# The Vector Partition Problem for Convex Objective Functions

@article{Onn2001TheVP,
title={The Vector Partition Problem for Convex Objective Functions},
author={Shmuel Onn and Leonard J. Schulman},
journal={Math. Oper. Res.},
year={2001},
volume={26},
pages={583-590}
}
• Published 1 August 2001
• Mathematics, Computer Science
• Math. Oper. Res.
Thepartition problem concerns the partitioning of a given set ofn vectors ind-space intop parts to maximize an objective function that is convex on the sum of vectors in each part. The problem has broad expressive power and captures NP-hard problems even if eitherp ord is fixed. In this article we show that when bothp,d are fixed, the problem is solvable in strongly polynomial time usingO(n d(p-1)-1) arithmetic operations. This improves upon the previously known bound ofO( ndp 2 ). Our method…
Momentopes, the Complexity of Vector Partitioning, and Davenport—Schinzel Sequences
• Mathematics
Discret. Comput. Geom.
• 2002
The lower bound νp,d(n)= Ω(n⌊(d−1)/2⌋Mp) on the maximum number of vertices of any p-partition polytope of a set of n points in d-space is established, implying the same bound on the complexity of the partition problem.
An Adaptive Algorithm for Vector Partitioning
• Mathematics, Computer Science
J. Glob. Optim.
• 2003
An adaptive algorithm for the vector partition problem that runs in time O(q(L)ċv) and in space O(L), where q is a polynomial function, L is the input size and v is the number of vertices of the associated partition polytope, based on an output-sensitive algorithm for enumerating all vertices.
Complexity and Algorithms for Finding a Subset of Vectors with the Longest Sum
The problem is, given a set of n vectors in a d-dimensional normed space, find a subset with the largest length of the sum vector. We prove that the problem is APX-hard for any $$\ell _p$$ norm,
A balanced k-means algorithm for weighted point sets
• Computer Science, Mathematics
ArXiv
• 2013
A generalization of the classical k-means algorithm that is capable of handling weighted point sets and prescribed lower and upper bounds on the cluster sizes is given.
The Complexity of Vector Partition
• S. Onn
• Mathematics
Vietnam journal of mathematics
• 2021
The complexity and parameterized complexity of the vector partition problem under various assumptions on the natural parameters p,d,a,t of the problem where a is the maximum absolute value of any attribute and t is the number of agent types is considered.
• Computer Science, Mathematics
• 2018
This work analyzes a straightforward matching-based algorithm and proves that this algorithm is a 3 2 -approximation algorithm for this problem, and further analyzes the performance of this algorithm on a hierarchy of special cases of the problem and shows that, in one particular case, the algorithm isA 4 -app approximation algorithm.
• Computer Science, Mathematics
• 2018
This work analyzes a straightforward matching-based algorithm and proves that this algorithm is a 3 2 -approximation algorithm for this problem, and further analyzes the performance of this algorithm on a hierarchy of special cases of the problem and shows that, in one particular case, the algorithm isA 4 -app approximation algorithm.
Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm
• Computer Science, Mathematics
ISAAC
• 2018
This work analyzes a straightforward matching-based algorithm, and proves that this algorithm is a (3/2)-approximation algorithm for this problem, and further analyzes the performance of this algorithm on a hierarchy of special cases of the problem,and proves that, in one particular case, the algorithm is an (5/4)-app approximation algorithm.
Approximability of the Problem of Finding a Vector Subset with the Longest Sum