Computer Science – Computational Complexity
Scientific paper
2011-01-10
Computer Science
Computational Complexity
Scientific paper
Quadratic Programming (QP) is the well-studied problem of maximizing over {-1,1} values the quadratic form \sum_{i \ne j} a_{ij} x_i x_j. QP captures many known combinatorial optimization problems, and assuming the unique games conjecture, semidefinite programming techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {-1,0,1}. The specific problems we study are QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2}, and Normalized QP-Ratio : \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum d_i x_i^2}, where d_i = \sum_j |a_{ij}| We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an $\tilde{O}(n^{1/3})$ algorithm for QP-ratio. We also obtain an $\tilde{O}(n^{1/4})$ approximation for bipartite graphs, and better algorithms for special cases. As with other problems with ratio objectives (e.g. uniform sparsest cut), it seems difficult to obtain inapproximability results based on P!=NP. We give two results that indicate that QP-Ratio is hard to approximate to within any constant factor. We also give a natural distribution on instances of QP-Ratio for which an n^\epsilon approximation (for \epsilon roughly 1/10) seems out of reach of current techniques.
Bhaskara Aditya
Charikar Moses
Manokaran Rajsekar
Vijayaraghavan Aravindan
No associations
LandOfFree
On Quadratic Programming with a Ratio Objective does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Quadratic Programming with a Ratio Objective, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Quadratic Programming with a Ratio Objective will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-455516