On Optimal Weighted Balanced Clusterings: Gravity Bodies and Power Diagrams

Abstract

We study weighted clustering problems in Minkowski spaces under balancing constraints with a view towards separation properties. First, we introduce the gravity polytopes and more general gravity bodies that encode all feasible clusterings and indicate how they can be utilized to develop efficient approximation algorithms for quite general, hard to compute objective functions. Then we show that their extreme points correspond to strongly feasible power diagrams, certain specific cell complexes, whose defining polyhedra contain the clusters, respectively. Further, we characterize strongly feasible centroidal power diagrams in terms of the local optima of some ellipsoidal function over the gravity polytope. The global optima can also be characterized in terms of the separation properties of the corresponding clusterings.

Constrained clustering via diagrams: A unified theory and its application to electoral district design

“…For functionsf i (x) := h(d i (s i , x)), this has already been shown by linear programming duality in [15] for a discrete set X, h = (·) 2 , and the Euclidean metric. In a continuous setting, i. e., for X = R n and balancing constraints defined w. r. t. a probability distribution on R n , this has been proven in [5] and extended to more general function tuples F in [30].…”

“…Another fact that roots in a basic result about extremal points of transportation polytopes has been noted in their respective context by several authors (e. g., [55], [44], [59], [34], [31], [39], [15]): An optimal basic solution of (P) yields partitions with a limited number of fractionally assigned points.…”

“…As already pointed out, power diagrams have been thoroughly investigated for the example of farmland consolidation ( [13], [11]) and a comprehensive theory on their algorithmic treatment has been derived ( [15], [14], [9]). Here, some trivial cases of clusters have to be excluded: A clustering C ∈ BC is called proper, if for all i = l it holds that | supp(C i )| = | supp(C l )| = 1 ⇒ supp(C i ) = supp(C l ).…”

“…However, this means approximately maximizing an ellipsoidal norm, which is an N P-hard problem with no constant-factor approximation unless P = N P (cf. [16], [17], [12], [15]). Thus, this will be problematic for huge k. However, as the number of representatives is limited and, particularly, voting is in effect often restricted to subpopulations (like the states within the Federal Republic of Germany), this remains tractable in practice (as demonstrated).…”

Efficient solutions for weight-balanced partitioning problems

“…One way to do so is to represent the lots by their midpoints in the Euclidean plane and to use the geographical locations of the farmsteads of each farmer as a set of sites. Then one performs a weight-balanced least-squares assignment of the lots to these sites [5,6,11]. The result is a redistribution where the farmers' lots lie close to their farmsteads.…”