André Lanka

Finding Planted Partitions in Random Graphs with General Degree Distributions

We consider the problem of recovering a planted partition such as a coloring, a small bisection, or a large cut in an (apart from that) random graph. In the last 30 years many algorithms for this problem have been developed that work provably well on various random graph models resembling the Erdos-Renyi model

Strong Refutation Heuristics for Random k-SAT

G_{n,m}

Recognizing more random unsatisfiable 3-SAT instances efficiently

. In these random graph models edges are distributed uniformly, and thus the degree distribution is very regular. By contrast, the recent theory of large networks shows that real-world networks frequently have a significantly different distribution of the edges and hence also a different degree distribution. Therefore, a variety of new types of random graphs have been introduced to capture these specific properties. One of the most popular models is characterized by a prescribed expected degree sequence. We study a natural variant of this model that features a planted partition. Our main result is that there is a polynomial time algorithm for recovering (a large part of) the planted partition in this model even in the sparse case, where the average degree is constant. In contrast to prior work, the input of the algorithm consists only of the graph, i.e., no further parameters of the model (such as the expected degree sequence) are revealed to the algorithm.

Certifying Unsatisfiability of Random 2k-SAT Formulas Using Approximation Techniques

A simple first moment argument shows that in a randomly chosen

The spectral gap of random graphs with given expected degrees

k

On the hardness and easiness of random 4-SAT formulas

-SAT formula with

Spectral Partitioning of Random Graphs with Given Expected Degrees

m

Partitioning random graphs with general degree distributions

clauses over

Techniques from combinatorial approximation algorithms yield efficient algorithms for random 2 k -SAT

n

The Spectral Gap of Random Graphs with Given Expected Degrees

boolean variables, the fraction of satisfiable clauses is