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

Finding Large Independent Sets in Polynomial Expected Time

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.We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let $G_{n,p}$ be a random graph, and let $S$ be a set of $k$ vertices, chosen uniformly at random. Then, let $G_0$ be the graph obtained by deleting all edges connecting two vertices in $S$. Finally, an adversary may add edges to $G_0$ that do not connect two vertices in $S$, thereby producing the instance $G=G_{n,p,k}^*$. We present an algorithm that on input $G=G_{n,p,k}^*$ finds an independent set of size $\geq k$ within polynomial expected time, provided that $k\geq C(n/p)^{1/2}$ for a certain constant $C>0$. Moreover, we prove that in the case $k\leq (1-\varepsilon)\ln(n)/p$ this problem is hard.Peer Reviewe

New Abilities and Limitations of Spectral Graph Bisection

Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana\u27s algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random planted bisection model - the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana\u27s algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model. Since the behavior of Boppana\u27s algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana\u27s approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana\u27s algorithm. As a consequence we get that Boppana\u27s algorithm achieves the optimal threshold for exact cluster recovery in the stochastic block model. On the other hand we prove some limitations of Boppana\u27s approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model

The minimum bisection in the planted bisection model

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 $\pm1$). 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 $p_- <p_+$ independently. The planted bisection model has been used extensively to benchmark graph partitioning algorithms. If $p_{\pm} =2d_{\pm} /n$ for numbers $0\leq d_- <d_+$ that remain fixed as $n\to\infty$, then w.h.p. the ``planted'' bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that $d_+ -d_- >c\sqrt{d_+ \ln d_+ }$ for a certain constant $c>0$