Semidefinite Programming and Graph Partitioning with Preferences Suely Oliveira1, David Stewart2 , and Takako Soma1 1
The Department of Computer Science The University of Iowa, Iowa City, IA 52242, USA {oliveira, tsoma}@cs.uiowa.edu 2 The Department of Mathematics The University of Iowa, Iowa City, IA 52242, USA
[email protected]
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Introduction
Semidefinite programming has recently been used as a powerful approach to approximating NP-hard combinatorial problems [2, 6, 13, 40]. Such a hard combinatorial problem is the problem of graph partitioning or graph bisection with or without preferences. The general problem of graph bisection with preferences is the problem [18, 27]: Given an undirected graph G = (V, E), split V into two disjoint sets P1 and P2 where V = P1 ∪ P2 so as to minimize the number of P edges in E joining vertices in P1 to vertices in P2 plus a weighted sum i∈V ±di where the sign ± is chosen according to whether i ∈ P1 or i ∈ P2 . Because we want to balance the amount of computation we would like (ideally) to have |P1 | = |P2 |. If we set the entries of vector d = 0 we have the standard graph bisection problem [18]. Graph bisection problems are important for partitioning data across processors in a parallel computer. Let xi = +1 if i ∈ P1 and xi = −1 if i ∈ P2 . The from P1 to P2 can be represented as P number of edges crossing 1 T 1 2 (x − x ) since if i and j are in the same subset then x Lx = i j (i,j)∈E, i
