Béla Csaba — Page 2

arXiv2007

On embedding well-separable graphs

Call a simple graph $H$ of order $n$ well-separable, if by deleting a separator set of size $o(n)$ the leftover will have components of size at most $o(n)$. We prove, that bounded degree well-separable spanning subgraphs are easy to embed: for every $γ>0$ and positive integer $Δ$ there exists an $n_0$ such that if $n>n_0$, $Δ(H) \le Δ$ for a well-separable graph $H$ of order $n$ and $δ(G) \ge (1-{1 \over 2(χ(H)-1)} + γ)n$ for a simple graph $G$ of order $n$, then $H \subset G$. We extend our result to graphs with small band-width, too.

math.CO↗

arXiv2007

A randomized algorithm for the on-line weighted bipartite matching problem

We study the on-line minimum weighted bipartite matching problem in arbitrary metric spaces. Here, $n$ not necessary disjoint points of a metric space $M$ are given, and are to be matched on-line with $n$ points of $M$ revealed one by one. The cost of a matching is the sum of the distances of the matched points, and the goal is to find or approximate its minimum. The competitive ratio of the deterministic problem is known to be $Θ(n)$. It was conjectured that a randomized algorithm may perform better against an oblivious adversary, namely with an expected competitive ratio $Θ(\log n)$. We prove a slightly weaker result by showing a $o(\log^3 n)$ upper bound on the expected competitive ratio. As an application the same upper bound holds for the notoriously hard fire station problem, where $M$ is the real line.

cs.DS↗