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Finite-dimensional reductions of the 2D dispersionless Toda hierarchy,constrained by the ``string equation'' are studied. These include solutionsdetermined by polynomial, rational or logarithmic functions, which are ofinterest in relation to the ``Laplacian growth'' problem governing interfacedynamics. The consistency of such reductions is proved, and the Hamiltonianstructure of the reduced dynamics is derived. The Poisson structure of therationally reduced dispersionless Toda hierarchies is also derivedComment: 18 pages LaTex, accepted to J.Math.Phys, Significantly updated  version of the previous submissio

Constrained reductions of two-dimensional dispersionless Toda hierarchy, Hamiltonian structure, and interface dynamics