We introduce a new method for filtering noisy chromosome conformation capture (3C) interactions that selects subsets of interactions that obey metric constraints of various strictness. We demonstrate that, although the problem is computationally hard, near-optimal results are often attainable in practice using well-designed heuristics and approximation algorithms. Further, we show that, compared with a standard technique, this metric filtering approach leads to (a) subgraphs with higher statistical significance, (b) lower embedding error, (c) lower sensitivity to initial conditions of the embedding algorithm, and (d) structures with better agreement with light microscopy measurements.