21 research outputs found
QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails
QPTAS and subexponential algorithm for maximum clique on disk graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time for \textsc{Maximum Clique} on disk graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time , unless the Exponential Time Hypothesis fails
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
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails
Computing a maximum clique in geometric superclasses of disk graphs
In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they
proved that maximum clique can be solved in polynomial time in unit disk
graphs. Since then, the complexity of maximum clique in intersection graphs of
d-dimensional (unit) balls has been investigated. For ball graphs, the problem
is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient
polynomial time approximation scheme (EPTAS) for disk graphs. However, the
complexity of maximum clique in this setting remains unknown. In this paper, we
show the existence of a polynomial time algorithm for a geometric superclass of
unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS
for intersection graphs of convex pseudo-disks
Maximum Clique in Geometric Intersection Graphs
An intersection graph is a graph that represents some geometric objects as vertices, and joins edges
between the nodes corresponding to the items that intersect. The maximum clique in a geometric intersec-
tion graph is the largest mutually intersecting set of objects. In this thesis, the primary focus is to study
the maximum clique in various geometric intersection graphs. We develop three results motivated by the
maximum clique problem in the intersection graph of disks in the Euclidean plane. First, we improve the
time complexity of calculating the maximum clique in unit disk graphs from O(n3 log n) to O(n2.5 log n).
Second, we introduce a new technique called pair-oriented labelling. This method is used to show the NP-
hardness of finding a maximum clique in various geometric intersection graphs, acting as a way to augment
the commonly used co-2-subdivision approach. Finally, finding maximum clique in two classes of geometric
intersection graphs are proven to be NP-hard. These are the intersection graph of disks and axis-aligned
rectangles, and the outer triangle graph. The former is previously known to be NP-hard, and so this proof
represents the use of pair-oriented labelling in a problem that was otherwise considered difficult to prove
NP-hard using a co-2-subdivision approach. The outer triangle graph is a novel intersection graph, which
therefore provides new NP-hardness results for finding a maximum clique in geometric intersection graphs
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Studies in Efficient Discrete Algorithms
This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity