Related papers: Counting Paths and Packings in Halves
Vassilevska and Williams (STOC 2009) showed how to count simple paths on $k$ vertices and matchings on $k/2$ edges in an $n$-vertex graph in time $n^{k/2+O(1)}$. In the same year, two different algorithms with the same runtime were given by…
We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m)…
We prove that in any $n$-vertex complete graph there is a collection $\mathcal{P}$ of $(1 + o(1))n$ paths that strongly separates any pair of distinct edges $e, f$, meaning that there is a path in $\mathcal{P}$ which contains $e$ but not…
Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every…
Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…
Given a graph $G(V, E)$ and a positive integer $k$ ($k \geq 1$), a simple path on $k$ vertices is a sequence of $k$ vertices in which no vertex appears more than once and each consecutive pair of vertices in the sequence are connected by an…
A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the…
We construct a family of planar graphs $\{G_n\}_{n\geq 4}$, where $G_n$ has $n$ vertices including a source vertex $s$ and a sink vertex $t$, and edge weights that change linearly with a parameter $\lambda$ such that, as $\lambda$ varies in…
In the paper ``Lower bounds on the number of crossing-free subgraphs of $K_N$'' (Computational Geometry 16 (2000), 211-221), it is shown that a double chain of $n$ points in the plane admits at least $\Omega(4.642126305^n)$ polygonizations,…
In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…
This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in all partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th…
An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…
The vertex set of the kth cartesian power of a directed cycle of length m can be naturally identified with the set of k-tuples of integers modulo m. For any two vertices v and w of this graph, it is easy to see that if there is a…
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time…
We give a randomized algorithm that determines if a given graph has a simple path of length at least k in O(2^k poly(n,k)) time.
Let $P$ be a finite point set in $\mathbb{R}^2$ with the set of distance $n$-chains defined as $$ \Delta_n(P)=\{(|p_1-p_2|,|p_2-p_3|,\ldots,|p_n-p_{n+1}|):p_i \in P\}.$$ We show that for $2\leq n=O_{|P|}(1)$ we have $$|\Delta_n(P)|\gtrsim…
We present an algorithm for evaluating a linear ``intersection transform'' of a function defined on the lattice of subsets of an $n$-element set. In particular, the algorithm constructs an arithmetic circuit for evaluating the transform in…
Graph embedding is a powerful method in parallel computing that maps a guest network $G$ into a host network $H$. The performance of an embedding can be evaluated by certain parameters, such as the dilation, the edge congestion and the…
Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice…
In the PATH COVER problem, one asks to cover the vertices of a graph using the smallest possible number of (not necessarily disjoint) paths. While the variant where the paths need to be pairwise vertex-disjoint, which we call PATH…