Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$k$-times bin packing and its application to fair electricity distribution

Dinesh Kumar Baghel, Alex Ravsky, Erel Segal-Halevi

TL;DR

The paper studies $k$-times bin packing ($k$BP) as a framework to allocate fixed supply $S$ among $n$ households with demands $D[i]$, by duplicating items across bins so that each item appears in exactly $k$ bins. It extends classical bin-packing algorithms to $k$BP, proving strong approximation guarantees for Fast and PTAS-based schemes, and demonstrates the existence of a finite $k$ that yields the optimal egalitarian electricity allocation. The authors provide both theoretical bounds—$FFk$ with asymptotic ratio $1.5+\frac{1}{5k}$, PTAS variants with $(1+2\epsilon)$ guarantees, and NFk with ratio $2$—and empirical validation on real electricity-demand data, showing superior egalitarian performance relative to prior heuristics. This work enables fair, efficient electricity distribution under capacity constraints and suggests broader uses for $k$BP, such as multi-copy data redundancy, with provable performance guarantees and practical algorithms.

Abstract

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{$k$-times bin-packing ($k$BP)}, where the goal is to pack the items such that each item appears exactly $k$ times, in $k$ different bins. We generalize some existing approximation algorithms for bin-packing to solve $k$BP, and analyze their performance ratio. The study of $k$BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by $k$-times bin-packing for some finite $k$. We also show that $k$-times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve $k$BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time.

$k$-times bin packing and its application to fair electricity distribution

TL;DR

The paper studies -times bin packing (BP) as a framework to allocate fixed supply among households with demands , by duplicating items across bins so that each item appears in exactly bins. It extends classical bin-packing algorithms to BP, proving strong approximation guarantees for Fast and PTAS-based schemes, and demonstrates the existence of a finite that yields the optimal egalitarian electricity allocation. The authors provide both theoretical bounds— with asymptotic ratio , PTAS variants with guarantees, and NFk with ratio —and empirical validation on real electricity-demand data, showing superior egalitarian performance relative to prior heuristics. This work enables fair, efficient electricity distribution under capacity constraints and suggests broader uses for BP, such as multi-copy data redundancy, with provable performance guarantees and practical algorithms.

Abstract

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into a minimum number of bins such that the sum of item sizes in a bin does not exceed the capacity. We define a new variant called \emph{-times bin-packing (BP)}, where the goal is to pack the items such that each item appears exactly times, in different bins. We generalize some existing approximation algorithms for bin-packing to solve BP, and analyze their performance ratio. The study of BP is motivated by the problem of \emph{fair electricity distribution}. In many developing countries, the total electricity demand is higher than the supply capacity. We prove that every electricity division problem can be solved by -times bin-packing for some finite . We also show that -times bin-packing can be used to distribute the electricity in a fair and efficient way. Particularly, we implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve BP, and apply the generalizations to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time.
Paper Structure (19 sections, 5 theorems, 6 equations, 1 figure)

This paper contains 19 sections, 5 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The k-times bin-packing problem
  3. Using existing bin-packing algorithms for kBP
  4. 1. Fast constant-factor approximation algorithms (Section \ref{['section:approx_algo']})
  5. 2. Polynomial-time approximation schemes (Section \ref{['section:ptas']})
  6. Simulation experiments (Appendix \ref{['subsection:ffk and ffdk bp algorithms experiment']})
  7. Electricity distribution (Section \ref{['section:experiments']})
  8. Related Literature
  9. First-Fit
  10. First-Fit Decreasing
  11. Next-fit
  12. Efficient approximation schemes
  13. Cake cutting and electricity division
  14. Definitions and Notation
  15. Electricity division problem
  16. ...and 4 more sections

Key Result

lemma thmcounterlemma

Let $W\subseteq \mathbb{R}^X$ be a nonempty finite linearly independent set. Then there exists a subset $Y$ of $X$ with $|Y|=|W|$ such that the set $\pi_Y(W)$ is linearly independent.

Figures (1)

  • Figure 1: The three blue points are (1,1,0), (1,0,1) and (0,1,1). The black line that originates from the origin intersects with the triangle (convex hull of the three blue points) at red point (2/3,2/3,2/3).

Theorems & Definitions (10)

  • lemma thmcounterlemma
  • proof : of \ref{['lem:Y']}
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof