On Quadratic Programming with a Ratio Objective
Abstract
We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an \(\tilde{O}(n^{1/3})\) approximation algorithm for QP-ratio. We also give a \(\tilde{O}(n^{1/4})\) 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: one is based on the assumption that random instances of Max k-AND are hard to approximate, and the other makes a connection to a ratio version of Unique Games. We also give a natural distribution on instances of QP-Ratio for which an n ε approximation (for ε roughly 1/10) seems out of reach of current techniques.
Keywords
Approximation Algorithm Full Version Ratio Version Constant Factor Approximation Label CoverPreview
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