A new separation theorem with geometric applications

TL;DR

This paper introduces a new graph separation theorem with geometric applications, providing improved algorithms for NP-hard problems by leveraging measure functions and clique cover properties.

Contribution

It presents a novel separation theorem for graphs with measure functions, leading to better algorithms for geometric problems and introducing new concepts of independent interest.

Findings

Existence of small vertex separators covered by few cliques

Improved algorithms for NP-hard geometric problems

Enhanced bounds for graph separation based on measure functions

Abstract

Let $G=(V(G), E(G))$ be an undirected graph with a measure function $\mu$ assigning non-negative values to subgraphs $H$ so that $\mu(H)$ does not exceed the clique cover number of $H$. When $\mu$ satisfies some additional natural conditions, we study the problem of separating $G$ into two subgraphs, each with a measure of at most $2\mu(G)/3$ by removing a set of vertices that can be covered with a small number of cliques $G$. When $E(G)=E(G_1)\cap E(G_2)$, where $G_1=(V(G_1),E(G_1))$ is a graph with $V(G_1)=V(G)$, and $G_2=(V(G_2), E(G_2))$ is a chordal graph with $V(G_2)=V(G)$, we prove that there is a separator $S$ that can be covered with $O(\sqrt{l\mu(G)})$ cliques in $G$, where $l=l(G,G_1)$ is a parameter similar to the bandwidth, which arises from the linear orderings of cliques covers in $G_1$. The results and the methods are then used to obtain exact and approximate algorithms which significantly improve some of the past results for several well known NP-hard geometric problems. In addition, the methods involve introducing new concepts and hence may be of an independent interest.