{\cal  {\cal  _1,...,{\cal P}_m, for the data reduction purpose, one way to summarize their topological features is to compute the {\em center} {\cal C} of them . Let P_i be the set of feature points in%DIFDELCMD < {\cal %%% P _i. Here we mainly focus on the two discrete versions when points in%DIFDELCMD < {\cal %%% C could be selected with or without replacement from P_i's. (We will briefly discuss the continuous case, i. e., points in%DIFDELCMD < {\cal %%% 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\geq 3. On the positive side, we show that this problem is polynomially solvable when m=2 and admits a factor-2 approximation for m\geq 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_\infty 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  .
Throughout this paper, a persistence diagram{\cal P is composed of a set P of planar points (each corresponding to a topological feature) above the line Y=X, as well as the line Y=X itself, i.e.,{\cal P =P\cup\{(x,y)|y=x\} Given a set of persistence diagrams {\cal P_1,...,{\cal P}_m, for the data reduction purpose, one way to summarize their topological features is to compute the {\em center} {\cal C} of them %DIFDELCMD < {\cal %%% first under the bottleneck distance. We consider two discrete versions %DIFDELCMD < {\cal %%% %DIFDELCMD < {\cal %%% and one continuous version. 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 diagrams are given. This implies that the general center problem for persistence diagrams under the bottleneck distance , when P_i's possibly have different sizes, is also NP-hard when m\geq 3. On the positive side, we show that this problem is polynomially solvable when m=2 and admits a factor-2 approximation for m\geq 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_\infty metric (i.e., for the Center Persistence Diagram problem). This is the best possible in polynomial time for the Center Persistence Diagram under the bottleneck distance unless P = NP. All these results hold for both of the discrete versions as well as the continuous version; in fact, the NP-hardness and approximation results also hold under the Wasserstein distance for the continuous version .
A
Throughout this paper, a persistence diagram
A
P
A
is composed of a set P of planar points (each corresponding to a topological feature) above the line Y=X, as well as the line Y=X itself, i.e.,
A
P
A
=P\cup\{(x,y)|y=x\} Given a set of persistence diagrams {\cal P
D
. Let P_i be the set of feature points in
D
P
R
_i. Here we mainly focus on the
first under the bottleneck distance. We consider
D
when points in
D
C
D
could be selected with or without replacement from P_i's. (We will briefly discuss the continuous case, i. e., points in
D
C
R
are arbitrary, which turns out to be closely related to the 3-dimensional geometric assignment problem).
and one continuous version.
R
3.
3 diagrams are given.
A
under the bottleneck distance
R
center persistence diagram
Center Persistence Diagram
A
for the Center Persistence Diagram under the bottleneck distance
A
as well as the continuous version; in fact, the NP-hardness and approximation results also hold under the Wasserstein distance for the continuous version