A Tight Lower Bound for Compact Set Packing
英文摘要：
This note is devoted to show a simple proof of a tight lower bound of the parameterized compact set packing problem, based on ETH.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
On the parameterized complexity of Compact Set Packing
The Set Packing problem is, given a collection of sets $\mathcal{S}$ over a ground set $\mathcal{U}$, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given $r \in {\mathbb N}$, is there a collection $ \mathcal{S}' \subseteq \mathcal{S}: |\mathcal{S}'| = r$ such that the sets in $\mathcal{S}'$ are pairwise disjoint? Unfortunately, the problem is not fix…
cs.CC
高多重性装箱问题的紧双指数下界
A Tight Double-Exponentially Lower Bound for High-Multiplicity Bin Packing
Kiel University（基尔大学）
该研究证明了在高多重性装箱问题中，除非ETH不成立，否则不存在在时间{{|I|}^2}^{o(d)}内解决该问题的算法，从而确认了Goemans和Rothvoss算法在参数化为物品数量d的上下文中是最佳可能的。
cs.CCcs.DS
Kernelization of Packing Problems
Kernelization algorithms are polynomial-time reductions from a problem to itself that guarantee their output to have a size not exceeding some bound. For example, d-Set Matching for integers d>2 is the problem of finding a matching of size at least k in a given d-uniform hypergraph and has kernels with O(k^d) edges. Bodlaender et al. [JCSS 2009], Fortnow and Santhanam [JCSS 2011], Dell and Van Melkebeek [JACM 2014] developed a framework for proving lower bounds on the kernel size for certain problems, under the complexity-theoretic hypothesis that coNP is not contained in NP/poly. Under the sa…
cs.CCcs.DS
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…
cs.CCcs.DS
