Abstract

In the planted bisection model a random graph G(n,p_+,p_-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to plus or minus 1). Any two vertices that belong to the same class are linked by an edge with probability p_+ and any two that belong to different classes with probability (p_-) c * sqrt((d_+)ln(d_+)) for a certain constant c>0.

Highlights

  • The minimum bisection problem is a well-known NP-hard problem [21], in which given a graph G one aims to find a partition of the vertex set of G into two classes of equal size so as to minimise the number of crossing edges between the two classes, called the bisection width

  • Running Warning Propagation on G naturally corresponds to a fixed-point problem on the 2-simplex, and the minimum bisection width can be cast as a function of the fixed point

  • There has been recent progress on determining the minimum bisection width on the ErdosRényi random graph

Read more

Summary

Background and motivation

The minimum bisection problem is a well-known NP-hard problem [21], in which given a graph G one aims to find a partition of the vertex set of G into two classes of equal size (up to ±1) so as to minimise the number of crossing edges between the two classes, called the bisection width. In the context of the probabilistic analysis of algorithms, it is hard to think of a more intensely studied problem than the planted bisection model. In this model a random graph G = G(n, p+1, p−1) on vertex set [n] = {1, . For a long time the algorithm with the widest range of n, p± for which a minimum bisection can be found efficiently was one of the earliest ones, namely Boppana’s spectral algorithm [6].1 It succeeds if n(p+ − p−) ≥ c np+ ln n for a certain constant c > 0. Our main tool is an elegant message passing algorithm called Warning Propagation that plays an important role in the study of random constraint satisfaction problems via ideas from statistical physics [36]. Running Warning Propagation on G naturally corresponds to a fixed-point problem on the 2-simplex, and the minimum bisection width can be cast as a function of the fixed point

The main result
Further related work
Outline
The core
Warning Propagation
The local structure
The fixed point
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call