A Polynomial Time Algorithm for Shaped Partition Problems

Abstract

We consider the class of shaped partition problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is fixed. In contrast, we show that when both d and p are fixed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of shaped partition polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is fixed; however, we show that when both d and p are fixed, the number of vertices of any shaped partition polytope is O(n d p 2) and all vertices can be produced in strongly polynomial time.

Convex Combinatorial Optimization

“…As the number of such circuits is n p 2 , the class of such families is edge-well-behaved with (improved) m(n) = n p 2 . Thus, while Theorem 2.6 with m(n) = O(n p ) and dimension dp implies a complexity bound of n O(dp 2 ) on the general shaped partition problem, in line with [30], with m(n) = n p 2 and same dimension dp it implies the improved bound of n O(dp) on the complexity of the unrestricted partition problem, in line with [42].…”

“…The problem is to find a p-partition π whose shape satisfies the lower and upper bounds l ≤ |π | ≤ u and which maximizes the value c(V π ). Shaped partition problems have applications in diverse fields such as clustering, inventory, reliability, and more-see [7], [10], [12], [30], [32], [43], and references therein. Here is a typical example.…”

“…If either the dimension d or the number of parts p is variable, then the shaped partition problem instantly captures NP-hard problems and hence is presumably intractable [30]. Therefore, it is interesting to study the worst case arithmetic complexity in terms of the number n of points with both d and p fixed.…”

“…In the more general case where arbitrary sets of shapes are allowed, the best upper bound to date is O(n dp 2 ) from [30]; while a matching lower bound is unknown, the lower bound O(n d( p 2 ) ) from [3] on the related number of separable partitions indicates that the quadratic term p 2 in the exponent may be unavoidable. We now show how to solve the shaped partition problem efficiently using our framework.…”

Algorithms for subset selection in linear regression

“…The equivalent formulation above allows us to reduce the problem to the Shaped Partition Problem [17,25], defined as follows.…”