广义拉普拉斯算子的有向Hamilton性和外分支
Directed Hamiltonicity and Out-Branchings via Generalized Laplacians
摘要：
我们的动机是一个诱人的开放问题，在精确的算法：我们可以检测是否$n$ -顶点有向图$G$有一个哈密尔顿圈的时间显着小于$2^n$？我们提出了新的随机算法，改进了以前的几个工作：1。我们发现，对于任何常数$0<λ<1$和总理$p$，我们可以计数的哈密顿循环模$p^{\lfloor (1-λ)\frac{n}{3p}\rfloor}$在预期的时间小于$c^n$的一个常数$c<2$，只依赖于$p$和$λ$。这样的算法以前只知道模2计数的情况[Björklund and Husfeldt，FOCS 2013]。2.证明了在$O^*(3^{n-α(G)})$时间和多项式空间中可以检测到Hamilton圈，其中$α(G)$是$G$中最大独立集的大小.特别地，这产生了用于二分有向图的$O^*(3^{n/2})$时间算法，其比[Cygan等人，STOC 2013]。我们的算法是基于代数组合学的“关联分配”，我们可以通过评估拉普拉斯矩阵的行列式捕获，启发的矩阵-树定理有向图。除了新的算法有向哈密顿，我们使用矩阵树定理推导出简单的代数算法检测出分支。具体来说，我们给出了一个$O^*(2^k)$时间随机化算法，用于检测至少具有$k$内部顶点的外分支，改进了[Zehavi，ESA 2015]和[Björklund et al.，ICALP 2015]。我们还提出了一个代数算法的有向$k$ -叶问题，基于非标准的单项检测问题。
英文摘要：
We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant $0<λ<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p^{\lfloor (1-λ)\frac{n}{3p}\rfloor}$ in expected time less than $c^n$ for a constant $c<2$ that depends only on $p$ and $λ$. Such an algorithm was previously known only for the case of counting modulo two [Björklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in $O^*(3^{n-α(G)})$ time and polynomial space, where $α(G)$ is the size of the maximum independent set in $G$. In particular, this yields an $O^*(3^{n/2})$ time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an $O^*(2^k)$-time randomized algorithm for detecting out-branchings with at least $k$ internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Björklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed $k$-Leaf problem, based on a non-standard monomial detection problem.
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