Some Partial Results on Linial's Conjecture for Matching-Spine Digraphs

Let $k$ be a positive integer. A \emph{partial $k$-coloring} of a digraph $D$ is a set $\calC$ of $k$ disjoint stable sets and has \emph{weight} defined as $\sum_{C \in \calC} |C|$. An \emph{optimal} $k$-coloring is a $k$-coloring of maximum weight. A \emph{path partition} of a digraph $D$ is a set $\calP$ of disjoint paths of $D$ that covers its vertex set and has \emph{$k$-norm} defined as $\sum_{P \in \mathcal{P}} \min\{|P|,k\}$. A path partition $\calP$ is \emph{$k$-optimal} if it has minimum $k$-norm. A digraph $D$ is \emph{matching-spine} if its vertex set can be partitioned into sets $X$ and $Y$, such that $D[X]$ has a Hamilton path and the arc set of $D[Y]$ is a matching. Linial (1981) conjectured that the $k$-norm of a $k$-optimal path partition of a digraph is at most the weight of an optimal partial $k$-coloring. We present some partial results on this conjecture for matching-spine digraphs.

A Novel Reduction Circuit Based on Binary Tree Path Partition on FPGAs

Due to high parallelism, field-programmable gate arrays are widely used as accelerators in engineering and scientific fields, which involve a large number of operations of vector and matrix. High-performance accumulation circuits are the key to large-scale matrix operations. By selecting the adder as the reduction operator, the reduction circuit can implement the accumulation function. However, the pipelined adder will bring challenges to the design of the reduction circuit. To solve this problem, we propose a novel reduction circuit based on binary tree path partition, which can simultaneously handle multiple data sets with arbitrary lengths. It divides the input data into multiple groups and sends them to different iterations for calculation. The elements belonging to the same data set in each group are added to obtain a partial result, and the partial results of the same data set are added to achieve the final result. Compared with other reduction methods, it has the least area-time product.

Linial’s Conjecture for Arc-spine Digraphs

A path partition P of a digraph D is a collection of directed paths such that every vertex belongs to precisely one path. Given a positive integer k, the k-norm of a path partition P of D is defined as Sum (p in P) min{|p_i|, k}. A path partition of a minimum k-norm is called k-optimal and its k-norm is denoted by π_k(D). A stable set of a digraph D is a subset of pairwise non-adjacentvertices of V(D). Given a positive integer k, we denote by alpha_k(D) the largest set of vertices of D that can be decomposed into k disjoint stable sets of D. In 1981, Linial conjectured that pi_k(D) ≤ alpha_k(D) for every digraph. We say that a digraph D is arc-spine if V(D) can be partitioned into two sets X and Y where X is traceable and Y contains at most one arc in A(D). In this paper we show the validity of Linial’s Conjecture for arc-spine digraphs.