For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.
Deep Dive into Trimming of Graphs, with Application to Point Labeling.
For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling.
In this paper we investigate the problem of deleting vertices from a given graph so as to ensure that all simple paths in the remaining graph are short. We assume that each vertex has a nonnegative weight, and we want to delete vertices of small total weight. Whereas there is an extensive literature on separators, which can be viewed as serving to destroy all large connected components, we are not aware of previous work on vertex sets that destroy all long simple paths. Let us make our notions precise.
Definition 1.1. For t > 0 and g ≥ 0, a (t, g)-trimming of a vertex-weighted graph G = (V, E) of total weight W is a set U ⊆ V of weight at most W/t such that every simple path in G of more than g edges contains a vertex in U . If G has a (t, g)-trimming, we also say that G is (t, g)-trimmable.
We say that a family of graphs is trimmable if, for every constant t > 0, there is a constant g ≥ 0 (that depends only on t) such that every vertex-weighted graph in the family is (t, g)-trimmable. Of course, it suffices to demonstrate this for t larger than an arbitrary constant. Not every family of graphs is trimmable. For example, if n, t ≥ 2 and we delete a (1/t)-fraction of the vertices in an unweighted n-clique K n , the remaining graph still has a simple path of n(1 -1/t) -1 edges. This expression is not bounded by a function of t alone, so the family of complete graphs is not trimmable.
With a little effort, one can show the family of trees to be trimmable. One popular generalization of trees is based on the definition below. Given a graph G = (V, E) and a set U ⊆ V , we denote by G[U ] the subgraph of G induced by U . The union of graphs G i = (V i , E i ), for i = 1, . . . , m, is the graph m i=1 G i = ( m i=1 V i , m i=1 E i ). Definition 1.2. A tree decomposition of an undirected graph G = (V, E) is a pair (T, B), where T = (X, E T ) is a tree and B : X → 2 V maps each node x of T to a subset of V , called the bag of x, such that
• x∈X G[B(x)] = G, and • for all x, y, z ∈ X, if y is on the path from x to z in T , then B(x) ∩ B(z) ⊆ B(y). The width of the tree decomposition (T, B) is max x∈X |B(x)| -1, and the treewidth of G is the smallest width of any tree decomposition of G. This standard definition is given, e.g., by Bodlaender [Bod98]. The family of graphs of treewidth at most 1 coincides with the family of forests. By analogy with several other generalizations from the family of trees to families of graphs of bounded treewidth, it seems natural to ask whether every family of graphs of bounded treewidth is trimmable. At present we cannot answer this question; we need a concept stronger than bounded treewidth alone.
Definition 1.3. The elongation of a tree decomposition (T, B) is the maximum number of edges on a simple path in T between two nodes with intersecting bags. For every s ≥ 0, let the s-elongation treewidth of an undirected graph G be the smallest width of a tree decomposition of G with elongation at most s.
Since every graph has a trivial tree decomposition of elongation 0, the s-elongation treewidth of every graph is well-defined for every s ≥ 0. The 1-elongation treewidth is the domino treewidth studied, e.g., by Bodlaender [Bod99].
Our main result about graph trimming, proved in Section 2, is that for all fixed s ≥ 0, every family of graphs of bounded s-elongation treewidth is trimmable. Ding and Oporowski [DO95] proved that the domino treewidth of a graph can be bounded by a function of its usual treewidth and its maximum degree. It follows that every family of graphs of bounded treewidth and bounded degree is also trimmable. We derive from this that all families of planar graphs of bounded degree are trimmable as well. This result has applications described below.
Our main motivation for investigating trimmable graph families arose in the context of labeling maps with sliding labels. Generally speaking, map labeling is the problem of placing a set of labels, each in the vicinity of the object that it labels, while meeting certain conditions. For an overview, see the map-labeling bibliography [WS96]. First of all, labels are not allowed to overlap. As a consequence, it may not be possible to label all objects in a map, and the goal is to make an optimal selection according to some criterion. When a point feature such as a town or a mountain top is to be labeled, the label can usually be approximated without much loss by an axes-parallel rectangular shape and must be placed in the plane without rotation so that its boundary touches the point. One distinguishes between fixed-position models and slider models. In fixed-position models, each label has a predetermined finite set of anchor points on its boundary (e.g., the four corner points), and the label must be placed so that one of its anchor points coincides with the point to be labeled. In slider models, the anchor points form anchor segments on the boundary of the label (e.g., its bottom edge).
Van Kreveld et al. [vKSW99] introdu
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