Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Greedy approximation algorithms for finding dense components in a graph
APPROX '00 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
On Non-Approximability for Quadratic Programs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Approximating the Cut-Norm via Grothendieck's Inequality
SIAM Journal on Computing
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Towards Sharp Inapproximability For Any 2-CSP
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Max cut and the smallest eigenvalue
Proceedings of the forty-first annual ACM symposium on Theory of computing
CSP gaps and reductions in the lasserre hierarchy
Proceedings of the forty-first annual ACM symposium on Theory of computing
Detecting high log-densities: an O(n¼) approximation for densest k-subgraph
Proceedings of the forty-second ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
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Quadratic Programming (QP) is the well-studied problem of maximizing over {−1,1} values the quadratic form ∑i≠jaijxixj. QP captures many known combinatorial optimization problems, and assuming the Unique Games conjecture, Semidefinite Programming (SDP) 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 problem we study is $$\begin{aligned} \textsf{QP-Ratio} &: \mbox{\ \ } \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2} \end{aligned}$$ This is a natural relative of several well studied problems (in fact Trevisan introduced a normalized variant as a stepping stone towards a spectral algorithm for Max Cut Gain). Quadratic ratio problems are good testbeds for both algorithms and complexity because the techniques used for quadratic problems for the {−1,1} and {0,1} domains do not seem to carry over to the {−1,0,1} domain. We give approximation algorithms and evidence for the hardness of approximating these problems. 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 nε approximation (for ε roughly 1/10) seems out of reach of current techniques.