Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On independent sets in random graphs   The independence number of a sparse random graph G(n,m) of average degree
d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability.
Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size
(1+o(1)) n ln(d)/d, i.e. half the maximum size. Yet in spite of 30 years of
extensive research no efficient algorithm has emerged to produce an independent
set with (1+c)n ln(d)/d, for any fixed c>0. In this paper we prove that the
combinatorial structure of the independent set problem in random graphs
undergoes a phase transition as the size k of the independent sets passes the
point k nln(d)/d. Roughly speaking, we prove that independent sets of size
k>(1+c)n ln(d)/d form an intricately ragged landscape, in which local search
algorithms are bound to get stuck. We illustrate this phenomenon by providing
an exponential lower bound for the Metropolis process, a Markov chain for
sampling independents sets.

16
The categories of this paper are: stat
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics