov chain
Partitioning into Expanders
Let G=(V,E) be an undirected graph, lambda_k be the k-th smallest eigenvalue
of the normalized laplacian matrix of G. There is a basic fact in algebraic
graph theory that lambda_k > 0 if and only if G has at most k-1 connected
components. We prove a robust version of this fact. If lambda_k>0, then for
some 1\leq \ell\leq k-1, V can be {\em partitioned} into l sets P_1,\ldots,P_l
such that each P_i is a low-conductance set in G and induces a high conductance
induced subgraph. In particular, \phi(P_i)=O(l^3\sqrt{\lambda_l}) and
\phi(G[P_i]) >= \lambda_k/k^2).
We make our results algorithmic by designing a simple polynomial time
spectral algorithm to find such partitioning of G with a quadratic loss in the
inside conductance of P_i's. Unlike the recent results on higher order
Cheeger's inequality [LOT12,LRTV12], our algorithmic results do not use higher
order eigenfunctions of G. If there is a sufficiently large gap between
lambda_k and lambda_{k+1}, more precisely, if \lambda_{k+1} >= \poly(k)
lambda_{k}^{1/4} then our algorithm finds a k partitioning of V into sets
P_1,...,P_k such that the induced subgraph G[P_i] has a significantly larger
conductance than the conductance of P_i in G. Such a partitioning may represent
the best k clustering of G. Our algorithm is a simple local search that only
uses the Spectral Partitioning algorithm as a subroutine. We expect to see
further applications of this simple algorithm in clustering applications
A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs
Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10]
introduced the technique of subspace enumeration, which gives approximation
algorithms for graph problems such as unique games and small set expansion; the
running time of such algorithms is exponential in the threshold-rank of the
graph.
Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11]
developed an alternative approach to the design of approximation algorithms for
graphs of bounded threshold-rank, based on semidefinite programming relaxations
in the Lassere hierarchy and on novel rounding techniques. These algorithms are
faster than the ones based on subspace enumeration and work on a broad class of
problems.
In this paper we develop a third approach to the design of such algorithms.
We show, constructively, that graphs of bounded threshold-rank satisfy a weak
Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan
[FK99] for dense graphs. The existence of efficient approximation algorithms is
then a consequence of the regularity lemma, as shown by Frieze and Kannan.
Applying our method to the Max Cut problem, we devise an algorithm that is
faster than all previous algorithms, and is easier to describe and analyze
Robust fuzzy PSS design using ABC
This paper presents an Artificial Bee Colony (ABC) algorithm to tune optimal rule-base of a Fuzzy Power System Stabilizer (FPSS) which leads to damp low frequency oscillation following disturbances in power systems. Thus, extraction of an appropriate set of rules or selection of an optimal set of rules from the set of possible rules is an important and essential step toward the design of any successful fuzzy logic controller. Consequently, in this paper, an ABC based rule generation method is proposed for automated fuzzy PSS design to improve power system stability and reduce the design effort. The effectiveness of the proposed method is demonstrated on a 3-machine 9-bus standard power system in comparison with the Genetic Algorithm based tuned FPSS under different loading condition through ITAE performance indices
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