5 research outputs found
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
A 3-Approximation Algorithm for Maximum Independent Set of Rectangles
We study the Maximum Independent Set of Rectangles (MISR) problem, where we
are given a set of axis-parallel rectangles in the plane and the goal is to
select a subset of non-overlapping rectangles of maximum cardinality. In a
recent breakthrough, Mitchell [2021] obtained the first constant-factor
approximation algorithm for MISR. His algorithm achieves an approximation ratio
of 10 and it is based on a dynamic program that intuitively recursively
partitions the input plane into special polygons called corner-clipped
rectangles, without intersecting certain special horizontal line segments
called fences.
In this paper, we present a 3-approximation algorithm for MISR which is based
on a similar recursive partitioning scheme. First, we use a partition into a
more general class of axis-parallel polygons with constant complexity each,
which allows us to provide an arguably simpler analysis and at the same time
already improves the approximation ratio to 6. Then, using a more elaborate
charging scheme and a recursive partitioning into general axis-parallel
polygons with constant complexity, we improve our approximation ratio to 3. In
particular, our partitioning uses more general fences that can be sequences of
up to O(1) line segments each. This and our other new ideas may be useful for
future work towards a PTAS for MISR.Comment: 41 page
Approximation schemes for independent set and sparse subsets of polygons
We present a (1+ε)-approximation algorithm with quasi-polynomial running time for computing a maximum weight independent set of polygons from a given set of polygons in the plane. Contrasting this, the best-known polynomial time algorithm for the problem has an approximation ratio of nε. Surprisingly, we can extend the algorithm to the problem of computing the maximum cardinality subset of the given set of polygons whose intersection graph fulfills some sparsity condition. For example, we show that one can approximate the maximum subset of polygons such that the intersection graph of the subset is planar or does not contain a cycle of length 4 (i.e., K2,2). Our algorithm relies on a recursive partitioning scheme, whose backbone is the existence of balanced cuts with small complexity that intersect polygons from the optimal solution of a small total weight. For the case of large axis-parallel rectangles, we provide a polynomial time (1 + ε)-approximation for the maximum weight independent set. Specifically, we consider the problem where each rectangle has one edge whose length is at least a constant fraction of the length of the corresponding edge of the bounding box of all the input elements. This is now the most general case for which a PTAS is known, and it requires a new and involved partitioning scheme, which should be of independent interest