On Computing a Center Persistence Diagram
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Given a set of persistence diagrams P_1,..., P_m, for the data reduction purpose, one way to summarize their topological features is to compute the centerC of them. Let P_i be the set of feature points in P_i. Here we mainly focus on the two discrete versions when points in C could be selected with or without replacement from P_i's. (We will briefly discuss the continuous case, i.e., points in C are arbitrary, which turns out to be closely related to the 3-dimensional geometric assignment problem). For technical reasons, we first focus on the case when |P_i|'s are all the same (i.e., all have the same size n), and the problem is to compute a center point set C under the bottleneck matching distance. We show, by a non-trivial reduction from the Planar 3D-Matching problem, that this problem is NP-hard even when m=3. This implies that the general center problem for persistence diagrams, when P_i's possibly have different sizes, is also NP-hard when m≥ 3. On the positive side, we show that this problem is polynomially solvable when m=2 and admits a factor-2 approximation for m≥ 3. These positive results hold for any L_p metric when P_i's are point sets of the same size, and also hold for the case when P_i's have different sizes in the L_∞ metric (i.e., for the center persistence diagram problem). This is the best possible in polynomial time unless P = NP. All these results hold for both of the discrete versions.
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