I encountered a problem simplifying set of polygons that are adjacent. If I simplify each polygon separately with the Douglas–Peucker algorithm (which is used by many open source tools), the resulting polygons are usually not adjacent anymore. This problem exists, for example, when simplifying borders of countries/provinces.
Does anyone have a solution for it using PostGIS?
Answer
A topological vector model will provide what you need. In a non-topological storage (such as a shapefile), a single edge between geometries is stored twice. In a topological vector, the areas are stored separately from the lines, so adjustments can be made without effecting topology. I couldn't find a good diagram, so I created this simple example, where the areas A, B and C are computed from the intersections of the lines (connecting 1-4) which separate them. 
This model is used by ArcInfo as coverages, in GRASS as its default vector model, and can be used in PostGIS with the experimental PostGIS Topology tool. Perhaps a simpler solution is converting your data into linework, removing the redundant segements, and then recreating your polygons after simplification.
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