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}
}
  • S. Onn, L. Schulman
  • 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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
Partitioning vectors into quadruples
TLDR
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
TLDR
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
TLDR
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
  • Vladimir Shenmaier
  • Mathematics, Computer Science
    Journal of Applied and Industrial Mathematics
  • 2018
TLDR
It is shown that, in the case of the ℓp spaces, the problem is APX-complete if p ∈ [1, 2] and not approximable with constant accuracy if P ≠ NP and p ∉ (2,∞).
...
...

References

SHOWING 1-10 OF 10 REFERENCES
A Polynomial Time Algorithm for Shaped Partition Problems
TLDR
It is shown that when both d and p are fixed, the number of vertices of any shaped partition polytope is O(n^{d{p\choose 2}})$ and all vertices can be produced in strongly polynomial time.
Optimal partitions having disjoint convex and conic hulls
TLDR
It is shown that if the number of vectors in each of the sets is constrained, then a weaker conclusion holds, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls.
Separable Partitions
Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Basis
TLDR
Using the Minkowski addition of Newton polytopes, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials $\mathcal{T} \subset K [ x_1, \ldots,x_d ]$, where d is fixed.
The Pareto set of the partition bargaining problem
Convex Polytopes
Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V =
Linear-shaped partition problems
Cutting Corners
The Pareto set of the partition bargaining game
  • Games Econom. Behav
  • 1991