Throughout the polygonal terrain of Utopia Planitia are circular grabens, inferred to overlie buried crater rims. Thirty-four of the 37 circular grabens discovered in Viking and Thermal Emission Imaging System (THEMIS) images southwest of the Utopia Basin (21¿--36¿N, 240¿--265¿W) have two nested rings, while the remaining three have a single ring. All 37 circular grabens bound topographic depressions whose surface relief scales directly with diameter, suggesting differential compaction of cover material within the polygonal terrain. The distance of doubled circular grabens from the center of the Utopia Basin scales directly with the spacing between the two rings, implying a relation between ring spacing and cover thickness. Numerical models of horizontal shrinkage during volumetric compaction of material over crater rims show two maxima of horizontal normal tension on the surface. These maxima, where we would expect grabens to form, move away from each other with increasing cover thickness. The modeled spacings of stress maxima within 1--2 km thick cover are comparable to the graben spacing observed in southwest Utopia Planitia, implying that cover thickness in this polygonal terrain may be 1--2 km thick. Where cover material is thin, the proximity of the two normal stress maxima may only produce a single circular graben, accounting for the observation that the three single-ringed grabens are most distal to the center of the basin. We contend that the correlation of shrinkage model results and observed ring spacing supports a wet sedimentary origin of polygonal terrain materials, consistent with the past existence of standing water in the Martian lowland.