41 research outputs found
The Lovasz number of random graphs
We study the Lovasz number theta along with two further SDP relaxations
theta1, theta1/2 of the independence number and the corresponding relaxations
of the chromatic number on random graphs G(n,p). We prove that these
relaxations are concentrated about their means Moreover, extending a result of
Juhasz, we compute the asymptotic value of the relaxations for essentially the
entire range of edge probabilities p. As an application, we give an improved
algorithm for approximating the independence number in polynomial expected
time, thereby extending a result of Krivelevich and Vu. We also improve on the
analysis of an algorithm of Krivelevich for deciding whether G(n,p) is
k-colorable
Planted Models for the Densest k-Subgraph Problem
Given an undirected graph G, the Densest k-subgraph problem (DkS) asks to compute a set S ? V of cardinality |S| ? k such that the weight of edges inside S is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a ?(n^{1/4 + ?}) approximation in time n^{?(1/?)}, for any ? > 0.
We ask what are some "easier" instances of this problem? We propose some natural semi-random models of instances with a planted dense subgraph, and study approximation algorithms for computing the densest subgraph in them. These models are inspired by the semi-random models of instances studied for various other graph problems such as the independent set problem, graph partitioning problems etc. For a large range of parameters of these models, we get significantly better approximation factors for the Densest k-subgraph problem. Moreover, our algorithm recovers a large part of the planted solution
Colouring Semirandom Graphs
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Peer Reviewe
Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap
Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
\phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut
Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"
We present an algorithm for strongly refuting smoothed instances of all
Boolean CSPs. The smoothed model is a hybrid between worst and average-case
input models, where the input is an arbitrary instance of the CSP with only the
negation patterns of the literals re-randomized with some small probability.
For an -variable smoothed instance of a -arity CSP, our algorithm runs in
time, and succeeds with high probability in bounding the optimum
fraction of satisfiable constraints away from , provided that the number of
constraints is at least . This
matches, up to polylogarithmic factors in , the trade-off between running
time and the number of constraints of the state-of-the-art algorithms for
refuting fully random instances of CSPs [RRS17].
We also make a surprising new connection between our algorithm and even
covers in hypergraphs, which we use to positively resolve Feige's 2008
conjecture, an extremal combinatorics conjecture on the existence of even
covers in sufficiently dense hypergraphs that generalizes the well-known Moore
bound for the girth of graphs. As a corollary, we show that polynomial-size
refutation witnesses exist for arbitrary smoothed CSP instances with number of
constraints a polynomial factor below the "spectral threshold" of ,
extending the celebrated result for random 3-SAT of Feige, Kim and Ofek
[FKO06]