可分解图问题的隔离方案
Isolation schemes for problems on decomposable graphs
摘要：
Mulmuley，Vazirani和Vazirani [Combinatorica'87]的隔离引理提供了一个自归约方案，允许人们假设问题的给定实例具有唯一解，只要解存在。自从引入隔离引理以来，人们一直致力于对特定类别问题的隔离引理进行去随机化。到目前为止，重点主要是在多项式时间内可解的问题。在本文中，我们研究了一个设置，是更典型的$\mathsf{NP}$ -完全问题，并获得部分去随机化的形式显着减少所需的随机位数。特别是，在参数化算法的进步的动机下，我们专注于可分解图的问题。例如，对于检测哈密尔顿循环的问题，我们建立在来自[Bodlaender等人，信息计算并设计隔离方案，该隔离方案使用-树宽最大为$t$的图上的$O(t\log n + \log^2{n})$随机位; -平面或$H$ -小自由图上的$O(\sqrt{n})$随机位;以及-一般图上的$O(n)$随机位。在所有这些方案中，权重在所使用的随机比特的数量上是指数有界的。作为一个推论，对于每个固定的$H$，我们得到一个算法，用于检测一个汉密尔顿循环中的$H$ -小的自由图，运行在确定性的时间$2^{O(\sqrt{n})}$，并使用多项式空间，这是第一个算法，以实现这样的复杂性保证。对于更多的本地性质的问题，如找到一个独立的最大尺寸集，我们得到隔离计划的树深度最$d$图，使用$O(d)$随机位，并分配多项式有界的权重。我们还补充了我们的研究结果与几个无条件和有条件的下限，这表明，许多结果不能显着改善。
英文摘要：
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for $\mathsf{NP}$-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - $O(t\log n + \log^2{n})$ random bits on graphs of treewidth at most $t$; - $O(\sqrt{n})$ random bits on planar or $H$-minor free graphs; and - $O(n)$-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed $H$ we obtain an algorithm for detecting a Hamiltonian cycle in an $H$-minor-free graph that runs in deterministic time $2^{O(\sqrt{n})}$ and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most $d$ that use $O(d)$ random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.
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