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We discuss hybrid Monte Carlo algorithms for odd-flavor lattice QCDsimulations. The algorithms include a polynomial approximation which enables usto simulate odd-flavor QCD in the framework of the hybrid Monte Carloalgorithm. In order to make the algorithms exact, the correction factor to thepolynomial approximation is also included in an economical, stochastic way. Wetest the algorithms for $n_f=1$, 1+1 and 2+1 flavors and compare results withother algorithms.Comment: 28 pages,14 figures,Late

ODD-FLAVOR HYBRID MONTE CARLO ALGORITHM FOR LATTICE QCD