On Approximating Four Covering and Packing Problems

Moving intervals for packing and covering

We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $κ$ distinct lengths, and another interval $B$, can we pack the intervals in $\mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $\mathcal{I}$) by moving $τ$ intervals and keeping the other $σ= n - τ$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $κ$, $τ$, and $σ$ as single parameter, but are FPT with combined parameters $κ$ and $τ$. We also obtain improved polynomial-time algorithms for packing and covering, inc…

On Polynomial Kernels for Integer Linear Programs: Covering, Packing and Feasibility

We study the existence of polynomial kernels for the problem of deciding feasibility of integer linear programs (ILPs), and for finding good solutions for covering and packing ILPs. Our main results are as follows: First, we show that the ILP Feasibility problem admits no polynomial kernelization when parameterized by both the number of variables and the number of constraints, unless NP \subseteq coNP/poly. This extends to the restricted cases of bounded variable degree and bounded number of variables per constraint, and to covering and packing ILPs. Second, we give a polynomial kernelization …

On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms

We give a lower bound on the iteration complexity of a natural class of Lagrangean-relaxation algorithms for approximately solving packing/covering linear programs. We show that, given an input with $m$ random 0/1-constraints on $n$ variables, with high probability, any such algorithm requires $Ω(ρ\log(m)/ε^2)$ iterations to compute a $(1+ε)$-approximate solution, where $ρ$ is the width of the input. The bound is tight for a range of the parameters $(m,n,ρ,ε)$. The algorithms in the class include Dantzig-Wolfe decomposition, Benders' decomposition, Lagrangean relaxation as developed by Held an…

一种放电方法：边三角形打包与覆盖问题的改进核

A Discharging Method: Improved Kernels for Edge Triangle Packing and Covering

本文提出一种新颖的放电方法，将边三角形打包和覆盖问题的核大小从$(3+ε)k$和$6k$改进至$3k$，并展示了该方法在分析其他核算法中的潜力。

On the tractability of some natural packing, covering and partitioning problems

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph $G=(V,E)$ and two "object types" $\mathrm{A}$ and $\mathrm{B}$ chosen from the alternatives above, we consider the following questions. \textbf{Packing problem:} can we find an object of type $\mathrm{A}$ and one of type $\mathrm{B}$ in the edge set $E$ of $G$, so that they are edge-disjoint? \textbf{Partitioning problem:} can we partition $E$ into an object of type $\mathrm{A}$ and one of type $\mat…

二维向量装箱问题不存在 APTAS：再探讨

There is no APTAS for 2-dimensional vector bin packing: Revisited

本文修正了Woeginger关于二维及以上Vector Bin Packing不存在APTAS的证明，并证明了其及Vector Bin Covering在特定近似比下是NP-hard的。