A spectral heuristic for bisecting random graphs

• Authors: Amin Coja-Oghlan
• Venue: Random Structures & Algorithms
• Year: 2006

The minimum bisection problem is to partition the vertices of a graph into two classes of equal size so as to minimize the number of crossing edges. Computing a minimum bisection is NP-hard in the worst case. In this paper we study a spectral heuristic for bisecting random graphs Gn(p,p′) with a planted bisection obtained as follows: partition n vertices into two classes of equal size randomly, and then insert edges inside the two classes with probability p′ and edges crossing the partition with probability p independently. If $n(p'-p)\geq c_0\sqrt{np'\ln(np')}$, where c0 is a suitable constant, then with probability 1 - o(1) the heuristic finds a minimum bisection of Gn(p,p′) along with a certificate of optimality. Furthermore, we show that the structure of the set of all minimum bisections of Gn(p,p′) undergoes a phase transition as $n(p'-p)=\Theta(\sqrt{np'\ln n})$. The spectral heuristic solves instances in the subcritical, the critical, and the supercritical phases of the phase transition optimally with probability 1 - o(1). These results extend previous work of Boppana [Proc. 28th FOCS (1987) 280–285]. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006An extended abstract version of this paper appeared in Proc. 16th SODA (2005), pp. 850–859.