2,824 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Discriminating Codes in Geometric Setups
We study geometric variations of the discriminating code problem. In the
\emph{discrete version} of the problem, a finite set of points and a finite
set of objects are given in . The objective is to choose a
subset of minimum cardinality such that for each point , the subset covering satisfies , and each pair , , we have . In the \emph{continuous version} of the problem, the solution set
can be chosen freely among a (potentially infinite) class of allowed geometric
objects. In the 1-dimensional case (), the points in are placed on a
horizontal line , and the objects in are finite-length line segments
aligned with (called intervals). We show that the discrete version of this
problem is NP-complete. This is somewhat surprising as the continuous version
is known to be polynomial-time solvable. Still, for the 1-dimensional discrete
version, we design a polynomial-time -approximation algorithm. We also
design a PTAS for both discrete and continuous versions in one dimension, for
the restriction where the intervals are all required to have the same length.
We then study the 2-dimensional case () for axis-parallel unit square
objects. We show that both continuous and discrete versions are NP-complete,
and design polynomial-time approximation algorithms that produce -approximate and -approximate solutions respectively,
using rounding of suitably defined integer linear programming problems. We show
that the identifying code problem for axis-parallel unit square intersection
graphs (in ) can be solved in the same manner as for the discrete version
of the discriminating code problem for unit square objects
Online Class Cover Problem
In this paper, we study the online class cover problem where a (finite or
infinite) family of geometric objects and a set of red
points in are given a prior, and blue points from
arrives one after another. Upon the arrival of a blue point, the online
algorithm must make an irreversible decision to cover it with objects from
that do not cover any points of . The objective of the
problem is to place the minimum number of objects. When consists of
all possible translates of a square in , we prove that the
competitive ratio of any deterministic online algorithm is . On the other hand, when the objects are all possible translates of a
rectangle in , we propose an -competitive
deterministic algorithm for the problem.Comment: 27 pages, 23 figure
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
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