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Online Node- and Edge-Deletion Problems with Advice
Algorithmica ( IF 1.1 ) Pub Date : 2021-06-30 , DOI: 10.1007/s00453-021-00840-9
Li-Hsuan Chen , Ling-Ju Hung , Henri Lotze , Peter Rossmanith

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class \(\Pi \) at all times. We consider only hereditary properties \(\Pi \), for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs \({{\mathcal{F}}}\) and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected \({{\mathcal{F}}}\)-Node-Deletion Problem, where all graphs of the family \({\mathcal{F}}\) have to be connected and almost tight lower and upper bounds for the Delayed \(H\)-Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the Delayed \(H\)-Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected \({\mathcal{F}}\)-Edge-Deletion Problem, where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from \({\mathcal{F}}\). We give a separate analysis for the Delayed Connected \(H\)-Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.



中文翻译:

带有建议的在线节点和边缘删除问题

在在线边和节点删除问题中,输入一个节点一个节点地到达,算法必须删除节点或边以始终将输入图保持在给定的图类\(\Pi \)中。我们只考虑遗传属性\(\Pi \),对于这些属性 存在最优的在线算法并且可以用一组禁止子图来表征 \({{\mathcal{F}}}\)并分析获得一个的建议复杂度最优解。我们对 D elayed C连接 \({{\mathcal{F}}}\) -N ode -D eletion P roblem给出几乎严格的界限,其中所有的族图\({\mathcal{F}}\)已被连接和几乎紧降低和用于d上限elayed \(H \) -N-D eletion P roblem,那里有一个禁止导出子ħ可以连接或没有。对于d elayed \(H \) -N-D eletion P roblem建议复杂基本上是在最大分量的大小的简单功能 ħ。此外,我们给出较小的范围在d elayed Ç onnected \({\ mathcal {F}} \) -E DGE -D eletion P roblem,我们有任意数量的禁止连接图。对于后一个结果,我们提出了一种直接从\({\mathcal{F}}\)计算建议复杂度的算法 。我们给出了d的单独分析elayed Ç onnected \(H \) -E DGE -D eletion P roblem,这是一般较少,但承认一个绑定更易于计算。

更新日期:2021-07-01
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