393 research outputs found
Polynomial-time algorithm for Maximum Weight Independent Set on -free graphs
In the classic Maximum Weight Independent Set problem we are given a graph
with a nonnegative weight function on vertices, and the goal is to find an
independent set in of maximum possible weight. While the problem is NP-hard
in general, we give a polynomial-time algorithm working on any -free
graph, that is, a graph that has no path on vertices as an induced
subgraph. This improves the polynomial-time algorithm on -free graphs of
Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on
-free graphs of Lokshtanov et al (SODA 2016). The main technical
contribution leading to our main result is enumeration of a polynomial-size
family of vertex subsets with the following property: for every
maximal independent set in the graph, contains all maximal
cliques of some minimal chordal completion of that does not add any edge
incident to a vertex of
A polynomial bound on the number of minimal separators and potential maximal cliques in -free graphs of bounded clique number
In this note we show a polynomial bound on the number of minimal separators
and potential maximal cliques in -free graphs of bounded clique number
High-dimensional structure estimation in Ising models: Local separation criterion
We consider the problem of high-dimensional Ising (graphical) model
selection. We propose a simple algorithm for structure estimation based on the
thresholding of the empirical conditional variation distances. We introduce a
novel criterion for tractable graph families, where this method is efficient,
based on the presence of sparse local separators between node pairs in the
underlying graph. For such graphs, the proposed algorithm has a sample
complexity of , where is the number of
variables, and is the minimum (absolute) edge potential in the
model. We also establish nonasymptotic necessary and sufficient conditions for
structure estimation.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1009 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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