474 research outputs found

### Simple PTAS's for families of graphs excluding a minor

We show that very simple algorithms based on local search are polynomial-time
approximation schemes for Maximum Independent Set, Minimum Vertex Cover and
Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic

### Minimum Cuts in Geometric Intersection Graphs

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph
$G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set
$\mathcal{D}$ in which two disks are joined by an edge if and only if they
intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for
$\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which
there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in
\mathcal{D}$ if and only if $D_1$ contains the center of $D_2$.
Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we
show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in
$G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$
expected time. To obtain our result, we combine an algorithm for the maximum
flow problem in general graphs with dynamic geometric data structures to
manipulate the disks.
As an application, we consider the barrier resilience problem in a
rectangular domain. In this problem, we have a vertical strip $S$ bounded by
two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of
disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$
a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve
from $a$ to $b$ that lies in $S$ and that intersects as few disks of
$\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in
disk graphs, we can solve the barrier resilience problem in
$O(n^{3/2}\text{polylog} n)$ expected time.Comment: 11 pages, 4 figure

### The Clique Problem in Ray Intersection Graphs

Ray intersection graphs are intersection graphs of rays, or halflines, in the
plane. We show that any planar graph has an even subdivision whose complement
is a ray intersection graph. The construction can be done in polynomial time
and implies that finding a maximum clique in a segment intersection graph is
NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and
Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure

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