[论文解读] Multi-Budgeted Directed Cuts

Stefan Kratsch, Shaohua Li+3 more|arXiv (Cornell University)|Jan 1, 2019

Optimization and Search Problems被引用 3

一句话总结

本文引入了多预算有向割问题，其中边被染色，每种颜色具有独立的预算。通过基于流引导的边容量增加和枚举‘重要多预算割集’的新分支技术，本文为三个问题——最小割、偏斜多割和有向反馈弧集——提出了固定参数可满足性（FPT）算法，参数化为总预算k。主要贡献在于，这些NP难变体在总预算参数下实现了固定参数可满足性，解决了关于加权和链-SAT变体的开放问题。

ABSTRACT

In this paper, we study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors 1,2,...,l to some edges and give separate budgets k_1,k_2,...,k_l for colors 1,2,...,l. For every color i in {1,...,l}, let E_i be the set of edges of color i. The solution C for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |C cap E_i| <= k_i for every i in {1,...,l}. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for l = 2. We propose FPT algorithms parameterized by k=k_1 +...+ k_l for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing k as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain l-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability.

研究动机与目标

• 解决加权和有约束的有向图分离问题的参数复杂性，包括加权st-割和Chain ℓ-SAT。
• 定义并研究多预算有向割问题，其中边被染色，每种颜色具有独立预算。
• 为多预算最小割、偏斜多割和有向反馈弧集问题开发固定参数可满足性（FPT）算法，参数化为总预算k。
• 通过引入‘最大推进’解，为Chain ℓ-SAT和加权st-割等开放问题提供理论上的可满足性证据。

提出的方法

• 引入图分离问题的多预算变体，其中每种颜色i具有预算ki，解必须至多切割ki条颜色i的边。
• 开发一种新颖的分支过程，通过增加边容量来衡量进展，而非减少k，从而实现流引导的搜索。
• 使用基于流的方法定义并枚举‘重要多预算割集’，证明其数量被限制在2^{O(k² log k)}以内。
• 利用这些割集的结构，将算法扩展至偏斜多割和有向反馈弧集问题。
• 将多预算框架与加权有向割和Chain ℓ-SAT的加权变体联系起来，证明‘最大推进’解的数量是有限的。

实验结果

研究问题

• RQ1 当参数化为总预算k时，多预算最小割问题的多预算变体是否具有固定参数可满足性？
• RQ2 多预算最小割的FPT方法能否扩展至更复杂的问题，如偏斜多割和有向反馈弧集？
• RQ3 多预算设置下‘最大推进’解的有限数量是否存在，能否为此类问题（如Chain ℓ-SAT和加权st-割）的可满足性提供证据？
• RQ4 加权st-割问题和Chain ℓ-SAT的确切成参数复杂度是什么？它们能否被归约到多预算框架？

主要发现

• 即使ℓ=2时，多预算有向割问题也是NP难的，表明多预算变体比经典最小割问题更难。
• 为多预算最小割问题开发了FPT算法，其时间复杂度被限制在2^{O(k² log k)}乘以多项式因子以内。
• 重要多预算割集的数量被限制在2^{O(k² log k)}以内，这使得最小割问题的FPT算法成为可能。
• 该框架可扩展至偏斜多割和有向反馈弧集问题，为这些问题在总预算参数下提供了首个FPT算法。
• 通过证明‘最大推进’解的数量有限，本文为Chain ℓ-SAT和加权st-割的可满足性提供了弱但结构化的证据。