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We design a deterministic polynomial time cn approximation algorithm for the permanent of positive semidefinite matrices where c = e+1 ⋍ 4:84. We write a natural convex relaxation and show that its optimum solution gives a cn approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices. We also show that...
We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n), where polyloglog(n) is a bounded degree polynomial of log log(n). We prove this by showing that any k-edge-connected unweighted graph has a...
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